slides/lecture10-dimensionality-reduction.ipynb

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    "<left><img width=25% src=\"img/cornell_tech2.svg\"></left>\n",
    "\n",
    "# Lecture 10: Dimensionality Reduction\n",
    "\n",
    "### Applied Machine Learning\n",
    "\n",
    "__Volodymyr Kuleshov__<br>Cornell Tech"
   ]
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   "source": [
    "# Part 1: What is Dimensionality Reduction?\n",
    "\n",
    "We will start by defining the task and providing some examples."
   ]
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    "# Review: Unsupervised Learning\n",
    "\n",
    "We have a dataset *without* labels. Our goal is to learn something interesting about the structure of the data:\n",
    "* Clusters hidden in the dataset.\n",
    "* Outliers: particularly unusual and/or interesting datapoints.\n",
    "* Useful signal hidden in noise, e.g. human speech over a noisy phone."
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    "# Dimensionality Reduction:  Examples\n",
    "\n",
    "Consider a dataset $\\mathcal{D} = \\{x^{(i)} \\mid i = 1,2,...,n\\}$ of motorcycles, characterized by a set of attributes.\n",
    "* Attributes include size, color, maximum speed, etc.\n",
    "* Suppose that two attributes are closely correlated: e.g., $x^{(i)}_j$ is the speed in `mph` and $x^{(i)}_k$ is the speed in `km/h`.\n",
    "* The real dimensionality of the data is $d-1$!\n",
    "\n",
    "We would like to automatically identify the right data dimensionality."
   ]
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    "Another example can be obtained on the Iris flower dataset."
   ]
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   "source": [
    "# import standard machine learning libraries\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "from sklearn import datasets\n",
    "\n",
    "# Load the Iris dataset\n",
    "iris = datasets.load_iris()"
   ]
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    "Consider the petal length and the petal width of the flowers: they are closely correlated.\n",
    "\n",
    "This suggests that we may reduce the dimensionality of the problem to one dimension: petal size."
   ]
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",
      "text/plain": [
       "<Figure size 864x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from matplotlib import pyplot as plt\n",
    "plt.rcParams['figure.figsize'] = [12, 4]\n",
    "\n",
    "# Visualize the Iris flower dataset\n",
    "setosa_flowers = (iris.target == 0)\n",
    "plt.scatter(iris.data[setosa_flowers,0], iris.data[setosa_flowers,1], alpha=0.5)\n",
    "plt.plot([4.3, 5.8], [2.8, 4.2], '->')\n",
    "plt.ylabel(\"Sepal width (cm)\")\n",
    "plt.xlabel(\"Sepal length (cm)\")\n",
    "plt.legend(['\"Size\" Dimension'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Dimensionality Reduction\n",
    "\n",
    "More generally, a dimensionality reduction algorithm learns from data an unsupervised model\n",
    "$$f_\\theta : \\mathbb{R}^d \\to \\mathbb{R}^p,$$\n",
    "where $\\mathbb{R}^p$ contains low-dimensional representation of the data $(p<d)$.\n",
    "\n",
    "For each input $x^{(i)}$, $f_\\theta$ computes a low-dimensional representation $z^{(i)}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Linear Dimensionality Reduction\n",
    "\n",
    "The transformation \n",
    "$$f_\\theta : \\mathbb{R}^d \\to \\mathbb{R}^p$$\n",
    "is a linear function with parameters $\\theta = W \\in \\mathbb{R}^{d \\times p}$:\n",
    "$$ z = f_\\theta(x) = W^\\top \\cdot x. $$\n",
    "The latent dimension $z$ is obtained from $x$ via a matrix $W$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "# Example: Discovering Structure in Digits\n",
    "\n",
    "Dimensionality reduction can reveal interesting structure in digits without using labels.\n",
    "\n",
    "<center><img width=40% src=\"img/aae_dim_reduc_2.png\"></center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Example: DNA Analysis\n",
    "\n",
    "Even linear dimensionality reduction is powerful. Here, in uncovers the geography of European countries from only DNA data\n",
    "\n",
    "<center><img width=50% src=\"img/dna_map.jpg\"></center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Other Kinds of Dimensionality Reduction\n",
    "\n",
    "We will focus on linear dimensionality reduction this lecture, but there exist many other methods:\n",
    "* Non-linear methods based on kernels (e.g., Kernel PCA)\n",
    "* Non-linear methods based on deep learning (e.g., variational autoencoders)\n",
    "* Non-linear methods based on maximizing signal independence (independent component analysis)\n",
    "* Probabilistic versions of the above\n",
    "\n",
    "\n",
    "See the `scikit-learn` [guide](https://scikit-learn.org/stable/modules/unsupervised_reduction.html) for more!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "<left><img width=25% src=\"img/cornell_tech2.svg\"></left>\n",
    "# Part 2: Principal Component Analysis\n",
    "\n",
    "We will now describe principal component analysis (PCA), one of the most widely used algorithms for dimensionality reduction."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "# What Problem Does PCA Solve? Intuition.\n",
    "\n",
    "In this example, the data lives in a lower-dimensional 2D plane within a 3D space (image [credit](https://doc.plob.org/machine_learning/14_Dimensionality_Reduction.html)).\n",
    "\n",
    "<center><img width=50% src=\"img/pca_example_plane.png\"></center>\n",
    "\n",
    "PCA finds (1) this linear subspace and (2) the coordinates of the data in it."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# What Problem Does PCA Solve? Math.\n",
    "\n",
    "Our first step is to make this goal more precise. Suppose we have datapoints $x \\in \\mathbb{R}^{d}$.\n",
    "\n",
    "1. Finding a linear subspace means finding an (orthonormal) set of $p < d$ basis vectors \n",
    "\\begin{align}\n",
    "W = [ w^{(1)}, w^{(2)}, \\ldots, w^{(p)} ],\n",
    "\\end{align}\n",
    "where $W \\in \\mathbb{R}^{d \\times p}$ is the matrix of stacked vectors."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "2. Finding projections of the data in in the subspace means computing\n",
    "\\begin{align}\n",
    "z = \n",
    "\\begin{bmatrix}\n",
    "z_1 \\\\ z_2 \\\\ \\vdots \\\\ z_k\n",
    "\\end{bmatrix}=\n",
    "\\begin{bmatrix}\n",
    "(w^{(1)})^\\top x \\\\ (w^{(2)})^\\top x \\\\ \\vdots \\\\ (w^{(k)})^\\top x\n",
    "\\end{bmatrix}\n",
    "= W^\\top x \n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "We can choose a basis $W$ for the data plane. The coordinates in this basis are denoted by $z$ (image [credit](https://doc.plob.org/machine_learning/14_Dimensionality_Reduction.html)).\n",
    "\n",
    "<center><img width=80% src=\"img/pca_example.png\"></center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Components of an Unsupervised Learning Algorithm\n",
    "\n",
    "We will define PCA in terms of the three standard components of an ML algorithm.\n",
    "\n",
    "$$ \\underbrace{\\text{Dataset}}_\\text{Attributes} + \\underbrace{\\text{Learning Algorithm}}_\\text{Model Class + Objective + Optimizer } \\to \\text{Unsupervised Model} $$\n",
    "\n",
    "The dataset $\\mathcal{D} = \\{x^{(i)} \\mid i = 1,2,...,n\\}$ does not include any labels."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# The PCA Model\n",
    "\n",
    "The model for PCA is a function $f_\\theta$ of the form\n",
    "$$ z = f_\\theta(x) = W^\\top x, $$\n",
    "where $\\theta = W$ and $W$ is a $d \\times p$ matrix of $p$ orthonormal column vectors denoted as $w^{(1)}, w^{(2)}, \\ldots, w^{(p)}$.\n",
    "\n",
    "Note that when $x = W z$, then $z = W^T x$ because $W^\\top W = I$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "This model enables performing two tasks:\n",
    "* __Encoding__: $z^{(i)} = W^\\top x^{(i)}$, finding the low-dimensional form of $x$\n",
    "\\begin{align}\n",
    "z^{(i)} = \n",
    "\\begin{bmatrix}\n",
    "z^{(i)}_1 \\\\ z^{(i)}_2 \\\\ \\vdots \\\\ z^{(i)}_k\n",
    "\\end{bmatrix}=\n",
    "\\begin{bmatrix}\n",
    "(w^{(1)})^\\top x^{(i)} \\\\ (w^{(2)})^\\top x^{(i)} \\\\ \\vdots \\\\ (w^{(k)})^\\top x^{(i)}\n",
    "\\end{bmatrix}\n",
    "= W^\\top x^{(i)}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* __Decoding__: $\\tilde x^{(i)} = W  z^{(i)}$, converting a low-dimensional $z^{(i)}$ to a high-dimensional reconstruction $\\tilde x^{(i)}$ of $x^{(i)}$\n",
    "$$ \\tilde x^{(i)} = \\sum_{k=1}^p w^{(k)} z_k^{(i)} = W z^{(i)} $$\n",
    "Note that $\\tilde x^{(i)}$ is the closest point in the subspace to $x^{(i)}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "# Principal Components Model\n",
    "\n",
    "Principal component analysis (PCA) assumes that \n",
    "* Datapoints $x \\in \\mathbb{R}^{d}$ live close to a low-dimensional subspace $\\mathcal{Z} = \\mathbb{R}^p$ of dimension $p<d$\n",
    "* The subspace $\\mathcal{Z} = \\mathbb{R}^p$ is spanned by a set of orthonormal vectors $w^{(1)}, w^{(2)}, \\ldots, w^{(p)}$\n",
    "* The data $x$ are approximated by a linear combination $\\tilde x$ of the $w^{(k)}$\n",
    "$$ x \\approx \\tilde x = \\sum_{k=1}^p w^{(k)} z_k = W z $$\n",
    "for some $z \\in \\mathcal{X}$ that are the coordinates of $\\tilde x$ in the basis $W$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "This model enables performing two tasks:\n",
    "* __Encoding__: $z = W^\\top x$, finding the low-dimensional form of input $x$\n",
    "* __Decoding__: $\\tilde x = W  z$, converting a low-dimensional $z$ to a high-dimensional representation $x$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# PCA Objective: Reconstruction\n",
    "\n",
    "How do we find a good subspace $\\mathcal{Z}$ as defined by a set of orthonormal vectors $W$?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "A natural objective is to minimize the reconstruction error\n",
    "$$J_1(W) = \\sum_{i=1}^n \\| x^{(i)} - \\tilde x^{(i)} \\|_2^2 =\\sum_{i=1}^n \\| x^{(i)} - W W^\\top x^{(i)} \\|_2^2$$\n",
    "between each input $x^{(i)}$ and its approximate reconstruction $$\\tilde x^{(i)} = W \\cdot z^{(i)} = W\\cdot W^\\top \\cdot x^{(i)}.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "In this example, if the points don't lie perfectly on a plane, we choose the plane such that the points' distance to it is minimized (image [credit](https://doc.plob.org/machine_learning/14_Dimensionality_Reduction.html)).\n",
    "\n",
    "<center><img width=80% src=\"img/pca_example.png\"></center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# PCA Objective: Maximizing Variance\n",
    "\n",
    "An alternative objective for learning a PCA model is maximizing variance. \n",
    "\n",
    "We start with some intuition."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "Below, we can project the data along the blue line or the orange line.\n",
    "\n",
    "The blue line is better because it captures the shape of the data and can be naturally interpreted as \"sepal size\"."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x12c073d68>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 864x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from matplotlib import pyplot as plt\n",
    "plt.rcParams['figure.figsize'] = [12, 4]\n",
    "\n",
    "# Visualize the Iris flower dataset\n",
    "setosa_flowers = (iris.target == 0)\n",
    "plt.scatter(iris.data[setosa_flowers,0], iris.data[setosa_flowers,1], alpha=0.5)\n",
    "plt.plot([4.3, 5.8], [2.8, 4.2], '->')\n",
    "plt.plot([5.05, 4.99], [2.85, 4.1])\n",
    "plt.ylabel(\"Sepal width (cm)\")\n",
    "plt.xlabel(\"Sepal length (cm)\")\n",
    "plt.legend(['\"Size\" Dimension', '\"Other\" Dimension'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "How do we automatically identify natural directions of variation in the data? Consider the following dataset (image by [Andrew Ng](http://cs229.stanford.edu/)).\n",
    "\n",
    "<center><img width=45% src=\"img/pca_projection_data.png\"></center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "One way to reduce the dimensionality of this dataset from is to project it along the following line.\n",
    "<center><img width=50% src=\"img/pca_projection1.png\"></center>\n",
    "Projected data is tightly clustered around its mean. It has *low variance*."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "An alternative projection is along the following line. Data is much more spread out: it has *high variance* around its mean.\n",
    "<center><img width=50% src=\"img/pca_projection2.png\"></center>\n",
    "Our goal is to identify this direction automatically."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "We may formalize this as follows. \n",
    "* Let $\\hat{\\mathbb{E}}[f(x)]$ denote empirical expectation for any $f$:\n",
    "$$\\hat{\\mathbb{E}}[f(x)] = \\frac{1}{n}\\sum_{i=1}^n f(x^{(i)}). $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* Assume that we have centered the data, i.e.\n",
    "$$\\hat{\\mathbb{E}}[x] = 0 \\text{ and thus } \\hat{\\mathbb{E}}[W^\\top x] = W^\\top \\hat{\\mathbb{E}}[x] = 0.$$\n",
    "<!-- $$\\frac{1}{n}\\sum_{i=1}^n x^{(i)} = 0 \\text{ and thus } \\frac{1}{n}\\sum_{i=1}^n W^\\top x^{(i)} = 0.$$ -->"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* The the variance of the projected data is\n",
    "\\begin{align*}\n",
    "\\hat{\\mathbb{E}}\\left[ \\| z - \\hat{\\mathbb{E}}[z] \\|^2 \\right] =\n",
    "\\hat{\\mathbb{E}}\\left[ \\| W^\\top x - \\hat{\\mathbb{E}}[W^\\top x] \\|^2 \\right] \n",
    "& = \\hat{\\mathbb{E}}\\left[ \\| W^\\top x \\|^2 \\right]\n",
    "\\end{align*}\n",
    "<!-- \\begin{align*}\n",
    "\\frac{1}{n} \\sum_{i=1}^n \\| W^\\top x^{(i)} - \\frac{1}{n}\\sum_{i=1}^n x^{(i)} \\|_2^2\n",
    "& = \\frac{1}{n} \\sum_{i=1}^n \\| W^\\top x^{(i)}\\|_2^2\n",
    "\\end{align*} -->"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Thus, the variance objective is simply\n",
    "$$J_2(W) = \\hat{\\mathbb{E}}\\left[ \\| W^\\top x \\|^2 \\right] = \\frac{1}{n} \\sum_{i=1}^n \\| W^\\top x^{(i)}\\|_2^2.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "# Equivalence Between PCA Objectives\n",
    "\n",
    "It turns out that minimizing reconstruction error and maximizing variance are equivalent.\n",
    "$$\\arg\\min_W J_1(W) = \\arg\\max_W J_2(W).$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "This image by [Alex Williams](http://alexhwilliams.info/itsneuronalblog/2016/03/27/pca/) provides intuition.\n",
    "\n",
    "<center><img width=80% src=\"img/pca_two_views.png\"></center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Consider the operator $W W^\\top x$. We can decompose any $x$ into a sum of two orthoginal vectors:\n",
    "\\begin{align*}\n",
    "x \n",
    "& = x + W W^\\top x - W W^\\top x \\\\\n",
    "& = \\underbrace{W W^\\top x}_\\text{projected data $\\tilde x$ (D1)} + \\underbrace{(I - W W^\\top) x}_\\text{difference between datapoint $x$ and $\\tilde x$ (D2)}\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "<center><img width=50% src=\"img/pca_decomposition.png\"></center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "We can compute the norm of both sides to obtain\n",
    "\\begin{align*}\n",
    "\\|x\\|_2^2\n",
    "& = \\| W W^\\top x + (I - W W^\\top) x \\|_2^2 \\\\\n",
    "& = \\|W W^\\top x\\|_2^2 + \\|(I - W W^\\top) x\\|_2^2 \\\\\n",
    "& = \\|W^\\top x\\|_2^2 + \\|(I - W W^\\top) x\\|_2^2\n",
    "\\end{align*}\n",
    "* In the second line we used the fact that $W W^\\top x$ and $(I - W W^\\top) x$ are orthogonal (easy to check) \n",
    "* In the third line we used that $\\|W a\\| = \\|a\\|$ for any vector $a$ and orthogonal matrix $W$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Thus we find that\n",
    "\\begin{align*}\n",
    "J_1(W) & = \\sum_{i=1}^n \\|(I - W W^\\top) x^{(i)}\\|_2^2 \\\\\n",
    "& = \\sum_{i=1}^n \\left( \\|x^{(i)}\\|_2^2 - \\|W^\\top x^{(i)}\\|_2^2 \\right) \\\\\n",
    "&= - n\\cdot J_2(W) + \\text{const.}\n",
    "\\end{align*}\n",
    "and minimizing the reconstruction objective $J_1$ is the same as maximizing the variance objective $J_2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Finding Principal Components\n",
    "\n",
    "Next, how do we optimize either of these objectives? Let's look at the variance objective $J_2$, which we can write as:\n",
    "\n",
    "\\begin{align*}\n",
    "J_2(W)\n",
    "& = \\frac{1}{n}  \\sum_{i=1}^n \\| W^\\top x^{(i)}\\|_2^2 = \\frac{1}{n}  \\sum_{i=1}^n \\sum_{j=1}^p ((w^{(j)})^\\top x^{(i)})^2\n",
    "\\end{align*}\n",
    "\n",
    "where $w^{(j)}$ is the $j$-th column of $W$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "We can further write this as:\n",
    "\\begin{align*}\n",
    "J_2(W)\n",
    "& = \\frac{1}{n} \\sum_{i=1}^n \\sum_{j=1}^p ((w^{(j)})^\\top x^{(i)})^2 \\\\\n",
    "& = \\frac{1}{n} \\sum_{i=1}^n \\sum_{j=1}^p \\left((w^{(j)})^\\top x^{(i)}\\right) \\cdot \\left((x^{(i)})^\\top w^{(j)}\\right)  \\\\\n",
    "& = \\sum_{j=1}^p (w^{(j)})^\\top \\cdot \\left( \\frac{1}{n}  \\sum_{i=1}^n x^{(i)} (x^{(i)})^\\top \\right) \\cdot w^{(j)}\\\\\n",
    "& = \\sum_{j=1}^p (w^{(j)})^\\top \\cdot \\hat\\Sigma \\cdot w^{(j)},\n",
    "\\end{align*}\n",
    "where $\\hat\\Sigma = \\frac{1}{n} \\sum_{i=1}^n \\left( x^{(i)} (x^{(i)})^\\top \\right)$ is the empirical covariance matrix of $\\mathcal{D}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Recall that the positive semidefinite matrix $\\hat \\Sigma$ has an *eigendecomposition*\n",
    "$$\\hat \\Sigma = Q \\Lambda Q^\\top = \\sum_{j=1}^d \\lambda_j q^{(j)} (q^{(j)})^\\top. $$\n",
    "* $Q$ is a matrix whose columns are orthonormal eigenvectors $q^{(j)}$ for $j = 1,2,\\ldots,d$.\n",
    "* $\\Lambda$ is a diagonal matrix of positive eigenvalues $\\lambda_1 \\geq \\lambda_2 \\geq \\ldots \\geq \\lambda_d$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Consider our optimization problem for $p=1$:\n",
    "$$J(w) = w^\\top \\cdot \\hat\\Sigma \\cdot w.$$\n",
    "How do we find the best projection vector $w$?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Using the eigendecomposition, we can write this as:\n",
    "$$J(w) = w^\\top \\cdot Q \\Lambda Q^\\top \\cdot w = \\sum_{j=1}^d \\lambda_j (w^\\top q^{(j)})^2.$$\n",
    "\n",
    "Note that $(w^\\top q^{(j)}) \\in [0,1]$ is the angle between $w$ and the $j$-th eigenvector $q^{(j)}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "The optimal solution to\n",
    "$$\\max_w J(w) = \\max_w \\sum_{j=1}^d \\lambda_j (w^\\top q^{(j)})^2$$\n",
    "is attained by the top eigenvector $w = q^{(1)}$, the optimum is $J( q^{(1)}) = \\lambda_1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* Let $a_j = (w^\\top q^{(j)})$ and note that $\\sum_j a_j^2 = ||Q w||_2^2 = 1$ because all vectors are orthonormal."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* Our objective $J(w) = \\sum_{j=1}^d \\lambda_j a_j^2$ is a weighted average of $\\lambda_j$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* The weighted average $\\sum_{j=1}^d \\lambda_j a_j^2$ attains a maximum of $\\lambda_1$ when all \"weight\" goes to $a_1=1$ and $w = q^{(1)}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "More generally when $p>1$, our objective is\n",
    "$$J(W) = \\sum_{k=1}^p \\sum_{j=1}^d \\lambda_j ((w^{(k)})^\\top q^{(j)})^2$$\n",
    "where $W$ is a matrix of orthonormal columns $w^{(1)}, w^{(2)}, \\ldots, w^{(p)}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "By analogy with the previous example, \n",
    "* $J(W)$ is maximized when $w^{(1)} = q^{(1)}$, $w^{(2)} = q^{(2)}$, ..., $w^{(p)}= q^{(p)}$\n",
    "* The maximum value attained is $\\lambda_1 + \\lambda_2 + \\ldots + \\lambda_p$\n",
    "* We refer to\n",
    "$$\\frac{\\lambda_1 + \\lambda_2 + \\ldots + \\lambda_p}{\\lambda_1 + \\lambda_2 + \\ldots + \\lambda_d}$$\n",
    "as the proportion of *variance explained* by the lower-dimensional projection $z = W^\\top x$. When $d=p$ it is one."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Algorithm: Principal Component Analysis\n",
    "\n",
    "* __Type__: Unsupervised learning (dimensionality reduction)\n",
    "* __Model family__: Linear projection $W^\\top z$ of low-dimensional $z$\n",
    "* __Objective function__: Reconstruction error or variance maximization\n",
    "* __Optimizer__: Matrix eigendecomposition"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Practical Considerations\n",
    "\n",
    "When applying PCA, the following tricks are useful.\n",
    "* Before applying PCA, it is important to normalize the data to have zero mean and unit variance.\n",
    "$$x_j^{(i)} \\gets \\frac{x_j^{(i)} - \\mu_j}{\\sigma_j} \\text{for all $i,j$},$$\n",
    "where $\\mu_j, \\sigma_j$ are the mean and variance along the $j$-th dimension. \n",
    "* This addresses scaling issues due to units (`km/h` vs `cm/h`).\n",
    "* In order to choose the optimal number of components, we can apply the Elbow method."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# An Example: Iris Flowers\n",
    "\n",
    "Let's look at an example over the Iris flower dataset. In its entirety, it has four dimensions; let's visualize it in 3D by looking at the first 3 dimensions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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wlCiVSrCEEjsRkYYo0N/b4tWaFosFIyMjUKvVUCgUCA0Ntfq7rcDZprdJgwA+nw+tVou0tDSvn1+wwOFwkJCQ4HLmniscXdQJBAIUFRVRE9wdOcY4WoNubRYM7T8rjaATPrFYTG3CBwKuTJa1Wi0mJiaCxmQ5mCohPXkuQqEQJq3KLs2rUs4hRCL2+LHn5ubw6RdfYVRlAosvgmbqFJLCJSguyINer4dGo6HS2yKRyGV6e3x8HDqdzuNjCFbi4uKg1WrR1taGnJwcjwTHWTZDIpEgJyfHqWOMszUYd5flI+iETygUQqVSLfnjemOyXF1djdTU1KDZjwwm4fMEmUyGlCgF+ns6EZuUBhaLBb1Oh7G+DpxTnLHwArAuUHr/4z0YgxzyyBhYCAIyWQTaek5A0T+AjWeewTjg+AjZ49fT04OUlBS37+fKpDo0NBQpKSmora1FaWmp09s5muLOuLssPUEnfGKxmNq0Xwz8abIcbIU4gdiU7y2eiviGtWVg19ThZNNxsDg8sM0GrM/LsDuxkgKnVCrtPExFIhGMRiOm9UD+ujLwaEUY4eFhONlyFOd7KHqrtdjIFWSPX01NDQQCAWJiYty630JVodHR0dDpdGhsbERhYaHD2zoSPoBxd1lqglL4/BHxLYXJcrDN5FutER8A8Hg8nLV+LUo1Guh0OgiFQuh0OgwPD9tV4NKj/7i4OKsCh5GREcjkYVaiBwA8vgA6ndHnCyWtVovu7m6oNBpEhIUhISFh1RS40CGrMskeP3eqMt0p3EpKSoJWq0V7ezuysrLcWoNxd1l6ll343nnnHTz88MNobW1FRUUFysrKHN4uOTkZUqkUbDYbHA4HVVVVDm/nqfDRTZbJf0tlshxMERIQXMLn7nNx1mJC//x4UoEbGhoKQq+GXqcFj3+6UGLy1Ajio8N9Er2RkRF8sPcADKJQcPli6Jr7EC9rxMUXnr8qreboPX7uVGW6W7GcmZmJxsZG9PT0IDk52a01GHeXpWXZX938/Hy8//77uOWWWxa87YEDBxAeHu7yNo6KW9w1WQ4LC0NiYuKSbTQH2zBacr8iGDEajXY2XaTA0VtMEhISnH5+yHlyUqkUPB7P4ePweDycVZqPr2oqEZaUBbFUhunxU1ANdeLCbed6ffwmkwmffHkI0pRChISS36EMdLfWo6a2HhvWr/V67UChu7sbXxz8Gl19g4gMC8WF556FkpISl99lssevvr4eZWVlLi8A3BU+giCQn5+P6upqCAQCREdHu7UGeZwGgwEAGPFbRJb9lc3JyfHbWhaLBXq9Ho2Njejo6EBMTAyKiorcNlleaoIt4guGdgZS4Obm5qBWq9Hf309lALy9QDKZTNi3/wAOV9bBxOKDbdbjvA1rcN45Zzs8CRYVFkAmlaC6sRWTg0rER4Wj7OItiIiI8Pj5kHt8p06dgtrCRVSo9YVjTHIG6luPBbTwufOZ6ujowFMv7kJI1jpErV+Lualx/OPNPbh2agZbNp/rcg1y7FBdXR3KyspcTojwZB5fSUkJKisrwePxqHaUhdagix+LxVqVaeilYNmFz10IgsAFF1wAgiBwyy234OabbwYAfPjhh9i9ezfa2tqg1WphNpsRExODc845B2vWrHFrhtdyEWzCt5JSnc5S3GQGAJg/IcbExPicAdi3/wC+bOhF6vqLwRMIodOq8Vn1YRAEgc3nnuPwPikpKR5VHC6E2WwGWPYXehwOB0ZjYGcdFirQsVgsePfjvQjN24ioxFQQBAG+SAyJPBTv730fZ525AVwu16WIhIeHQ6vVUj1+zgpTPPkc0FOphYWFbvcWk4/BuLssHkuiCFu2bMHIyIjd7x999FHs2LHDrTWOHDmCuLg4nDp1Cueffz6ys7OxadMmpKSk4O6770ZWVhaEQiF27dqFtrY23H777f5+Gn4nGIUv0J6PMxcceoo7PDwcSUlJVuXk5LRvX/e+dDodDlfWUaIHAHyBCCnFG/HV0T3YdNaZi5Z50Ol06OnpgV6vn49k9HPQqlUQiE73Fo4M9CInLWlRHt9fLCR8BoMB3YOjyC2+yOr3ArEUZr4Mw8PDSEhIWFC04uPjodFo0Nra6rDHzxtXItsGd3chH5txd1kclkT4vvzyS5/XICcfR0ZG4tJLL0VFRQU2bdqEoqIiq9v5q6pzKQg24VvOVKc7Ake64DjbX6Pjr+hVqVTCxOJTokciEIlhABsqlQpyudznx7GlprYW7312ACrwYdDrcPDbWpxRmo+G5goII5MgFEswO3EKAu0Eys+50O+P708WEj4OhwMhjwOtRgWxREb93mw2w6hVQSwWuy1a6enpaGxsRG9vr11hiretIRKJhBqR5MlninF3WTwCMwdog0qlgtlshlQqhUqlwhdffIGHHnrI4W0Xu4/PnwSb8C1FqlOv19sJHN3mzVOBW2wkEgnYZj10WjX4AhH1e61KCT7L7Ja/o6cMDAzgzc8OI7H8Qpgs8/uWFoMWB49/gxuvuRTdfQOYnhtCXmokcrI3QCQSLbzoMrKQ4LBYLJyzvhT76o8ia8MFYLPZsFgsGGirQ1ZCBCIjI6HT6dweHOusMIX8uzcoFAqkpqaioaHBo8iRcXdZHJZd+D744APceeedGBsbw0UXXYTi4mJ8/vnnGBoawk033YQ9e/ZgdHQUl156KYD5L/H3vvc9XHih46tUJuJbPvwlfLZVuM4ELioqatGKlPz1XPh8Ps5dvwaf13yNlJKN4AtF0KqU6K77Gt89wzuz5IWoqKmDOD4HQrEUc3NzAAgoImIw0R+N8fFxnLvpLL8/5mLiSvjIC6HC/Fy0tXfgxL43wAuNgVE5hfgQHm66+UYAnqUp6ZMX+Hy+25MXFiIqKgpcLhcNDQ0oKipyW8SYyM//LLvwXXrppZSo0YmNjcWePXsAzFsM1dfXu7UeOZ1hJRCMwufJ8yGrcOktJnSBk0gkEIlEiypwS8Hmc88Gi0Xgq6OfwgA2uDDhu2eV45xNGxfl8San5yCS2BfGcEUyzMzNLcpjLiYWiwUWiwVTU1MOL4TIz8lVl+2YNwCQySCXy5GamkqJhKdpSi6Xi+LiYtTU1KC4uJgqePIVsrqc9Ap1B/K4ySHHjLuL7yy78PkbiUSyYoSPzWYHVR+fsz0+usDR/5FtJmQEF0hG3f5M27JYLGw+9xxsOutMKJVKSCSSRX2O6clx2NfSj9DoOKvf6yaHkHDW2Yv2uP7AUaSv1+uh1+sxNjbmNNIn3yuTyQSxWGxXEetNYYpQKERBQQHV4+crZFVoRkYGGhsb0d3d7Xblrq27CyN+vhF0wrfS9vjIZtVgQa/XU6OWnAlcTEwMxGJxwLaZLBZcLtdvaTNXlJaU4JvKV9F7ogHS8FgYdDqc7GlBkpwTMOOJbA0BbNtJ6Hu1JpMJnZ2dyMzMdLkm6cFZXV1NZQpIvC1MkclkyMjI8LgwxRH0kUT5+fkee4WSxz81NQWhUBjwo9cCmaA780gkkhW1x7cSIz6LxQKdTmfnZKLVasHhcBAWFgaxWIzY2FiIRKIVKXArqSfRFqlUilt/9D18degIjhz7CCyCwAVnn4FzNu3w2RDdbDbjxIkTqG5ogdlsRlFuJvLy8py+x3TPW1tLN7ohgG07CR21Wu3V/pxAIEBISAh13N72w0VERECn06Gtrc0n02/bkUT0Ce5hYWFurUEQBEZHRxESEgKhULgiv1uBQNC9agKBYMXMHwv0PT5HAkdW2PL5fOrEFRcXB5FIhNHRUQCnW08Ylo/Q0FBccel2nLm+HEajEQkJCT6vabFY8N7uj1F5chSKhCywuGw0769GdkMzvnfV5VbpbKVSaedZGhoa6tLSzdXjero/V1RUhNraWqxZswZCodDnKRXx8fFob2/3aG/OFlvxpTe4FxQUuB3Bmc1mcDgcxt3FB4JO+FaSbVagCJ/tsFxyZA59lqBIJEJcXBw1S9ARgdjA7i0rOeJbLHp6elB5YgDJ5VugNxqh1+kgTcrD15UHIPrwQxQUFEAikTicOuELnjqmAPNFbnl5eZQNmS8RHzD/HeHz+TAYDA7Np93B0THw+XwUFRWhrq6OEml31mGz2SAIgnF38ZKgEz6SlTCHbKmFjy5w9BSl7bDchIQEiEQij9NijFgEHt5+D8jBykqlkvqsHDp8BFNGHsZr6zE+Mwc2m43YyHDEZxfDzNYgNzd3EZ6B+8/B9jZyuRwpKSmor69HUlKST+cDUmzIHj+hUGi1h+juGo4ESiwWIy8vD7W1tSgrK1uw/9RkMlF7hQCoNgdG/Nwn6IQv0MWOzmIJn8VisZsG70jgQkNDvRI4ZzARX2AxPDyMmpoacDgcyGQyyGQyu9uQAke/GCKLw8gUJTl3cnB4GF/sqYYkLRlxOfOz5qYnx9H57dcQpkpw8uRJpKSkeHQCHhsbw5cHDqGpvQshUgnOO3MtSktLrb7HvlzERkdHQ61Wo7e316diENJcmj7Hj8/ne+S642qWolwuR3p6Ourq6lBaWuryO0kXUHqPH2Nt5j5BJ3wrCV+FjxQ42wjOYrFQw3LJ4gGhULjok95XUpo5mDGbzfh07xeoaO2FjhcCs1GPY42vYvvmM5GYkGD1WQFg9VmJjIyEUCh0KF4cFguzo4OILj0fBJsFs9GI0RNV6GtvhpRfgNE39yBaCNz8w+9T0whI+vv7sf/QEfQPn0JCTCQ2n30W+Hw+Hn/6X7BEZyGiYAuUqjn8e/dB9A8O49IdF1P39TV7k5KSgomJCczOznq9Bj3dyuVyUVJSgpqaGpSUlLjtfLNQupV0mGloaEBxcbHT52wroIy7i+cEpfBxuVwYDIaAsKxyhbvC50zgAFhFcOHh4RCJRMuW8giGKIlkpT4Xi8WCuro6HGzoQkLhGVCrNTCaTDDoNHjprY/w0+suR2RkJCIiIhwKnNlsxsDAAAwGAzU8l0StMyArJwsjVXvBC0/CqZ5WTM9pkLLxUkTFhSErOxv9HU14ddc7+Pntp+drtra24ulX3oYwuQghyevQMD6C48+8hBgZHxppHDJz1sxPppcpEBIejS/2v4mzN55pNcrHl5M5QRCIj49HR0cHRkdHPU5Rkq8L/bUSCoXIz8+n9hDdOde4s1eZkJAArVbr1CgbOJ3qpMO4u3hGUAqfSCSCSqUKeOFjs9lWwucq7SQUCiESiSCRSJZd4JwRTKnOQMe24pZekPTl4aMQRWWBx+PDAgIcNhtyRTK6tLMwGo2IioqCyWTC8PAwuFwuIiIiQBAEBgcH8e/X3sS4xgIWhweObgZXb78Q69aWAwBCFXJERcchv3wjhvtOYrThFHLPvgYmkwkiwbxAxqfn4cSBFkpgLBYL/vvexwgvOg+h0fEAAC5fiK+rjmLvvv2ISs9HR0sjSjduQWxaDjhcHjiKWPT29vpN+EhiYmJw8uRJqzYHd3EUrYWEhFA9fq7m+JGYTCa3si7p6elobm5Gd3c3UlNTHR6L7Tp0dxeDweC0NYRhnqAUPtKvcymahb2BFLjZ2VmoVCo0NTVZCZw7aadAJJhSnYEi4qTrDb3IRK1Ww2Qygc/nQyKR2BUk1bedhDY8ElKZFObpmdOLcbgwGo1obGzEa+99DJWJA7PRgMQIKb53+Q784+X/gpe2DpkJ8ydb9ew0Xt39GSLCw5CamorC/DzsO1IJQ3Qc0vNLUVtxHFyBGMaxPkRnFwKYf93YfDGVkZicnMTYnBZZUfMtLmaTCYd3/xe60FTINyQgPD0P0KvwzZd7sFkkQWhMAix6jVV1oz+Ez2w2UzZk9DYHT+7v6HsYEREBrVaLxsbGBf033R1kSzbi19bWYmhoCLGxsVZ/dxTxkfcj/04QBOPu4oKgFL5A8et0J4IjCAIpKSkrSuCcsVLTg4EAacxNFzjS9ca2Z9JVSwkA5GWm4vP6LsjDo2DB/PtiMhlhmBwCl5uIZ//7AWLKLkBc2HxENtp9Ar/781MQRmciPuF0hCGSySFNKcKho8eRmpoKmUyGG6/egbd2f4aT7QQIzRSm2iuw/sxNEAnn97l0ahVY2hnKjYTH4wEmI8xmE9hsDkZ7O6A0cxCdWQZDex20M+NQxKdCn1aOE3XHkWE2QWJRIT093eq18aaq0/b11el06Ovrg1gsRk1NDdatW+d2A7irY0hISIBarUZ7ezuysrKcruGpUXZRURHV4B4eHu7WsdCtzUjxY7AnKF+VpZ7QYDab7aZ5a7VaAKcr4yQSCaKioiAQCKgPv8ViwcTEhN8McJebYBK+xXwuzkYr+cvWbU1xEeqaW9FZexQcaTjMRgNGJgdxZmE6mk90QJRYAFnY/D4XQRCITs1GZ8NxaGdn7NYShygwPtxP/ZyUlIRf3nEzRkdH0bexGLs+/QqayVFo+Twopycx2laJKzefBYFAAOB/8wjnxnDk3RexYcf1UM1MghMSBdXUOOKiIiDg8zDU3wGThUBPUxUShXrc8eMfWD1vf0R833xzFO/u/QqisHiYDDqIocHU1BS2bdvmlhgtJFqZmZloaGhAX18fEhMTHd7GWaTmDHqDO4/Hs6rKdfV6kH8j7RAZ8bMnKF+RxYr4HAmcRqMBQRB2pd9CoXDBL2uwpSECpSE/UHA2HNfT0UqenvgFAgFu/P41qG9oxJGKGgj5XGz+zpnIzMzEX595HpKIfLv7SCLiMdVZa/dY0yP9KEg+7cQzNzeHw98cQ31rJ9hsFjbkp2NaOYr+miaEK+S45PLzUVBQAAB4/M9P4PXdX4AdkQLlTBeq778RmTn5mNGaERIRg6z8XPD5AsTHqXCy/ltkrC/EQw/cZ/dauJsidEZbWxt27f0ayRuvRnj0fNrwVG8Hdn24B7GxsSgpKVnw9V1I+AiCQEFBAWWVFhkZ6fEajuDz+VR61pMKUrr4Me4u9gSE8L3zzjt4+OGH0draioqKCqdO6Hv37sXdd98Nk8mEm266CTt37nR4O7FY7JPwmUwmhxEci8WCSCSCSCSCTCZzW+BWC8EU8XmCreGyUql0OP09JSXFo6kMLS0t2LP/MHoHhxERqsAFZ5+BdWvL3fq8CQQCrFtbjtiY6PlG8//tE6UkxOJI3yD4QjFOVH+DoYEeiMVS8E1qZMSG4mT110jIKwOHx8NI1wmwxjqx6Xs/BQBoNBq8+N+3oRRGI2bN+bBYLDhxshnRLDZ+d//PrdKvH3/8MV7bexTJl98HvkQOC4Cxrhb0738BawtyIGBrwPnfwFjd7CT4k1340c9vcfj6uFMN6epz9/lXhyDNKINAerqgJTIpA1O9yWhra4NCoVhwSoI7osVisVBSUkL1+NkW0Ljq43OFSCSyqiB1F/I1Y9xd7AkI4cvPz8f777+PW265xeltTCYTbr/9duzbtw/x8fEoLy/H9u3bHbpFuCt8CwmcWCxGSEgIYmNjIRAIGIFbgGASPkfPxWg0UpZu9IkCpOEyWXGblJTkc0VxQ2Mj/vX2HkQVnImsggugnBrH658fhVKpwpbN53r8XEg2nbkBnx98Au/t/RCC1FKI087EyGg/9CebsOPGK8HjC/D11+9CbzCiMDsdO269kTJQbmxqxgxLjrScYmq91IK1aK/4Cp2dnVb7W6+/uxuy/HPBl8jnjwFAZGoulN2lKMqJBVugQs2n/waLw0OYhI97bvqewwpGwPdU5+j4JESKPNiuwBIpoFAoMD4+bjfNwRZH4qvX69Ha2goAyMrKgkAgsCqgsY3QyAIbb6BXkHryHWPcXRwTEMLnjulrRUUF0tPTqS/HNddcg927dzsVPvoeHylwZMk3I3CLQ6BUQvqKyWSiipI6OzutJgqQLSVhYWFITExclJ4pi8WCjz4/gNjisyGPiAYAyMIikbbuAnx26EOcdeYGag/NUyIiIpAQFYo2tQic0DiY9DokpWYgZt06vL/3Vbzwtz/ish0XO0wvdvUPIiTS3oCcHxKBz7/cj4amFkRFhKG0tBSTM0oIksLtbssWh0Gt0eK3996Lubk56PV6hIaGLliY4strnJOWjI+bexEWc9qo22KxwDQ5gKSkM5GRkYHKykqXbQ62r0dFRQWe+OeLMAgUIAgWWKox3PPTH+GMM86wi9DIiyBf/UK9nRLBuLvYExDC5w6Dg4NWDvPx8fE4fvy41W1UKhVaW1vR2NiIyclJvPbaa1izZg0uvfRSKuXkbwNdhtOstHYGVxE/i8UCm82GQqGgGrmX6vOi1WoxOjmD7HLrCIQvFMHCl2J8fBzx8fFer3+iux/rzvshOIL5qmLyZMwNS8CJEyewdu1ah89VLpWg+5S1+4lyZgpfvL8L8qh4RKpDYKjrxa4P9yA9PhIVPc1QxGdQtzVbLNAPtSD3O5di19vv4vDxaphMZqwryccl393mdDSPr8K39fzN+HjfQxiSyJCSXwqDTove+qPIjJYgOzubsiFz1eZAF63h4WE89vd/I/qsqyCLmK9enZsYxZ+ffQV/T0hAQkICQkJCKAuysrIyav/b14grLi4OHR0daGlpQW5urtuvC+PuYs2SCd+WLVswMjJi9/tHH30UO3bs8Gntv/3tb3j55ZchFouRk5MDo9GIuLg4/OEPf0BqamrAh/crwVDbHQI11emsKMlVxH/q1Cmo1Wq356T5Ex6PBwGPDZ1GBYFIQv3eZDLCpJ3zqArY0WdLwBfAoNOAL7b2rjQbtFZOLbaUFObjm9fegzomASLJfIXhwU/egSUiDcXbrgCXx4XFYkHtF+9hqKUep0bHMDvUheSzLweLw8NIzX4kiExoaDuJU5xIxG+6Biw2G9Xtjah7/En86NrLER4ejoSEBDuvTnf73xwRGRmJ2390LQ4erUDbR8+Cw2bj7PWl+N7VV1Lr0qc5lJeX21VC0kXrwKHD4MTlU6IHANKwKIwnFGD/gUP44fXXUY+r1WrR0NCAoqIimM1mGI1GvPDSy3jvk88xNzeH8pIi3HrTDxccsktiMpkoYe7q6vJosDDj7nKaJRO+L7/80qf7x8XFob//dFn1wMAANfft5ptvxt133029ke+88w7q6+uteoECFfJKbLF9NJeC5RY+TwyXA7koic1m49z1pdhbfwwZ5eeAzeHCbDajt6kKhemJPhszXHjOmXjr6BFknn3p6aneQz0Q6KddTliIiorC1d85Gx98/hXMAjmMeh26Wuqx8Uc7weXN713Vf7UbvUOjYGWdh03b0tFccQgtbzyKiLBQbDvnDJx91oV4Zc83yD9/LXgCIdVfd7yqCR2n5hAqESIxXIKf33Yztec2OTmJ4xWVMFuA3OxMrF+/Hn19fTh89Dhm5pQoyslAeXm5SzeWqKgo3HDtlXj7/d346pvjePvjL1BZ14g7brqBqkIlpzmQI4LoYksXvrGJKfCl9u+BQBaGsclJq98lJiZCo9Ggvb0dJpMJf/rr31B/yoDYc36EKLEMJzrqccs9D+DFp//idI+TDnmuyMnJQV1dHQYGBtyO/hl3l9OsmFRneXk5Ojo60N3djbi4OLz55pt44403AMCuxHep+/h8IZiEb6naGejepWShia3zjaO+yZXEBVs2Y3buIxz96m1wZREwKKeQmxSNa664wue1L/7uNrS0d6Jxz0vgRKbArJkFd2YAD95z+4LFF/l5echIT8fAwAB0Oh06W+ohl8+LwNzkGLpOtCLmgh9DOdqPyPgkJGfdip6qXGxK4IDH4+GPz7yIaUjQ9eKTSM8rQlh0AprqaxGz5ceQ8oHMnCz0NVbijnvuw0Vbz4deq8aujz4HN74AQkUkPjj6PvhPPwtReAzkGWvBE8ej/kAj9uw/jIcf+KXTiwKj0YhH//IURnlxSLvk5+Dw+BjvPYH7//BX/P2x3yA5ORkNDQ0YHx8HQRBoaWlBXl4eJQz0yDk/OwMHP/gayFlj9RjK4ZPIv7Dc7rEzMzNRX1+Pjo4OVLX1IOuq+8BizX/fY/PWod+gxX/eeBMPP/irBd878lxBb3Dn8/mIiIhY8L4A4+5CEhDC98EHH+DOO+/E2NgYLrroIhQXF+Pzzz/H0NAQbrrpJuzZswccDgfPPPMMtm7dCpPJhBtvvBF5eXkO1/O1nWEpCabeN39HfPT5gaTA2ZpzSySSRbF2W+7olcPh4JorL8PWLVMYHx9HSEiIw96whXCU6uTz+fjNzl+itbUVJ0+ehEwmQ1lZmdspVD6fT6XYclITMdLVipj0PEwOdoMXmQKjwQguy0JdkEak5eHd9/4PYRklSLnwJvTP6CCSKdBVtRcnqo8idP3l0Ot1mNNp0dnSiLZv9kFt4UBdfwqtBz6EvHwHCso2QCaTQW/YgK/feh4pECE9NRdGgwFcaThOVh/ES6+8int+drfDE3l7ezt6p/TI2XEh9feI5Gxo5ybx0n/+i9HxSZzSscGWRcIw3odoCQv33HErFQHTL07PPPNMvL17D7oqvkRc3jqAIDDYXIFQ4zg2bdpo99hkj9+ePXvADk+mRI8kLDkXdd+87tZrT2+CZ7PZKCkpodxd3PUfJZ//8ePHsW7dOq8rTVcyASF8l156KS699FK738fGxmLPnj3Uz9u2bcO2bdsWXC9QLMvcgRE+1wNy6d6lgWrOvZgoFIpF8Zwl/SB9HR57/TWX4/dP/hPds5PQaFRQT46APz6A/Kw04H8nWM3cNE6NT6DsB9+BhcVG/2gjzBI5Itacj4ZXj8B8ahhGFg+KyFgc3/8BODE5UMRngItpiBJywI3NQndvP1KTkzA9Mw1hxjr0VO9GwbmXgMPlQiAUIqt8E6oO7UJfXx+SkpLsjnNoaAic8AQ7UVTEpeKTV99E1gXfQ0bBBgDzn8fOrz/Cq6/vwj1334GoqCirVgSBQIDHf/cgdr3zHr7a9yLMZgvOPWMtvn/Nb51ePJC9lJpjbTAajVZ7iOqpMUSGu7eXbFsgw+PxvGpwB+ab21ertVlQPtuVmOoMBhZqZ7CdKED+W+wBud6w3BGfv/C1cKqurg673v8IJ3v6ERMVgSsvvhAbN26k1kxOTsbjv7kXXx06jPaTUxjUDCItUgL5/8TaZDRiqP4wwiMjwRPMn5QLstPR0n4SRoIDk8mMuZEepGy8FCyTHkadBrK0UkwPdUImMsAMFngiGdSnpsDhcCCVSMATCKFnEZDRIhyN2YTw0FCMjo5CJBLZpf4UCgXMMx12z2+svxs6kxnx+eup3xEEgaSyzWj59J9ob2+HQCCwExyFQoHbbr4Jt918k9uvZV5eHsI/3IPehqNIKT4TLBYberUSE7Vf4K47b3BrDUfbIs7aJ1xBfi4IgoDBYJg3Fw+C7RZ3CUrhk0gkTMS3DND3Qxz5UXpjuMywfFRUVOCxf/4H4SVbkFKwHXPjI/jrfz7E1Mwsdlz8Xep2ERERuPqKywEAV2zfhj88+Qw6e5vAEkigH+nEljU5qKhTQauaA08khlAkQkFuFkYH+9ADPXizQ1APdUIgn0/lGucmweOyEZuehd6qAzCpp6kWE6lUCmVnFaKTTheCWCwWjLRW44oz16K4uJiyDaNPXM/KykLY/sMYaDyKuLz1IFgsKCdHMd10EDHxSXYXB1yhGAaTGbm5uWhqaoJCofB4gvvIyAiam5shlUqpYpmnHn8E9z/0ezS2HoE0PBrGiX786OpLce657pkSOPP7DAkJQWZmJmpra1FaWrpgBEeOSKI3uK8md5egFD6xWEwVOwQ6tjP5Vhq2AqdWq1FZWWlluBwbGwuRSLSi0inBEvF5i8ViwUtvvIvo9RdDETOfOlTEJkMovQr/ff8VbD1/i8Mm+pycHLzw9BOoqanB1NQUYmLOglwuh8lowBefvY648gsgkYfBoJ6Dqv1bZCYnIXnz1TjZUInxlgMwTw2BrRyFODwObB4fRedchMov/g1xUj5mw3iY6z+BWG0PFNJonKw6CK5QCvXISWSE8nDhBeeDx+OhsLAQ9fX1KC8vp6IfNpuN3z94P55+9gW0ffAt2HwRBGYtHrjtBvzrtbehnpmAKOR0unG85wTSk+IRGhqK3Nxc1NXV2U2Vd/Xa/f0fz+KdPfvBj06DWTMHqen/8IOrLsWOHTvwwZuv4euvv0ZXVxe2b3/Yo5YZV72A4eHh0Ol0qK+vR0lJiUsRo88GXI3uLivnTOQBTMTnf2wNl5VKJYxGo5XhcnR0NGZmZrB27drlPlwGH1GpVBiZmEJ2tPWkAYFUDotAhqGhIaSmplIVtrbzAnk8HlJTU6nPxk9v/glS9+3Hh3s/x4jRAhGXhRsvPh9j43n4un8KGy+7EWazCRMDXTi8+3XMjMfAIN8EmI1ICOGgKMYI3mwz1pyZgwsfuR1GoxHHKyoxq1Qh59zvID8/nxJiiUSCrKws1NbWory8nPqOxcbG4m9/eQzDw8NQq9VITEwEl8uF2WzBM//dhdCCcyENj8HUYBeUbV/j3gd+BoIgoFAoIJPJ0N3djcjIyAWFYe/evXj3YA3Sr/glOLz5Yxrvasb/Pf8SLrroInC5XJx99tlITk7G4ODggs41dBYaZhsXFwedTmdXlbrQOqvN3SUohY/H41EjOQIdFosFk8m03IdBQRou009kpOEyOfQ0MjLSqeFysHxhgiXi83aPTyAQgMsioNeowP9fE73JZIJep4Vy8hSGh4cxMTEBi8VC9UiS0yac9UhesuNifPei70CpVEIikYDD4WBiYgLVf/wrOr/VQJGQCZ1aiZRIOVKjuBDOtSA5OQbZ59+DgoICyOVyaq3R0VFwuVxIxfOG8bZiFB4eTg15LigooAR690efYN/XxwAAW85aj4u/uw3bvnMhIiPC8c7uPRg6+TUK0lNw1e92QiqVoru7GwkJCRAKheDz+WhtbV3QMeXt3XsQseZ8SvQAIDw1D4PVkaiqqsLGjfOVn0lJSWhra0NnZycyMjKcLWeFO+4vKSkpVMWus15mR+usJneXoBS+lcRyRXy2EwVIw2XbiQLJyck+Gy4zrCzIie/rivPw5YEPEZK9ATq9HgKBAJr+ZhRmJKGoqMirClsOh2MlYGFhYfjjQztx4NAhNLQ2IzREis0P3IXs7GzqNidOnLA6CX918CBe370P3KgMsDhcfHDwdWwsTMOPf3i91fEkJSWhpaUFXV1d0Ol0+O2jf0aPTojo3AsAgsAb31TgaGU1/vLo71BWVkZNPhgaGsIfn3wabd2DYPEEELOM2HHBOdixYwf6+/vR29uL5ORkp89xcmoaknz7tChLJMfc3JzV77KyslBXV4f+/n4rS0ZnLBTxAfMXbQs1uDtbZ7WIX1AL30qwAlts4TOZTHYCRxoukwK3mIbLK5VgifhcYRvd09PXEokEm8/ZhN17fo2TTdXgRabAMDkABUuLp156DhKJZOEHWACz2Yyuri6YTCZsu/BC7LjYcT8Z/Xt86tQpvL57H1LOuYqKRM35ZTh84D2UOhjbk52djZqaGtTV1aFr1oysC0671Ug37UDHl7tw9OhRqrjEYDDgl79+GOqYNci6/GoQLBZmx4bw/JsvIDc3F6WlpaiuroZQKHQ6zaGsuADfnGyEqPR0wYrZaIR+uAPZ2XdZ3ZYgCBQWFlIFOQs1orvr90lfl8fj2fWAuhqKS6Y9g9ndJSiFbyW9Uf4SPleGy6TAKRQKJCQkMAK3irBYLLBYLJidnXV48UOmrx3NC3z8r/+H5I2XYU1iJpSTpyCUKTA72o//vPkuHv7VfT4dV3t7O/7y9+cwpWeBYHPANyhx500/wLp16xw+B/LzWldfD25UOiV6AMBic6BIK8aR49V2wkc6nLz02hsQxlinKAmCgCQxF1V1TZTwVVRUYBISZBRuoG4ni4iFJGcTPvx0L8rLy1FcXOxymsP137saX991L/pZbERmFEGnnMFI9RcozUlxaEtGb0Tn8/lWk9ZtMZvNbmdgbBvc6ZG2K7eo1eDuEpTCRxKMEZ+t4bJSqWRGLC0CKzHiI71K6fuzMzMz4HA4CAkJ8ejiR6lUoqqxFRk77gCbw4U0bD66kUXEoPHDf2ByctLtKkdbZmZm8PCf/w+S4guRmTi/tzU3PoI/P/cq/hoVZZdGpH+PjUYTCLb9acsCC3Z/+AGeef5FaHQ6ZKYk4PcP7sRZZ50FLpcLhUyCjrFRxP8vxTc52I2poR7olDMIKTk9aunUqVMgZPaRnCQsGv297QBgNXPP0TSHpKQkPP+3P+Hl197A8b3PQiaV4LZLL0RUZAQmJychkUjsxIvH46GoqIjyCXU0IQJwHak5gsfjoaSkBDU1NSguLqYa7BdKmZ5+vY3Ucw4mglb4hEIhtFqtR04GywGLxaI+XHTohst0P0qCICiBk8lkiImJYQRulUH3KqVXUgLWZtwxMTEYGhpCSEgIwsPtZ+O5QqfTAWwuWDYiw2KxweLw5v/uJd8cPQZjaArCE08XdEjDoyFMLcMX+w/g5h//yOr29OkMOdlZeHf/tzAZy8HmcKm/f/Lso9CLoxF+zs0IkYWiv6MaV914O156+k8oKytDWkoyDlbsRodUjqZ9b2F6YnzenmyyH4auXFz//WshEAgQHx8P8+R+u2OeHelFSWoy9bNIJKLaHBxNc0hJScEjD/2aOr4Pd+/GzocfBTgCcFkWXLl9G2695SdWgiIWi6k1y8rKnE6j93RfVSgUoqCgAPX19SgtLQWfz3d7rxBAULq7+PxMurq6MDs7i7m5OZjNZoyOjmJ2dhazs7Mwm834xS9+sSwnZZFIBJVKFfDCRxAEtFotTp06ZTdRgLTrWgkTBYKNQHidSSMA+h6crZWbRCJBRESES69Sb55LaGgoohVSTA52ISz+9Oib6ZF+hIo4LqeVL8TYxAS4MvveNZEiAkOjbXa/p0d8ycnJOGdNNg4efB8hqYXgcHjoOLYXczoj0q+9C2ze/FilkPQSTDd/jR/c+nNIJFJo1EpEJaaj4eXHIcw8A1EX/AIsDgcckx5NB17Czbfegf+8/ALWrFmDBOnr6D7+BRJKNoHN5WO8tw2a9qO48vbHrY5LoVAgOTl5wb65jz/5BH/61y5IN/0QManZ0Cmn8dbh9zCn/Bt+vfNeuzXJCRGlpaV2a3praC+TyagWj7KyMo/2CgFQVfLBIn4+P4utW7dCq9VCJpOhtbUVpaWlUCgUCAkJwXvvvYdbbrnFY8cDf0AKn7uu5YuNs6t08mqK7IdbDMPlpWYlpJjdYSlTnQaDwSpFqVQq7ZxuEhISlszKjSAI3HLDtfj9//0LqqlyyKMTMHtqCOrO47j/1ht8+nympyTj44p9ADZY/X52qBtbSuz3wMxmM/V5IggCP7j2ahTm1uObimroNAao+DpIkovA5s6LnlmvRf9nz0OUezbYIVGQhEaAPzWC8ePvguAKID/jGrA5HIhEIhCEGJFn/wB7334IWq0WAoEAf3n0YTzz3L9x6L0nAYKF5Lgo3H7DVQ5n38XExECtVqOtrQ05OTl2n3uLxYLnX/kvojdeDYtIDgDgS+RI2vJ9fPz2Y/jpT260a2CPjo6GVqulWjHoa3qa6qQTFhZGNbiHhoa6vVdIFz/SQWel47PwdXSc9r87++yzcejQIernzZs3Q6PRLIvwLdeEBk8Nl6enpzE5OYmUlJQlP9bFgNwbW+nCt1jHb9tGolQqqeo5uhGAWCxe9qvrgoIC/OnX9+CTvfvQ3XEQJfEx+O4Dd3s0/NQR5eXliH7/Y3RXfoX4gg1gcTgYaqsFb6ITWzZfZ3d7288TQRAoLi5GcXExAOCll17C3sZPqL/PdtWCo4iDKL0cuvEBCEMiwJOGQXeqB5jaC65YCpNOC4vFDIJggyePhJlgY3R0FElJSVAoFPjNA/fhlxoN9Ho9ZDIZjh8/7vSEn5qaiqamJodtDjqdDqNjE8iNTsTs7Onp9RyeAHxFDAYGBhw6tyQnJ6O1tdWux8/XKe6xsbHQ6XQYHh726JwTbO4uPn+zyKnCPB4PHA4Hu3btwvr16zE5OYnx8XGf9gJ8YbEnNPjLcHmlW5bZshKLQpzhy/Mwm812AmfbRhIeHo6kpKRF7ZP09SIkJSUFd956sx+PaL7g4vcP3o/Xd72Ngx//AyaTCeXFBfjhQ/c7LJhZ6DlcddVVePSpf2K6swbyjDXQT58CJywRhtkJcDlsECw2YDZDHJeFsW8/hEkzB7B5MJnMIAgWtGN94LFADbYmEQqFVJGJK8EhCAJ5eXkO2xz4fD7kIVIoJ0YA3unJDSajAbrpUZejprKzs+168fwxuzM5ORkjIyMYHR1FTEzMwnf4H+R3OxjcXXwWPhaLRX1xd+7ciVdffRXffvstRkZGcN1113nkQ+dP/BXx2Rouk/sstmmo+Ph4r9JQgebc4itkD9BKN552V8Btp74rlUpoNBqrKlu5XI64uLgVf7LwJwqFAnfedgtu/+lPYLFYXH5eFhI+iUSC5596HD+9ZyfmGvbBoNPAxBEiPjEdFohgNs7vTxmmhiAJkWP8i+chP+MqcGJToBw+icmDr+DGqy5ZMMJ2dQwsFsthmwNBEPjhtVfimbfeQdiGKwCZDCaDDn1ff4BN5SUuhce2xy88PNynVCd9XblcjtnZWfT19SExMXHhO9HuGwwN7n7JpahUKnR2diIpKQkXXXQRVCoVtm3bhq1bt7q9xuTkJK6++mr09PQgOTkZb7/9tsM5ZGw2GwUFBQCAxMREfPTRRw7X82Y0ESlw9L0Wk8lkZbgcFxfnV8PlleLV6S4sFitoIj46tils8gIIsJ/6zhQhuY87J3F3otbNmzejtfoY3nnnHQwNDeHjfQchNs+AHZmGwbFRGJWT0LR/g3VX/BQ17/4TU7sfwzSHBx4b+M6Z6/DIw7/1+bk4a3P4/rXXYGpqGi+/8RRmw2NhUE5h81nr8Kv7frHgmmw2G8XFxaiurgafz/c51UliNpuRkZGBzs5OCAQCj4YcB4O7i89nb71ej2eeeQYHDhxAW1sb1q9fj9tuuw2bNm0C4H6q5fHHH8fmzZuxc+dOPP7443j88cfxpz/9ye52QqEQdXV1C67nKuKzLSRQqVRWjhVisRgxMTFLss8SbMK30lOdFosFBoOBavhubW11mMIOCwtbdUNxlwtX5xCTyYTKykqcOHECsbGxuPLKKyEWi3Httdfit4/+CW31n8Oi0WL21BCiIyPB6zuOf/71UVxw/vmYnJxEVFQUTp06hcbGxgUnGriDozYHgiDw4x/dgLI1xQgPD0doaKhHmTA+n0/1+LHZbL9kU0wmE7hcLtXgzuVyPRp4vNLdXbw+q5NXHk8++SQaGxvx7rvv4q677sK6devw+uuvQ6vV4oILLnB7vd27d+PgwYMAgBtuuAHnnHOOQ+FzF5FIhImJCXzxxRfg8/mIiYlxaLgcFRUFsVi8bA2awSh8K+X5OJs4wePxqPS9vyP8pSYYCo2cPYfZ2Vncfe8D6J7UgRWVDkLZiCf/+W/846+PIzs7G/954TmMjIxAr9ejt7cX69evt3IhIdOM0dHRUKlUOHHiBHJycnw+XkdtDiaTCWKx2G0zalvIHr+qqiq/fL/I8zcZpdbU1KCoqMhtK7qV7u7i9beZvKpvbm7GZZddBolEAp1Oh40bN6K/vx9DQ0PU7dx5QegbrdHR0RgdHXV4O61Wi7KyMnA4HOzcuROXXHIJAKCtrQ1HjhxBc3Mzmpub0dLSAplMhry8PGzfvh2FhYUBabgcbMIXiKlOW79SpVJpZ8htO3FCqVSir6/PpX0Uw9Lg7Bzy3AsvodcShtTtlwEEARbBwtjJetz3m0fw4Vuvg8ViITo6GhaLBQMDAy4vblNTU9HQ0ODxnpczbNsc/JGiVCgU4PF4aGxspAbbegu9gV0oFFIzDMvKysDn891ag3xPqqurUVpauqLcXXy+jJVKpejp6QEwf1XyzDPPgM1mU5VN9A/sli1bMDIyYrfGo48+avUzQRBOxbK3txdxcXHo6urCeeedh4KCAqSlpaG9vR0ajQYXX3wxdu7ciS+++AJdXV245557fH2Ki0qwCd9ypjpd2bnRq2zdsexa6SnbYMNRf9ynX+xH4mX3z79X//t9RHoRuhq+RGtrK/Ly8qjbLnTxTRAE8vPzUVVVBZFI5LHTjSPobQ4hISF+SYtzuVyEhYWhubkZ+fn5XkdZts4tUqmUMvR25ETjDIIgoNFoVpy7i9dHSb7gZWVlVH/Kpk2b8Le//Q333XcfvvOd71jdDgC+/PJLp+tFRUVheHgYMTExGB4edrrZSpYcp6am4pxzzkFtbS3S0tKwfft2q9utlCnsjPB5jq0ZAFlJCYCxc7MhGFKdjrBYLNDr9CA487M3zWYz+AIBCBBgceftCkncjbbIQhKyitLXCRT0Ngej0ei3/WCyx6+rq8vrnkpHr0lYWBiSk5Mpv1BPjpcgiBXl7uL1EZIvyo033ggAUKvVOPfcc3HZZZehr68PBw8eRGFhIcRisVuh8/bt2/Hqq69SLRE7duywu83U1BREIhH4fD7Gx8fxzTff4L77HLvESyQSj6s6l4NgOyn5c4/PX5Zd3sBEfIEFvSCN/G9yYhz6679BTP4GWDBfXc4yaGBRnrLaq/Mkzcjn81FYWIiGhgaUlZWBx+P59Dkg2xyOHTvmUfGIK8h5e7W1tRgcHLTrP3QHZ69JTEwMdDqdQ9eYhY4JWDnuLj5Ls1KpxP79+9HS0gKDwQCRSIT33nsPk5OT+OUvf4ktW7a45RCwc+dOXHXVVXjxxReRlJSEt99+GwBQVVWF5557Di+88AJaW1txyy23UFHSzp07kZub63C9xW5gZ3CMt3t8gWbZFUyspIsrcryWUqm0+jw0NjbaVVz//je/wk/v2Ykxix4hCVmYGu6DqvkgHvrZT6w8ej2NeqVSKdLT0ymzaMC315DL5SI5ORmdnZ3QaDROJy94gm2Pnzf90s6eU3JyMtra2tDR0YHMzEyXa9C/6+R6Op0OfD4/oMXP56rO9957D3/+85/xox/9CARBICwsDGlpadSVEzkGYyHCwsKwf7+9K3pZWRleeOEFAMAZZ5yBxsZGt9ZbLsuy1c5CkdJKsewKlogvUJ8Dma6mR3H0xn+JREINSK6vr8eaNWvs1khPT8erz/4f3nr3fdQ3f4ms2CgU/fganLFhg91jeXoSjoyMhFqtRnNzM7Kzs30+iXO5XERFRTmd5uANHA6HakcoLCz0qzVkVlYW6uvr0dvbi6SkJKe3s90rpFubBbJhg8+vvlAoxCWXXIJf/vKX1O/S0tLw7bffOhwquVRIJBJG+JYBMtUZKJZdDMvLQulqeluRs8Z/VyfPuLg43HP3nQDmxcVgMKCqqgpCoZASArrJtSckJSWhubkZvb29Pp/AzWYzVVy10DQHZzi6iCF7/MiRQwKBwKfjJCEjyurqaggEAqfTOByNN1oJ7i4+7/Ft2rQJMTExlANAfHw80tPTfRpb4g+8cW5h8Bxby66JiQmMj4+Dy+WuaMuuYIn4lhIymqeLHNkXSQrcYqWryfeKx+NRpfnl5eXg8XheCx9BEMjNzUVlZaXDmZmeQGbIFprm4M4atkgkEmrPz1/RJHB6f5Kc4O5oj9LZXL9Ad3fxSztDdXU1Pv74YxQWFuK2224DQRCora1FVlbWslWVSSSSFVHVSRLo1XfuWnbp9XqEhoYGzDio1c5ifK7IthG6wNGjebLgiN4XuZRIJBJkZmZSUZA3qU4SFouFnJwcVFRUYHZ21uu+Trp/bWpqKhobGx1Oc3B3DVtCQ0ORlJTkVjTpyQUd6e5SXV3tsMHd1UDbQI78fGpgJwgC9fX1eP7557F//368+uqreOKJJ7Bz5048++yzuOaaa5bNsFgkEq0Y4QukUT6kZZdtBZ27ll1KpXKZjty/MBGf9cUO+TmwnfQeEhKy7NG8o/cpIiICSqUSra2tiI+P92mPjs1mQy6Xo7Gx0aMGbzpms5m6CCB7BqurqyESidz2yVzIoDo2NhZarRYtLS3Iy8tz+n542kwvEAicplMXEj7y8YxGY0C5u/jcxxcaGoqcnBzExsbi2muvxa233goA1MSB5XqiK2ncD5kWWOoqKFeWXd6acq8kyzKG0zi62DGZTNTFjkQioWZIBmK1niPTi+TkZDQ2NmJ4eNin85DFYgGPx6Mmo5eVlXl8MW8bANCnOfD5fGqaw0JrLPTap6SkoKWlxWWPnyuxcgY9nVpWVkaJ+EJrka87mSoOFPHzOdUZFRUFkUiE2267Dbm5uTh58iTeeustajLDcj/JQImkXLHYIu2NZZe3BKJlmTcES8Rn+/mnfxZIgSM/CxKJBBKJZMkM2hcbMrI6duyYTy0E5B5hWFgY1Go1GhsbUVRU5PP+HH2agzuFKe5kz+g9fkNDQ4iNjXXrWNwhNDSUEv/S0lLKg9SdYwIQUO4ufhlE29jYSBW4nHfeeRAKhdi5cyeA5RO+QBc7Ov6ayedPyy5vCRbBWOmQ7QIajQaDg4MwGAxUuwA9XR3oVbW+fpZYLBbS0tLQ3NwMlUrldnsVHbpQJCQkUGPYPDGcdiY25DQHdwpT3J3Fx2KxUFRURPX42Q739SbiI4mOjoZOp0NjYyMKCwvdXove4A4sv7uLz48eFhaG+vp6u9+TJcsWi8Wv/SXBiKe2ZYFs2RUsqc6VIuDO2gUsFguEQiGMRiPlPbkS5wT6I2PDZrMRERFBVXp6mtWwFa2srCzU1NRQFovu4Eq0HE1zcOc4XMHhcKg5frZFKb4IHzDf5qHVatHe3g6xWOz2WoHk7uKz8FksFrz66qswGAzQarXQ6XTQ6XQwGo3QarWwWCx4/PHH/XGsHkNGUst9dbEQzoRvOS27vCXYvEcDCW/aBVpaWhAaGmrlZLKS8IfwkRcBkZGRaGhowJo1azxa07YqlCAIFBUVobKyEkKhEHK5fME1FhItepuDMzcqTwsFBQKBw6kLvgofAGRmZqKxsRFqtdqjCu5AcXfxWREIgsCBAwfA4/HA5/MhEAggEAggFAoRERHht4ZKbyDdWwJ9tAyLxYJer8fU1NSKt+xaKZHSQgTClAlH7QKkwPlrTzbQ8ZfwkSOKlEolTpw4gezsbLfv76gPkIyoampqrKatu1pjoZM82ebQ09PjsM3B3VQnHXLqAlmUwuFw/FJIR+6fHjlyxONzPN3dhcfjLYv4+SUUevXVV/2xjN8RCoVQqVQBJXyOLLuUSiVmZmYgl8sDxrLLW4JF+JYC23YB25S1RCIJiHaB5cQfwkcXrrS0NNTX12NgYADx8fFu39/RyVkoFCIvL88tGzJ3xGahNgdvBSssLAw6nY5Kpfoj4gPmL9jJCfaTk5N2e4muoPf4Lcdn2y9nVjK15eiEZ7FYlu0Evpx+naTZrjPLLrI8PCkpCf39/ZDL5V4ZzQYawZLq9LeA6/V6qzQlvV2AjOIWI2W9EqqaXeHu8bu6DX0NUlwqKyshFovdmpjgSnDkcjmSkpLQ0NCAkpISn3vnbAtT6BftRqMRn332GV57810MDw1izZoS/OaB+1FaWrrgurGxsdBoNGhtbYVCofBb1shisSA9PR2tra0e+4WS54rl6PX2iyIFwt6SI0Qi0aLbltladjky213IsitYxAJgIj6yXYCepqSbcEskEsTGxnrUG7ma8VfERz9H+TtNGRsbC5VK5TKF6km0xuPxHLY5PPV/T+M/734M4RnXgVuagMquGmy9aDt2v/c2zjzzzAXXTU1NRXNzM06dOuXWvqQ7mEwmCIVCr/1Cl6vIxW8Rn8Vicfhfo9EImUy2LFed/oz4XLlY0AtNXJntOoMRvsBjofePfsFDfh7o7QLkdIFAbxcIdPyd6iQRCoXIzc11K03pzjGkp6e7TKF6mqa0bXOYmZnBiy+9jPAbngFHNp8Z4pbGghCIcd+vfoNvDn214Jqk9+jRo0f9dtFFpk0lEgl1vPQG90DFL89+7969GBgYoCo5dTodDAYDDAYDxsbG8NBDDzlspKTf/+6774bJZMJNN91E9QCS6HQ6XH/99aiurkZYWBjeeusttzzuvBE+spLStsmXbtnlbxeLleQysxDB0s4AzH8WHFXWklkE8oJHKpUiOjo6INsFVvpFiL+KWxyl0hQKBRITE9HY2Iji4mKf0pQEQaCgoICq9LTdtvBmf06hUFD+m6dOnYI4LgMcqfW64uyzUPvp3zxKpcbExGBwcNDjfTlH0PcLFQoFUlNTrRrcAxWfhI98sSsrK9HV1YXY2Fjw+XyqUkcul0Mmk7lUf5PJhNtvvx379u1DfHw8ysvLsX37dquS3hdffBEKhQKdnZ148803cf/99+Ott95a8PgWmtBAt+wiT2h0yy6JRIK4uDiPelW8wV8N7IHASnZuMRqN1Odgbm4OKpWKspQiPw8robLWlkATY0/wxWDanTXi4uKgVCpdNqTTfTZdwWazqd654uJiq2Z5bwtTyL25jo4OGOYm7Z14lFMQiiUev8cpKSlobW11aDztCbaFMlFRUdDpdGhoaPDY3WYp8Un4yDfyt7/9LQCgv78fOp0OPB4PiYmJbq1RUVGB9PR0pKamAgCuueYa7N6920r4du/ejYcffhgAcMUVV+COO+5w60qQnMI+OzuL6elpcLlcO5smeopSLBYvS4jOYrF8HnsSKKyEVCe9XYD8PNi2C0RFRWF2dhZr165d7sNd1Xg7UsiTNTIzM1FTU4ORkRFER0c7vL+7oiUQCFBQUOCwWd7b55Gamoq5uTmI2Baomr+CpGAzAMBiNkH9zWu47gfXebQ2WVRVWFiIhoYGlJaWemW8Ta5lexGYmJgIrVZLjV5aiOUQR7+kOnU6HZ5++ml8+eWXaG9vR3Z2Nm644QZs2bIF4eHhLkVqcHAQCQkJ1M/x8fE4fvy409twOByEhIRgYmIC4eHhVrfT6/Vob29HU1MTmpqasHv3bszNzeGpp57CDTfcgO9+97vUVOdAGpPB7PEtDuS+rO2UbwBUb6RCoUB8fLxd4dFKr4YMFvxd1ensvkVFRaioqIBQKLQzjPY06pTJZEhLS/Nbyo8cCvvwgw/g1w/9DsqOQ4A8DvreOuRnpeOx3z/i0XqkWEmlUmRlZVn1+HmKs9aIjIwMNDY2oru7GykpKS6f23Lgl1Tnc889h6qqKrz99tt46KGHkJqaSg0vvOyyy5bsJPLss8+isrIS+fn5WL9+PaRSKYxGI+66665Ff2xfCCbhW67n4mgfjtyXJc2XPWkXCBbRW+kCvtDxm81mqFQqcLlcp/267kRsHA4HRUVF1PQFegTkTdQZFRUFtVqN1tZWp04snsBisbBmzRq89sqLGBkZwfT0NEpK7sH69es9Pja6WIWFhUGr1S7YjuEKR/ch20ZqamogEAjctnZbKnwSPvLKvr29Heeffz5CQkKg0+lQVlYGs9mMnp6eBdeIi4tDf38/9fPAwADi4uIc3iY+Ph5GoxEzMzMOe97uvvtuq5+VSiWam5u9eGZLSzAJ32JHfK7aBcg05VLsyzIsDaTwWSwW6HQ6q/Q0fTagSqXCmjVrHFqzuStcYrEY2dnZdqOHvN2fS05ORlNTE3p7ez2+rzPy8/PB5/Nx2WWXee2KZft84uLiqB4/T6fCu8J2gnsg9Sn7JdUpk8moN1cmk+Ef//gHYmJisHHjRgCur57Ly8vR0dGB7u5uxMXF4c0338Qbb7xhdZvt27fj1VdfxYYNG/Duu+/ivPPOc/uDvBKG0TLCZ49tuwA5aYLNZlP9kUy7QHBC9ySdmJjA7OwsZmZmwOfzHUbv5J4tua9mm7LzJFUZFhYGlUqF5uZmFBQUUFXK3ggfQRDIy8tDVVUVNZXAF8xmM8RisdvTHJzhKD2ZlpaGpqYm9PT0uExNegqHw6EmuBcUFATMwAKfhI8Un/Lyciq627hxIx577DFccMEFuOSSS6xu5/AAOBw888wz2Lp1K0wmE2688Ubk5eXhoYceQllZGbZv344f//jH+MEPfoD09HSEhobizTffdOv4ltO5xROCTfg8nTSh0+nsXE2A+XYBiUQCqVS6LJMmgoFATnWSFzf06J28uCGjd4VCAT6fj6ysLJdryWQyJCUlOWxN8HSPLjExEXNzc5QI+FJZSkY9Bw8exNzcnE8nftLhhN7m4KnhNn0dOqRI19TUQCgUOizy8RY+n0+lkW0NA1bkHh/5YbjssssAgEo37N27F3w+H9PT0245BGzbtg3btm2z+t0jj5zesBUIBHjnnXc8Pj6yqjPQCSbhc9XOQLaP0AXOaDRatQuEhYUF7JRvBu8h92DJf/TRSRKJxOkYrcnJSeh0OrceIzY2FrOzs3bTx73Zo8vJyUF1dTXEYrHPlaVcLhdCoRANDQ12+4eeQDepjo2NpaY5uFM56WwdOrZT4Reyc/MksyMWi5GXl0cV0ix3lsYvqc7Gxka8+eabkEqlMBgMmJiYwNDQEL7//e9jx44dfnED94aF+vgChWASPjLiI/vg6O0Ctu0jEomEse0KMpztwdJ7Yz3phfQ0YiVbE0ZHRxEVFeXVGoC1Z6av43MsFgu4XC7S09Pt9g89wfY8mpaWhsbGRvT29iIpKcntdVyZVNNTk7a9iJ6s4wi5XE69BqWlpcu6B+9zcQtBECAIAjqdDiEhIZBKpdDpdJicnERbWxt27Njhr2P1GIlEsiIivpXq3EJvF6BHcVqtFv39/ZBIJFAoFIs68Z3BNYuV6nT03qvVar9btnnazkAKVmVlJUQiEaRSqdcX3jweD4WFhfj22299MpggI6zw8HCo1Wqr/UNPsE1RkpWTVVVV1LxBd1godUvvRXQVnRmNRo/FKzIykmpwd+WWs9j4ZY8vPz8fTzzxhNXf6uvrqQG0y9XXtZL2+ALducVZuwDdpzQyMhI8Hg+NjY1+KeFmCAwMBoPde0+aE/viUesO3gg3l8ulmrPJCnNvj4vca2xra8PatWu9ElC68CYmJkKpVNqlY93B0WtBr5y0nebgCzKZDJmZmVRq0pHAeTtVISEhAVqtFq2trcjPz/fH4XqMX/JMer0e3d3d1B6OTqfD7t27qXLb5VL1lRLxBVKq09d2AZPJFDAN7Aye4czRhsPhUNWUMTExSzor0tvCEolEgrS0NDQ0NPg8AYDH40EqlaK1tRV5eXke39824szOznbpFOPN8ZHFI55OR3BFeHg41ePnKDrzZa5feno6mpub0d3djczMTH8crkf45dPb39+Pm266CdHR0eDxeNBqteByubj//vsBLN/YIj6f7/bG+HKyHMLnqqKOdDUh5wV6kqoKJOcWBsdRAt14m15sApwegBsoKWpforWoqCjMzc1haGjIp3OQ2WxGUlISTpw4gb6+PrftGOn3pz8+PR3ryCnGG/zR5uCI+Ph4aDQah0U0RqPR68chJ0Usl1WjX16dhIQEvPjiixAIBJicnKTKbZeblVIZuJgnFtt2AfpJbqGKOm9ghC+wsFgsmJ2dtfoMkJW0ZAQfyJW0vu5RpqWloa+vD6Ojo16fk8iUHtmTR75mntzf9rXlcrkOZ+75gq9tDs5IT09HY2Mjenp6rKbiOKsOdRdyZuly4LPwaTQaCIVCxMTE4PPPP0d/fz8MBgPkcjkuu+wyOz/N5SCQe5n8iW27gFKphMlkWtKT3Gp4nQMRi8VCRfD0hn+tVotTp05BJpMhMjISKSkpAT8rjY6v312CICCRSDA4OAi5XO5VdEUKF30/baGKR0f3t4WcuUdWevojSluozcGbi1KyiKa6uhpCoZCqll2Oyen+wqdXemZmBg8++CD+8pe/4Ne//jWqqqpw0UUXwWAw4NChQ2hra8OTTz7pr2P1mGA9CZP+hPRKSvpejFgsRnR09JLuxTAsHY4KjcieOHI+IBnB19XVISMjY0WJHR1/mVQXFBR43UdHPwY+n+90+oIzXFWVkinlhWYCeiJYrtocvN2Xo4s+n8+HXC73KdW53Ph01Gw2G8eOHYNAIMDHH3+M7u5uq7+npKQsq/CRrNSIj7yKt536TqYIAmkvhsH/mEwmq2IT20IjiUSC+Pj4FTcf0BP8NY+P7sNZXl7u8Zr075ZMJkNqaioaGhrcSiku1E5BzgTs6OhwWujhyTnMVZuDLz3VZHq2pqYGJSUlPhW30I91OfBJ+AQCAfR6PQYHB6nBhuSVAFkJtNzweDwYDAav3RKWCrPZjMnJSZftAotVMs6wvDgbn0QQxKr3JfXHRStZIBMWFoa5uTm0tLQgLy/Pp3Wjo6MxNzeHEydOIDs7e8HHX0hsMjMzUVdXh8HBQTuTfnINT0TGWZuDr2IlFApRUFCAuro6REZG+q2CdKnxSfg4HA4uv/xy3HzzzUhOTsY111yDLVu2wGKxYN++fSgtLfXLVYEvCIVCqFSqgBE+0oCXLnAGgwFarRYTExNLNvWdYXkgp7zb7sPajk8SiUR+ucBZqdkOEn8dP7lGUlISmpqa0N/f73F1pi2kC4kzsSJxR7QIgkBBQQGqqqogEons7MK8KSRx1Obgj/OxTCZDRkYGWlpanE6tD3R8TtD+9re/xbfffouamhps2LABBEGAx+PhggsuwODg4LKfvEnbstDQ0CV9XHq7AHmSo7cLkCc4stigqqoKaWlpAVlZx+A59J44UuBsbduYfdiF8Ueqkw5pxlxZWQmJROLTeYEUK9Ilxpm3pbvpRQ6Hg+LiYlRXV6OkpMSq4tHbQhKxWIycnByqzcFfgUhERATEYjH6+/sRExPj9cXJikx1kqxfvx7r16/HzMwM9QU3GAxoamrC+++/j4yMDOTn5y/Lk1xs9xb6nDDyBEfviRKLxQgJCUFsbKzLdgGyl48RvpWFbU8cfboEmaYMCQlBXFyc3ZR3hoXx1SDaEWQfXXV1td20AE8hxaqmpsbpWp48B4FAgPz8fMoujCye8aV1IDQ0lGpzSElJ8ds5RiwWQ6/Xu5XuDTT8dqm5Z88efPTRR1AqlSAIAmKxGK+99houvvhiXHXVVcjOzl6WyjJ/GlXTrZts01Rk07e37QKB5N7C4Bi6qw35L9CnS6z0nkp3Up3eCKNAIEBeXh5V7OJL1C0UCqm2BEdreRplhYSEICUlherH88dFMdnm0NPT43YbxkKYTCYkJSWht7fXY6Ps5cZn4SPfkAceeAC33HILzjzzTOh0OsTGxuLw4cP4/ve/j3POOWfBN37v3r24++67YTKZcNNNN2Hnzp1Wf3/llVdw7733Urn0O+64AzfddNOCx0dOZ/YEejUdeaLT6/WL2i7ACF/gQFbTGgwGdHV1QaVSQaPRgMViWe3DrZSeuJUcZS7mHqVcLkdiYiKamppQVFTk0+MoFAokJiY6bEswm80ef06io6OhUqnQ1taG3Nxcv/TMpaWlobKyEnNzcz6tQ2IymcDhcKi9SU+MsklWdKoTADZs2IArr7wSERER1O9uvfVWbN26dcFKNJPJhNtvvx379u1DfHw8ysvLsX37djuj46uvvhrPPPOMR8flagq7bbsAWU1HbxcIDQ1dknaBYBO+lVJUodfrrd5/ejWt2WyGVCpFdHQ0U027TLjzOfIlqo2Li3M4w8+btePi4jA3N4fOzk6rog9vo7XU1FQ0Njair68PYrHY5ywCQRCIi4tDZ2cnTp065bFI2UJGsiwWCyUlJdQcP39YsC02Pgsf+Wbcc8896OvrQ3d3N+XenpCQgLfeegujo6P43ve+h9jYWIdrVFRUID09HampqQCAa665Brt37/aLwz8Z8fX29kKv14PP51OpSvIER0Zxy9kuEEzCR9qWBZJQ2Db9k1G8K/PtyspKqwu5lchqSXX68lnLyspCdXU1pFKpnRh4WlyTlZVlZ0DtrfCRhThVVVUIDw/3S/rcYrEgPj4enZ2dPk9zoKdw6RZstoU5zljO84PfUp2vvPIKDh8+jLCwMHA4HIyMjIDFYiEzMxNGoxGzs7NOhW9wcBAJCQnUz/Hx8Th+/Ljd7d577z0cPnwYmZmZeOqpp6zuQzI3N4empiY0NTWhsbERn376KUwmE6KionDDDTdgy5YtAdn0u1Jn8jliOf066cVG9Cie3hO3VFF8oLCSn6Mr4SP7H1UqFSIiIrxOO9OHzpKfERJPi2sIgkBhYSFV6SmTyXzan2Oz2SguLsa3337rc4QGgLIw9Mc0B9u9S5FIhPz8fMqCLZB7Tv0W8f3iF7/A7bffDi6XCy6Xi66uLuzatQvbt2/Hpk2bfJ43d/HFF+Paa68Fn8/H888/jxtuuAFfffWV3e1ef/111NfXIz8/H5dffjni4uLA5XLxk5/8xKfHX2xWwkw+d1kq4SN74uipSnqxEbkXJxQKA6bYhMEzSOEj+1/pFzTke83lcjE6OurTYFMej0fZmtGtyLwRLS6XSwkLOQ/Ql88fn89HQkICent7kZaW5pOgkPuNtm0O3tQqOHpeISEhVpPmA/V757c9Plu38tLSUjz55JNoaWnBpk2bXH4g4+Li0N/fT/08MDBg1xBKX/+mm27Cfffd53CtW2+91ernnp4eq7UDlWBLdfrzubjqiSSLTZieuODA1mx7YmICk5OT4PF4Vg5GaWlp1HttsVjQ1dWF7u5uaruEvp67SKVSOysyb0WLbpEmkUh8FgAej4fIyEifBYUepZFtDg0NDSgpKfHqosHRfSIjI6k5fq6KhlZ0qpOkq6sLjY2NMBqNUKvVGB0dhVardauzv7y8HB0dHeju7kZcXBzefPNNvPHGG1a3GR4eRkxMDADgo48+cug87giRSMQMo11iWCyWVxEfvSeOvh8HWPdEMj1x7hNoe6106C42tpPdJRIJpFIpVCoVkpKSFtyLyszMRE1NDaRSqdW+rKfPn7Qia29vR1ZWlk/RWlhYGFQqFXp6elw6u7iDyWRCSEgIxGIxmpubve6Ltk1PLjTNwVsSExOh0Wio1zHQ8Fn4yBfys88+w9NPP43k5GTweDyEhYXhlltuwebNmwG4no3H4XDwzDPPYOvWrTCZTLjxxhuRl5eHhx56CGVlZdi+fTuefvppfPTRR+BwOAgNDcUrr7zi1vExU9iXHndSnc4mvfN4PKrYJCEhwS/VbAzLi7ORSfSI3dlk97GxMbfef/pwV7FYTBVXeCNc6enpqK2txdDQEGQymU8XDomJiejp6cHIyAjkcrnX65ApyqSkJLS0tKCnpwcpKSlerWNb2+BqmoMvZGZmor6+3qvhvYuNz8JHvoi33347br/9dq/X2bZtG7Zt22b1u0ceeYT6/z/+8Y/44x//6PG6TMS39NBTna5aRsjUVXh4OJKTkwOyJy6QI6ZAhDR5cNQeQkZxngw99qSqksfjUa4n5L6VN+8fvUAlOTnZ5wsvqVSK6elpn1oISMEiCAI5OTmorq6GWCz2eD1HDjD0aQ4ikchvlcykpVt1dTUEAoFfinP8RdBviCy2ZZm/CAbhI096Wq0WJ0+ehE6nC6iWkdXKYrzWtvuudC9SMorzh9m6p1WVISEhVk3p3qYqORwOVenpa1+axWJBbm4umpqa7KpG3YUuWOTkhcrKSo9bEpy5yNjO21toTXffF7IqlVyX/loGxR5foLKShM9oNC73YbgF3YCZ/Ed3tiEIAhEREYiIiAiolhFPCcR+xOXAWRS3FF6k3rz+ZFN6d3c3YmNjvT4mkUiExMREdHd3+7TXZzabIRAIrIbhelqZafv49MpRT1oSXD0PR9McnOGJDRuPx6N6/Hz1RvUXQS98EonEb16di0kgRny2PXHkIFwAVJrS0SDc1tZWZqzSCsT2gkalUi1KFOcJ3l54ZGVloaqqCjwez6dUpVQqhUgkQmtrK/Ly8rxagxQbqVSK9PR01NfXo7S01KPjcrQ3Zztc1533ZCHBItscyOpRZ1XSnvqPikQi5OXlUe0Ty72tEfTCt5IivuUUPts+KZVKRRkw0/0p3emJW+7n4i+WsxF/saFPlHBUPevogmY58Fb4yGKXiooKnyIMi8WCkJAQ6PV6r4s06FFWZGSklQenuzibzhAWFga1Wo3Gxka3/EbdEazQ0FAkJia6bHPwZryRXC6nevxKS0uXtfVoVQifM6/OQILNZi9JAzu5N0MXOXqFHbkPJxaLvb4qC2bBWGlYLBYrgSP34sjqWYlEEtDVs76kmvl8PtLS0tDa2gqj0eh1kzabzfZphp/tflhycjLlwemukLpKUSYkJEClUtl5hDrC3fFGsbGxlEA7anPwdq5fZGQkNBoNGhsbsWbNGo/v7y+CXvh4PN6K2DtbjCjJ9qperVbDYrFQxSYymcyjCjt3CRbhW2nPgz4Xcm5uDmq1Gmq1Gr29vU7T0oGOr3us5MWctxMYSMEhizS8neFHf1y6Byc5ysyd43AlNKRH6NDQkFNrSBJ3L3DS09PR0NDgUKB9GWiblJQEjUaD/v5+r1oy/EHQC99KwRfhI3vi6FEc2RNH7sUlJCQsmT+pv51bGKyhG27bvt9kFJeUlASRSITq6mqv96YCAV/n8ZETNgiC8Kr3jR5pCQQC5ObmUu0SvnyX6NWO7pg6LxSpEQRB9TGKRCKfegbpa9JHDtHbHHyd5J6VlcWkOpeCQK/Oc0f4SFNe22ITek9cWFgYkpKSltUg1lvnlkBjuT8vttPdyagdQEC934uJr99b8v7Z2dmoqqqCVCpFeHi414+vUCgQHx+PpqYmFBYW+nRsfD6fMnVeu3atSyFwZx6fO9PgPcVZm4OvwkcQxLKm1oNe+Jb75OUutsJHn/Zua+dE9ywMxJ64lZYidMVSPQ9n093pUVygTXdfCnydx0dGbLYTGNwZm0O/P534+HjMzs567Z5CJyQkBKmpqdS0dWfP1d29OaFQ6LfJ8iSO2hx8Fb7lJuiFj8QfE4wXA7KEfGpqCnNzc6ivr7cqIReLxU7tnAKVYBG+xbigcDQ2yTZqj4iIoKz/GHx7H+jCyefzkZubi7feegs8Hg/x8fE466yzXJ4XnJ03srOzUV1dTb1fvmDrD7rQ81gIuVyOpKSkBcXUE2zbHBjhWwEIhUJotVqIxeJlOwZXaSuRSAShUAg2m43s7OwVVXzgiGBpZwB8i/icRXHetIgweAc9YpudncWPf3oHmnuGwY5IBWtuBOG8x/H6y/9GfHy80/s7qm6me4OSLSC+QJb5Dw4OOjW09uScQFZlnjhxAtnZ2QA8d8Gxhd7moFAofL4wY5xbFhlyCvtSCR+9J478L/2ER1Zy0dNWRqMR09PT4PP5S3KMi8lqi/hs914d+ZFGREQgJSVl2Rt3gw0yEnI1rJb82+/+8Ed06KRI/v5tMOj1IFgsTNXvxx333IsP397l8P4LOZ0UFBSgvr7e6R6du98DspCEFFKFQuHW/VxBNsv39/cjISHB57mAwGlBHR0dRXJyss/HuFysCuETi8WL4t7iynWePOFFRka6dcILpigpWIQPsD9xuRqISkZxgbr36i+USiWOHDmCweERpCYn4YwzzgjYCzbyZK/VavHJZ58j/gePgyAI8Hh8aLVahBadh7b/7kdPT4/DE/lCKUaZTOZyj84TsfF3cYqtmEokEr+kJ9PT0zE6OorJyUlER0f7vN5ysGqEz1f3Fkdz4sieOLFY7LHrvC3BUgkJBEc7g8VigdlsxsTEBLUnR0ZxdIGjD0RdDXR1deGXDz4CnSwBHEU0DN/sRejrb+Gpx3/vN1d/f0IKl06ngxkEOIL/ZX2I+T0/rVYLjkiK2dlZh/d3R7iio6MxOzuLjo4OZGZmenx/OkKhELm5uX4rTmGz2SgpKUFVVRWys7P9InwEQSA0NBRTU1MYGxvz+n1nUp0AbrzxRnzyySeIjIxEU1OT3d8tFgvuvvtu7NmzByKRCK+88orbnf+ejCYymUx2BswGgwFcLpc64cXHxy9ZT9xKZCUZbgPWA1HJCxuTyURd7Mjl8qCP4tzBYrHgsb/+H9g5m5GaVUL9vq/6AJ5+9t/4/UO/WsajcwwpPDKZDAnxsZjpboQ8tRAAQLAIWFST0E2OIC0tzeX9FyIjIwM1NTUYGRmxioK8SS+SRgPeNt3bwufzqZSsVCr1aS0Ss9mMrKwstLe3QyAQ+G3dpSJghO+HP/wh7rjjDlx//fUO//7ZZ5+ho6MDHR0dOH78OG699VYcP37crbUdpTrJfRnbOXEEQVBpgWDvkVosAjXV6c5A1OjoaKqCtrGxEYmJiW473wc7g4OD6B+bRcamYqvfxxWeiW/f+RM0Gk1AOO/Toe8B/nbnL3HLz3fCoNwBaWIO1Kd6MXv8fdz505tw8uRJFBQUeJ2qJBvIKyoqIBKJqH43b6vJ4+PjoVQqcfLkSaSlpfksfjKZDPHx8ejp6fHLXh+Z3vdmQgSw/G1mASN8mzZtQk9Pj9O/7969G9dffz0IgsD69esxPT2N4eFhxMTELLg2m81GbW0tKisrkZWVhbi4OLt9mYiICIhEomV/Q4KBQEh1GgwGu704bwaiBqKALxcGgwEsDtfu9SLYbJgtCMgony48Z599Nl7/19/x9+deQOsXXyAxIR4/fexBbNmyBU1NTejr67ObQO7JIFxyhh+ZpuTxeD5VUtJtyPxR9RsSEgKhUIiWlhbk5eX5dK4j2xmEQqFb0xwCjZVxlJi/2kxISKB+jo+Px+DgoEPhO3LkCPbs2YOGhgYMDAxgZmYGmZmZOOOMM5Ceno7MzMwV8watRJZyv9JisdiN0rGN4lZCH+Tk5CSqauowfGoc8TERKFuzxucBqP4mMTERErYJM8M9CIlJpn5/qrMBOenJS5ruIi+sFopebItTysrK8OoLZXa3y83NRWVlJaRSqZUJtafCJRaLkZmZSY0e8iW6IifBV1RU+EX4TCYT5HI5DAYDent7farKpPfxhYaGIiEhweU0h0AjcM8EPnLWWWfh1ltvRXx8PJ566inw+XynaVQG/7JYqU53BqLGxsb6zXR7qVK2PT09eO71d0GEpUCsiEJryygOHHsBt//oe25lNJYKNpuNX95xM377xDOYSVsPSUQs5oa7Yemvw12/f3DRHtdsNsNisVD/6JDvs7OGaneFh+zLszWh9ka4IiIioFQq0dbWhri4OJ9qAbhcLnJyclBTUwOdTudT9azZbAaHw0FmZiZlkO1tYYrt6x0XFwe1Wm3VN+iK5RbHFSN8cXFx6O/vp34eGBhw2uh51llnWf0sFosxMzOzqMfnLwLdU9QdfBUMcnSS7Sid5RyIulhYLBa8+eGnCMnagLCY+YxGZHwKhnvC8MGnn+O2m364vAdow7p16/DsnyOx+5M96BuswdnZKbj4F3/ym0CTIkf+12g0wmg0Ut6OLBbLyueRzC6QAmX73fHk++TIhNrbiC05ORkNDQ0YGRnxOVrj8/mQSqVUCtXb9UixovtveluY4uh1cTXNIdBYMcK3fft2PPPMM7jmmmtw/PhxhISEuP1lk0gkGB4eXuQj9B2yl2+ln8w92eNbaHRSSEgI4uLiwOfzl/yCYCkivsnJSYzN6ZAVk2D1++ikdJz4smpJjRfcJSUlBT+783af1qCnKm3TiaSwEQSB6Oho9Pf327UJ0CHfJ/LETl/LU+FSKBSIjY1FS0sL8vPz7fb4TCYTPvvsM7z/8WcwGI3YfuEW7Nixw64AjiAI5Ofn4+jRo1Shi7eQe9MhISFoampyWITjDvQojcfjobCwEPX19SgrK/MqkrTb6/3fc3Y0zSHQCBjhu/baa3Hw4EGMj48jPj4ev/vd72AwGAAAP/3pT7Ft2zbs2bMH6enpEIlEePnll91ee6VNYV/pwudoj4/0JF1oIOpqaxMhCAJwIK4WsxkgAiP6P3nyJN758BM0tXUgVCHHjq3n4rzzznP72BZKVZJRCD2KI0lLS0NdXR1OnTqFyMhIp49BfndsIz9vMigJCQmYnZ1FX1+flSibzWbc8bNf4FB9J8T554HgsvHws2/ivd2f4j8v/ctO/NhsNlJSUtDe3g6tVut1dTB5TkhMTERLS4vX5ti25xaJRIKsrCyqMMUf3zt632AgtzkEjPDt2uXYMoiEIAj84x//8GptkUi0YqawL3c1pD8wGo3QaDTo6+uj9uIAUHtxK2Ug6lJEfKGhoYgPl2Gk7ySiE0/3kg11tyE3Jd7tKQKLRWdnJ379+N8gzt6ImC1nQTM7hX++9xUGh0dx/XXfs7ot+dkloy/b95YUN9t0pSvIKKK6uhpisdhl9EvPNJAncW9SlQRBICcnB1VVVTAajdT9jxw5gkM1rYi/8tdgsedPnYqMUjTufhKfffYZduzYYbcWh8NBZGQklab0RlzIyQzkcXlrjm0ymezEOTw8HGq12i9jlkgcTXOwZbm/9wEjfIvJYlmW+RsWiwWTybTch+E2zgaiks+Dy+UiISEBYrGYMWF2wTWXfhfPvrILJ8eHwJeFQTt9ClLDFC698brlPjTseu9DiLM3ITarCADAF0kg2XwVPvr4eXxn6/lQKBTUxQG9Z45MOZInel/efy6Xi9zcXDQ1NbmMTOiRGSms3u6Zs9lsFBUV4fDhw9Dr9eByudj31UFw09ZRogcABIsFYdYZ+Hjvlw6Fz2w2QywWIyQkBM3NzV6lKeniTTfHJrcD3MVZAVBiYiJUKhW6urqcNvLTcedi0HaaQ6BVVAfW0SwSKy3VGWh4OhBVqVSir68voCoSvWGpqjpjYmJw/123oLGpCaNjE4jLz0V+fv6yNM7r9Xqo1WrIZDKwWCw0tnYg5btbrNKULA4PnNBYdHV1obS0lEpVLuZwUbIBu7W11WUPGr3Kk4wAvT0mgUAAgUCAxsZGrF27FjwuFzDaZ47MBj34EscmF+Tjk+lTb9oIbAWL3J9raGhAWVmZ2wYbrl6LrKws1NbW2jnPOMLdi4lAbnNYFcInkUgY4XMTZ5ZtngxEDVTnlkBGLBZj/bp1y/b4er0e/33zbXx24AgMZgIKiQDXXb4dMqkE6tlpyMKirItGNHMIDw+niiLa29ux9/MvoNPrce7Zm1BaWur3E11cXBxmZmYwMDAANpuNI0eOwGgyoay0FBkZGdTj0YtdfB3Fw+FwEBMTg5aWFlx80Xfw3w/vhLH4XHCE85GWyaCDtvUQLnvkPof3p4sNmT6VSCQeTYF3JFhSqZSavlBaWuqWuLuaocdisVBYWEhFkq56SD2ZxeeszWG5RXBVCB8T8dnjbCAqQRAOozhPCBbhW+4v52JCbxsAgL8/+y8c651F8tYfgyeUYnZsCM+88RHOyk3C0bqDkJ53Jdjs+QkjQydqESvjIiMjAwDw/L9fwDMv/Re8tHUgODy8vvs32LKhGH/546N+jwCzsrLw5FN/w8dffQNBUjHA5uLfb32C7567Dvfcdaed+PnDnisxMRHffPMNDn/9NcLEfDQ/ewfkhZvBFYdA3/kttm85C+edd57D+9Ln+dGnwJPm9u7g7DlERkZS/YK5ubkLrrOQYHG5XBQXF6O2ttalBZmnQ2gDsc1hVQifRCJZMXt8iyF8Sz0QNRAsy/zFShXwiYkJfL7vS3xx4GvkHjmGCzefg+zsbIdtA2NjYzha34rMi28D538naUV0PCzlF6K362ucX5iGfR8/B354IoyqKUQJgF/d9zMQBIH29nY889J/EX/Fr8ATz0cJ5rIL8OUHf8UXX3yBCy+80K/Pa3h4GJ8c/BahZ/8QEXGJYLEImAyb8cknz2Nd6TcoLS3F3Nwc9U+pVPpcrdjV1YVb7vgZtKHp4CdugljYg6naL3D59m340T+fcBnd2ooWn89Hfn6+yxl+trgSmpSUFDQ2NrolKmSRjCtEIhFyc3NRW1vrdDqE0Wj06DUNxDaHVSF85AT2QMdX4XNkvK1Wq5d8IGqwjFhaaREfGcUNDw/j/ocfgz4sE0TSGahXG/HNX5/Hzdd8F9suvNCubWB8fBw8eTQleiSKmCQ0H34Lf//LY7hsx3fR09MDmUyG7Oxs6r6ff7EP/LS1lOgB83uAkvxz8f4nn/ld+A4eOgR+cgkUkTGYm50FX8CHyWgCO6EIL/7nDepCju7FCvhmDHHvrx6CJW8rYku2wqDXg1t4DuZSinG8+h38/en/c7muo2gtJCSEanB3Z+/L2RR4YP4zmpeXRzmxhIWFuVzHHcFSKBRITExEY2MjiouL7Y7P04gPsG9zoNvCLQerQvjYbPaKOBF7Inwmk8kqgqMPRKWL3HIYbwdLqhMIzIhvobaBt97fDXN8CdJKNkKr1UKn1SIyKRMvv/kyzjvnHLtKwIiICBhmx2E2m8BinT6hTY30Qchl48SJE8jIyHBY9KA3GAC2/UnZDBZOdnXj9TfeROH/inW8zSbo9XoqgjvZ1Q2N3gy1Rg1YAJ1OD5FICJkiDHJuKIqLi+2PxWx22NzuDmq1GpVVVUi6+ccgCAIcLhcGgwHSlGIMH92FEydOICcnx+n9naUpY2NjMTc3h87OTipl7Azye+0MsgK1uroaJSUlTltgPN2bU6lUaG9vR1ZWltXfvO01pjfNr1+/fllbdVZVjXkgnsTosNlsu3YGcpTO2NgYuru70djYiIqKCtTW1mJ4eBgWiwVRUVEoLCzE2rVrUVhYiLS0NERFRUEsFi9L1BIswufpa2exWDAyMoLh4WG/pXrJk7bRaITBYIDBYIDJZILJZILFYgGbzQaHwwGPxwOPx4NAIMDxmnrEZs9PAxcKhSBYLJg5fLDkMThx4oTdY8TGxmJNVhK6vv0CJuO8aUR37df4/Pk/YEgN/PJP/8T1N9+B5uZmu/ues2kj9F1VMBv11O/m5uYwULEXUCTg8JAZf37lAzz5939ShhTOsFgsUKlUGBkZQUdHB2pqanDs2DE0NjZiYmICPB4PF2zZDM5EF2RSCeQKOQhi/jWaPVmLc85c63Bduq2Z15/L/30WWCwW2CwWDEYD4EZK39UeY2ZmJmZnZzEyMuL1GiQCgQD5+fmoq6tzOiXDU8HKyMiAWq3G4OCg1e89TXXSkUgkyMnJgV6vX/jGi8iqiPhWUspKpVJhcHDQLoqjT/0O9IGogVCd6i8WOlEODg6iuqYWwyPDaG7vgo4jBkGwECpk48ZrL0d6erpbj0OP4hy9dnSHk4XaBvg8PowGPciSJJlMhomJCRi0Gqd7SvfceRv++a+XcHT3P6AzWdDW0or8S25FRn4xAAKj3W24+4GH8Ydf/QJr1qyhCp7WrFmDC88qxefvPwFx/rkAi43+43uhELCw8aqfgsPjw5KzBnWHPsDRo0dx9tlnAzg9/Je+F0cajkulUsjlciQkJNhZ1cXExODskgM49Om/ocg7CwSbh566Q8gKseCC8893+poQBEFdVHpy0haJRCgpLsbJxoMIK97yv/tzMNtTCyFhXNCQ2ZVokdMXKisrIRaLnbqcuCtYISEhSElJQX19PdasWeMwRelJ1E0QBAoKCqi9OTI96U2qk05oaOiibrW4w6oQPiCwoj1nA1FJZwWBQICoqCikpaUFXOOnOwRTxOfqeXz86R7s+mQ/WJHpaO/ug2l2DNm5USjdvB2TI/342wuv43f33mm377KQhRdd3Lzpjdt67pn4sPow0jfuoBrKLeopGCYH7NJWJGKxGPf+/E5MT0/jH8/9C4LEQiTnl2BufBiNxw9h5GQLlCNduOzGO6EQ8/GT66/F7T+9GVwuF48/+nts3rcP73/8Gfr7BiBIjMWm790JDm++1cFssUCakIvdn++HXC6HRqMBi8WCVCqFVCpFbGwspFKpWydTgiDwmwfux5lffYVP9x2E3mDA9ovXI2WB6mPydaQ3t7vLXx57BJdceS1OjfeBG5cNw3gfDCe+xn0/ux2zs7NQKBRO77uQaHG53AV78jwRrJiYGCiVSocpSsDzIIDD4aC4uNgqjeoPW8XlvnBfeWdVL2Gz2TAajUt+pUFe2bozEPXUqVPQarWIj49f0mP0N8EifK7o6+vDrk8PIO2C63FqYgoKQQIkinB0f/0O4rrbEJOag9nRdBz7tgLfufACh1EcKWzk3pOnAmexWNDb24vh4WEIBAKkp6dDKpXiyssuRXPrn3Dis5fAjUqHWTUF9mQP7rvzFpw8eRJ5eXlO15TL5dAYTBApotCw7220fHsARHgKjEYLLCwuZKUXQRibgRf37IHFYsHP77oDLBYLW7duxfnnn49PP/0Ubxw9CaVKA/30zPwoHDYbRtP8/iHptevLiY/NZuP888/H+bQIb2BggCrrd7e5faFjIAtisrKycOjLvXj9jV2ob2pFxtpkXPe3jxAVFYWamhqXpf/upCklEonLnjxPWzLS09NRV1eHwcFBpxNsPIGeRi0vL/cp1RkorBrhI3v5Fmu4p7MozpOBqMGSIlzuqzl/4ep5VFXXgB+XA55ABI1mGCweHwSbC1FyEbpbGxCVnAmBLBRDo2OUw78vUZwtWq0Wz730Kqpbe9B64gRODfaBTViwacNa/Ob+X+CPjzyEhoYGtHd2QhGShvXr74BUKkVTUxOGh4dduurkZ6bj6w8OoLd/AIptP4dRqwJbFgnD9DAmv/o3IrJKgbwteOfj97Dl3LNBEATVLiSVSqEd6QCn6AzIwsLBYs/vr020HMNVW85atEkTZHP74OCgywtHV5McbKE3v0dGRuKen91td5vs7GxqjJGj99Rd0YqMjMTc3JzDYhlPIywyRUmmUOVyudv3dUZISAhSU1NRX1+P0NDQZXEV8ierRvhEIhFUKpVfhM/ZQFRHUZwnIhAswhdMOIpczWYzdHo9wOLAZDJBLBJieHYKkCnAYnNg+t/JTj05grSyFJ+Ghzrj4z170TxmQENzC5BUjtTz74JBp0FDZwVuuftevPHScygpKUFJSYnV/Uj3kJCQEKdVdVsv2II//OUp8PMuAsHhwWJRAgC4IVEQJJVgvKUCosQcmPlSTE5OYs2aNVZ+rBNTM3j/qw8hSS4Ehy/AbF8b0hVsbNp4lsPH8wcEQSA7OxvV1dWQSqUuv+fOJjnY4o5ohYWFYW5ujrJSs8WTNCUpLAMDA1bi7eneHHA6RVlTU2M1WNcXoqOjoVKpMDo66tV0CDrLfXEcUFWdN954IyIjI5Gfn+/w7wcPHkRISAiKi4tRXFyMRx55xO21vXFvIU2YR0dHcfLkSdTX16OiogKNjY0YGxsDm81GXFwcSkpKsHbtWhQUFCAlJYVqBvf0zWWEL/Agqyr1er1VVWV+bg60g22AxYzIyEjwTDoopyag7G9GQkoGhjqbIVAOYcN6/9uQmc1mHDxWBb3JAqM0BmFF54HF5YEvCYEgqQi6iGx8+NHHDu/L4XCQk5ODpqYmu8+ayWTC9PQ0VCoVkhITwOdyYZgZhdmgA0xGcHh8sEUyEEYt2CYD+BYdCgoKIJVKrU7Ml12yHb+5/XqsDTcgmzOGWy/ZhAd+cfeiXADQYbPZyM/PR2tr64JVg6TJgqvvm7vRWlJSEoxGIwYGBrxegzym/Px89Pf3Y3p62qs16AiFQmqwrrNKT09JTU2FyWTCxMSEX9ZbLgIq4vvhD3+IO+64A9dff73T22zcuBGffPKJx2svNKHBURS31ANRGeFbWiwWCw4fPowP9+zDxPQMCnMyccUlF1P7IiwWCyqVCnK53G5mXFFREc4ursLXB96BPK0EcSE8tH77MYiJPqgVFmRkp+GqO25elHlkZrMZOoMRU6pT4EamWv+RxYIoJg3NJzqd3l8mkyE0NBQNDQ0ICQnB3NwcZXRAZiy+s/ls7Dp2EkkFa9HTNwiDZgYWDgfavgaIstbC3HUMWzdtcNiITEZfC1U8LgYikQhpaWloampy2RxOn+TgLOpzV3BIwaqoqIBEIrFKLXoqWvRIjdw79KWYRKFQID4+Ho2NjV7d3xaCIKBQKDA9PY3x8XGPPEfpayw3ASV8mzZtQk9Pz6KsLRKJoFarqWZYUuhUKhV0Oh04HA61FxcXFwexWLzkG7iM8C0+9LaBV1/7L945UIXIonMhygjFt72tOPLrR/Dk7x9EamoqkpKSUFNTg5iYGIepolt/ciPWVlfj0LFKGGDED+/8HkpKSsDhcHzeA7FYLKisrMTeLw9Co9Ph7A3lOO+888Dj8cDhcJCVkojxtkEYp4dPPzeTEYTZBMP0KSRlxVLPV6VSUS0Dc3Nz0Ov14PP5UKvVlFDYFpz84Lrv493d30Pr5zMgojJh0GoweeIbWKb6wOdqcOWOi3D3HbfaHffw8DA+2P0xmttPIiUhFpftuBipqal2t1uIzs5O7HrrHfQODGFtaRGuvPxyl9WTdCIiIjAzM7PgmB16sYuj/T7b6euuIJvIa2trrSaaexOtCYVC5OTkUNPRvUl10omPj8fs7CxmZ2e9XoOO2WxGdnY22traqFarlUZACZ87HDt2DEVFRYiNjcUTTzzhskJtbGwMDQ0NaGhowNGjR3HgwAGwWCz8/Oc/x5lnnhlwA1FX2jy+QMdV28DU1BTe++wrZGy/A1yhCAQISMIiMcAVYNe7H+KhB+4Fm81GdnY2mpubHfoxslgslJeXo7y83K/HbbFY8Myzz+HDQ9WQZqwHm8dH5a7P8cnn+/HEH38PgUCAK7Z/B60nn4OmqwYzMZkQxmTAoJmDUDcJ9YkjKLj0Zzh+/DgsFovVVI3k5GSqZF6v16O6uhqJiYl2z21qago6oxkCthbqE4fAZrERFiJBZloJXvnXPx2KUHt7O27/5a+BhGJIY0vQ2TeIj++8D48/+Ats2LDB7ef/+eef4677HgQ/+2xwFck4/uFR/OvF/+CDt/+LhIQEt9ZIS0tDbW0txsbGXHpD0lOetpGfp5MdRCIRVexSVlbm0zzA0NBQaiqEr8IHzHt6joyMuDV2aCFMJhP4fL5Xo5EChRUlfGvWrEFvby8kEgn27NmDSy65BB0dHQ5v+8gjj+Drr79GYWEhioqKsG7dOmzcuBGXXXbZEh+1+wTLBPalhi5uztoGbCd/9/b2QhCZDJ7IusowMj0fVZ/8g/pZLpdDLpejp6cHQqEQJpMJsbGxXrm6NDc3Y3p6Gunp6S5PPh0dHfjgy6PIuOQusLl8mC1msOSxOLDnZfz4lttw8403YM2aNbjrxmshFxD45POXMcURQiyRgM8Hfv2zn2L9+vULZi14PB6ysrLQ3Nxs1/D8+q63Icw9Bxllp1sGpgc6Uf/WE9h+5fdRUlSAG39wrZVF2BNPPwt+wVZE55QBAEKTsjETk4rHnvw7PnxzLWZnZ/Hk//0dH+35HEajAeds2oj7fn4XkpKSqDV0Oh1+8cBvEHbRzyCO/l+kmHsGRr/djd//8U/41z+fcev1pk9ul0gkLos7HE1uB7yL1sLCwjA7O2s1McHbi2pyhp9er/f5wtxisSAkJARdXV0QiUSQyWRer2UymaisRkZGBjVs1pO9zOVmRQkf/c3atm0bbrvtNqd55oceesjq58HBwWW3yVkIJtW5MPQozvaK3JPmb5FIBKNmzu73evUcpBJrMWSxWLj3wYcxodSDw+UhUibEL++8xa5i0hlDQ0O478HfYXBGB64sHNpT3di+ZRN+ftcdDoWpsqoKvISCedEzm3GioxOzWiPYKetQc+Igfvevt1EYswe333ITdv7yHvz+t7/B8PAw1Go1enp7cay6Hk1tHbjgvLMXnIsXGhqKqakpu7RgQ8sJhORfTP081lGH+k9ehiBnC4xJGajVTOFHd96LJx7eic2bN0OpVKK5/SSyrrOeGh8Sm4IxEwft7e345QO/wQg/HjFXPAQWl4/KxsO46gc34uN3d0Eul+PQoUP4+uuvoeOFwCyOwPDwCExmM3g8LmS5m/Dla/d6FEHxeDzk5uaisbGRGpjrCPp+H3Ba/LwtKklOTqaqM32BIAjk5uZi//79mJiYcGlAvRAmkwlcLhfZ2dmUUHlbbER3bomIiIBKpUJzczPy8/MDQtTcIaCqOhdiZGSESllVVFTAbDa7/WFYqLglEGCEzxqyotJgMFBVlfThohwOx8qnks/ng8fjgcvlUsUozsjLy4OcpcOprhba45kwWL0fO76zhfqdUqnELx98BIbUjYje+lOkX3YPzDkX4IE/PIH+/v4Fn4PFYsF9D/4OUxFFSL/0Z0je8gNkXHk/Pqk4gXffe9/qtjqdDuPj45icnIRGOYepqSkMDA5gRmOAKDwWXB4f8ogYlF3+U/TMzJuUR0ZGQiwWIyYmBs+++Cqe31OBPlEW6rThePBvL+GFl19Z8BhTU1MxPT2NyclJ6nfxsdFQjQ/NPwezGa1fvoWQs66DMKUE/NAYRBVuROSWm/CHv/yNVnxhhsWB16zFaEBFRQWGtBzEnn0teFIFOAIRossvhDGmEM8+9y98Z8fl+NX/vYr3q3owOTWFqVklLHwJOBIFjAQPY+MTXlUmhoSEICYmBm1tbS5vRza00/08Pdnjs12LrM70deuCxWJBKBSira3Np5mipFiJxWJkZWWhrq7O63ONrWVZUlISWCwWuru7vT6+pSaghO/aa6/Fhg0bcOLECcTHx+PFF1/Ec889h+eeew4A8O677yI/Px9FRUW466678Oabb7p9hbESprCvVuEj91iMRqNd2wB972VkZASTk5NWAsfhcKyqLd2FzWbjkV/fC3Prl+j84jV0HfkY7R88gzPSwnDpju3U7Q4fPgyNJA7xeesgFAqgVqsRmpABbkoZPvp0j8O1jUYj6uvr8cobb+OPTzyFzpFpROdvoD6rLA4PkaUX4j9vvYf29nZUV1fj2LFjaG5uxtTUFM7YsAHEWDuEPBYMRjP4UjnYANQnK5CStwYsFhuixDxU1NRRj/nZ3s/RZ5Ai47yrEZGSg9icUqR/50a8t/cw+vr6XL4W5GibEydOUFmR666+HHMNX0I9OQrV1BhUc7OwyKJh1GkxNatE/+AQRFHJmNWZ0dfXB6FQiDPL12Co/rDV2mMddYgLk2JwZBTsWHtXFXFSPl5/5wNoks9Ewva7Eb/xchimhmGcHQcIFkCwwOIJoOr4FmKJd6br8fHxsFgsdmbLjl4HAJQBuC/T2zkcDoqKiqDRaKDT6bxag4TFYlEz/LxtS6DvE4aHhyM6OhrNzc1eOSzZRt0EQSAnJwcTExMYHR1d8P6BEBUGVKpz165dLv9+xx134I477vBq7ZUwhT3YrL4cpaUW8qm0bRtgsViorq7G31/4D2b0gMVoQFpsOH5++80+W7ulpqbi1eefQW1tLaanp5GRcbVdBWL/4DA4ofMVkkKhEDMzMzAY9BBHJOBkb73dmiaTCW+++yFax9QIi8/AkH4CM2Y+BgYGEaqQn44A+GJMTk0jNDQUSUlJdmmn235wFZ5/41nMiBKgIgQwnzqJ6KhIxOfN75+ZDXrwuWKMjIzgyDfH8Nwr/4VozQ6r9ByHJwA3Lge1tbULDiklLc+am5tRXFyMNWvW4IE7foS//uNZDE5pYZgbB6GbgyQiFmwOFzrVLE6NjMCk01D7Z7+463bc+rN70fV5P7gRKTBOD4E71YPf/flRfHP0KCxVVXaPqx4fhFqjQVTRuQAA1UgvhLFZmPz8GYgyN4AXFg9dfxOMp7qgh3dbFeSJmWxud7W/RW9u97WoRCQSQSAQWBW7eEtISAiSkpKczshbCNuWiMTERLS0tKCnp8erZnRHhV7FxcWorKyEUCj0aQ9xKQioiG8xIdsZAplAuBLyFwRBwGg0Omz+pu+lcLlcypibjOToUVxPTw/+8PS/ISjahqyLb0HWJbdjLCQbv/79n3weLmyxWFBbW4v3Pvkcb3zwKd754GN0dXVZ3SY5MR6GCXKvhoBUKp1vCxjpRmZqktVaGo0GFRUV+PZEP8QxGVDqjJBGJsAyNwalWguzxYyQkBDI5XLoxvpQWlyE8PBwh3stV115OV548lHsyA2BsO9brN9yEdZf9mOw2BzotWpo+hoQIhXjrl//AR/UD2Ncz0Xv8Ck0tbRaRwUmg9v+tOT8RjJC3H7xxdj10nOIkvERl1EI02AL2BweAAI8kRTDtV8hP2t+BBYAREVF4Y2X/4X7r9uG7RlC3LXjTLz96gvIyMjARdu2wTxQj9n+02ORNJMjUDXugzwsEsT/RIErCQEBC6Iv/zVgNoB1qh0RKdlI334bIiKj3HoejiCb21taWhYcj0SmPP1RTcnhcBAeHu5wHJQ70C8OY2NjIRQKcfLkSY/XsU1PkhcDY2NjGBsb8+rYbOFyuSgqKkJjY2PAD/4OqIhvMRGLxdBoNMt9GEGJoyiOzWZDrVZTlYVkNAfAo5PJJ3v3QZhWDnn0fBk7QRCIzS5Gx1AHjh8/To258Ybdn3yCF977AhGF50KaFoH6/k4cf/hxPPare6jm640bN+KF19/CQP0RxOavB8FiQzXcjbnWI1j7g9+ira0Nc3Nz1PioxtYTUCRkIDIqElwuFwQI9K8/EzWVH0Fcfj7EqVmY6G7GTO0e/OGxh1weX0pKCh781QNISEjEB/uPQD0zAZjN0A+24qrvnI139+xH7MYrIQ2LAsHho662BuqoFAwODSEpMRGa2SmYRk6gvPxmt1+TjIwMVFVVQS6XIyQkBGazGbKIGOSe+30ceuVPGB0+AW5ECgzjvTAMtuD+D960ur9AIMDWrVux1WbdiIgIPPe3v+Cue3+FWb4cLC4Pxol+PHzv3Xj6uRegHO2FJCoJIQnZYFuM0PY3I6x8O2Kio2Ay6DD0ydO48/tXu/08HCESiZCamoqmpqYFoya9Xo/Z2VmfpqMMDQ3h5MmTKCoqQkdHh1em0bYFNllZWaiursbo6Ch1weHuOrbFPWSURo4d8kc/nlgspgpoysvLA9bMelUJX6BHfIHOQjPj6G0DaWlpOHny5IJVhQvRNzQCSXSp3e858iiM+nClqlKp8J+3diP1wpsgEM+nZUQhYRgRiPDyG2/jT4/MixKHw8HDO3+Bp597AXWv7IHZAsSGy3HLD64Cj8ejxkeRUdXI2DjGRozgcU/3NW25/AcwG1/AYOU76KoiUJCbg5sfewgFBQULHidBEPjh9dfh7I1norq2FiyCQHnZpejp6cGsiYN40fzJKqlwHYa7T2C4ajfmZJEwD0VD39+Iu3/8fZd9bLaQ+0lkf1Z0dDRELCNMBj223vEYRtrrMDc2BMSGQZEa6rKP1pZ169bhyP69qKmpwdTUFEZGR9HS3ol1JQXY99k/IVvzXUiikxBdsAG9B14Dt+sIRiMToO5vwcUXbsHNN/3Y7cdyRmRkJGZmZtDd3Y3U1FRq+C05F5De4C+VShEREYFTp05hZmYG8fHxbnlejo+P44Yf34yj33wDrkSOn/3iXvzinp/jnE0bKRcod7EVPoIgUFRUhMrKSmp2oTs4i155PB4KCgrc7sdzZysmLCwMGo0GjY2NKCoqsvv+B0Jma1UJX6BXdQYSvs6MUygUkMlkGBgYcLvp2BGZyQn4sqcXobHJVr83Tg4iPrbQ63W7u7tBSMMp0YPFAqPRCFlMCioOvY3KykoYjUbweDxIJBLceN016OrqglQqxVlnzU8ZIK+U6anEgtwcfF23G4akdHD/N4/OZDQgJS4aT9x/K2JjYz0+Vp1Oh/aODtQ2nYDFYsGBQ0dQd6ILwyoLTj7/R6TmlaDovB3YcOmPMNBag5P7/ovvXb4OG3/xPa8eTyQSISkpCS0tLSgoKMCtP7oOf/7X6wgrPh+h8elgsTmYbvgSd9z9E49TgVwuF1FRUXjosSegD0uDIDIFulkNBGwgabYJ00PHcU5GGn649yOo1WqMj4+joKBgwT1KdzCZTJibmwOfz0dPTw9GRkbAYrEoAbHdb52ZmcHDjz6O43Ut4EnkgHYGN33/Knzv2mtcnrwvueJqdFkiEH7zS2Bx+TBMj+Bv/3oMoQo5LBYLysvL3W74djT01Z0Zfo7WcZbylkqlLsci0XG3xSM+Ph5KpRKdnZ3IyMhY8PZLzaoRvpVQ1blckCJH/pf+paYLG73gxB3S0tJQWVmJ8PBwr93ht114AT7/1SMYlUcgMiUHZpMR/Q3HEMHWoKyszKs1jUbjvJXX9CSmp6dhMhphAcBhs2HSayCViFFQUAA+f76P7oVX/oOjjV3gRKYAhil8sO+PuPNH1yIrKwvHjx9HWFgYBAIBkpOTERcXh61nFOGLI3vBCZsvvjFODODCjWu8EiG9Xo/fP/4ETkwDEZklaDryGYYnlEhaexUidAbw5FHord4Dztd7UXDOd2FQzeCH112Dq6+60qvXhiQmJgaTk5MYGhrCls3nQSEPwRvv7UZv636kJsbjV7/+OYqKirxa+09P/R3I2ITkwtPTGsbjMqBr/xKfvud+pbYrSGvC2dlZh16kubm5aG9vR1FRkdPP5oO/exQntDJkXP0A2BwutLOTePbt16CQy7Ft23cc3qeurg4nOruh+NF9IFjzgsWVR0N49k/w1NP/xJGDX7olMCTOhIac4dfQ0IA1a9YsuBaZindGZGQklEqlVeO9s3XcTV9mZWWhtrYWQ0NDXn32F5NVI3wrKdXprc3RQpDpSbJqzdvmb3dhs9nIyspCa2urS8NgV8TGxuLRX92D515+Da3Ve0HAgvXFefjJXTvdSstotVorn0qNRkP1M8WG8DHbfwIJ+WvnK2rNZnQePoBLt11AnSSOHTuGo+2nkHXB98H63xd+bnIMz7y8C9/dshH7jtZCEBYLiYALseVL/OCKHdh05pnIy86mihDS0jZ63Xz87bffon3KhKzzroRBq8bE6AgStv4UyslTiIkIxfDYKYgy16Pl0GvgmLWIZilx+SU7vXosW7Kzs6kRRqWlpSgttU85e8rc3Bxqm1qRdd01Vr8PS8lDZ+XH6O3tRXJystvrWSwWqNVq6v2dnZ2FTqcDj8ejJryTRTu2n2cOh+O0ub2rqwuNJweQceV9ADH/OAJZKKI37MArb77rVPi6urogjEmjRI+EF5OBwb5uREREYHZ21uHcPUe4MqiOjIzE7Ows2tvbFzQEdydSS0lJQWNjI/r6+pxG2J4IH0EQKCwspCo9SZs7JtW5hKwU4SPLqX3dFKZHcbaQzd90cfNV5JyhUCgwOjqKoaEhr6dBZ2Zm4sk//h5zc3Ngs9kO58iZzWZqsga5V2MwGCAQCKgTYHR0tNW4qEd/+ys8+Ic/oWPoBDjSMOjHelGSHo9rr7qCWverbyoQmV1KiR4AzPx/e+cdHlWZ/fHPzKQnM+m9d1KAkEIHARVFwYoFCyqWLXZdu6joqqui666V3bXs2suqiLIIghUR0knvIb33PuX+/sjvjjPplUzI/TwPjxIyd97JTO73Pec953vqq/n5WAqHvvuewIgFBNu6ErFkDR1N9fznk93c/ccbcHZ2npTThkhiWiYq//7+t56ONuSWdigsbVDY9KdoF0aGU9fQQI9C4PKVEZyzYQM2NjaUlJSw538HKDxRgZe7CxvXrxt23NdwKBQKIiMjycrKIj4+fvoLFWTDnyEVFBSwb98+ent7SUhIwN3dXV9UZG1trZ/B5+PjM+YJKg4ODnh6epKXlzcoyqmpqcHCyVN/HUHWn/K3c/WmoKp6qMsB/ZuFzopcrLQaZIrfbq+9FdkEhIQB/W00aWlpY4qERqssDQ4OHtO1xiJYYj9nUlIStra2Q35+xyN8MD1zAaeCOSN8lpaWk24kPRmMV/gMC060Wq2+F1DfLD3FUdxECAkJISkpSZ8SnCjiQb6YxhIjOfHs1tbWFktLS/r6+ggICBh1ZIqnpye7/raTjIwMmpqa8Pe/kODgYKObZk9vH2YWv7UbFKf/yo//+xzC1uLuE0RvXyfff/MF1cV5nH3ZtTTJ+0Vnqs41rK0s0TT1l4Zb2zui6+1A092BoNOiUFhgY2uDc7c5HSprggIDMTMzIycnh6deegPb0MU4LVpARWMdT732Lr+/7FxWr141rudXKpV4e3sPKQ7QL0jvf/I5Gbn5ODrYc+GGMzn7rPXDfs6USiUxUfMoyT6G1/zl+q83ncjBXWWtj/bUarX+PX751df4ePdeLEKWIpMpeOWt97hq8wU88vCDY27VGA4fHx8yMzMHCYefnx+99WXotBrkCjNkyBBkAs0VBQQHBgx7vcjISBLiYkn/9hVsV1+HwlpFb00h3d/tYufzTwPGE9Lt7OxG7HkbLVIbOG19uMKZsQqWOGUiOTmZRYsWDdpkjlf4oL//NSoqSl/pOdn3bCqYM8I30qRlU2Ik95ahCk5EkRPbBSbaNjCdmJmZERoaSm5u7pBVXsMhprEMRa6np0dfcKJUKvH398fWtt/R44vdX/Luf/egM7dF093GmqVx/OHGbcNOGhfXNpLnZsLCSL4+noO9iweCTsexA1/itOJyOrq6sbC1p10rwzJmEyk//ofmjl783B3oXh0z3h/RsJy2YimH/vYm6pAozC2tCVu0lPzDn2DltwDnwMWUJP1A8lf/xsM/hAdeeAOL3lacHFQ4JWzC1bffe9PG3gmlsxv/+fS/LFu2dNw3Hm9vb5qamgaV0Ofn53Pfn3diG7Eanw3r6G5t4rXPD1JRVc1N11837PXuu/NWfnf7PZQ2VWLlHkh3fQWa0mQeuOMPpKen093djbm5OUqlkpKSEj7f9x2+Vz+Lxf8XImmWX8iHnzzGOWevH9fUh6EQ/TCTkpL0mQHoF8RV8Qs48t1H+CzbhIWNkraaEzQc/ZK777tlxOOITz54l1vvuIvP/3UTcjMLbKyteGbHdi699FL995iZmbFgwQK9GAyXth/LJlh0iRk4Emm81xGxsrIiOjqatLQ0Fi9ebNTSMRHhg/7oWvQwXbp06bgfP9XMGeETma7zs6lCLpej0WgwNzcfMVUpOpzMVBQ3XpydnfVjUTw9PQf9u0ajMUpTdnR0oNPpsLGx0Q/39PX1HTaN9c3+A7yx+zsCztiGtdIBTV8vh499Q89Lr/LQfX+a8LrPWLeWXxL/SsGv32Lt5EZHZwdWMgUOdja0NDVi5eSJpbMXHQ4e2EWspCB1P83NzfrHt7W10dzcjIeHx4RMgSMjI7n4jKX8939vYe4RiqWZHPvuahQFNRSc+IGi0jIiL7yFoMgYZEBzVSk//OdpLl29xeg6tvZOVCpsqKqqMpqGMBYMnU9UKpU+XfXux59hG7kGr/AYACysbLA9/XK++vIVLjxv46A2CnE2oKWlJX9++F6+OXCA4hOp+Pl4cM4N2wkMDESpVGJlZaV/j99+5z2sItfpRQ/AzNoOy4h1fPLZF5MWPvituV087xM3Bg/ffw+v//NNvtz9V9QCuDmqeOT2G1i2bNmIQqJUKnn7jX/yxGOPUFRUxKpVq4b8XltbW8LCwkYsdhlrE704EkkU0oGPGW8zvr29PYGBgaSnpxtN7pio8EF/hqWnp4eOjo4Zd3aZM8JnqmI3MIqTyWT6w/npKDiZScLCwvTpHbHoRCw4May48/Lyws7OblzNwx9+/hVeS87FWukAgJmFJYHLzuHXL16murp6SLEdC0qlkkfvv4vvfviRoynHMe9rx9fVHh0yWisb0AGCug9tdzvotLiELCA5I4eEhAR2vvh3vvnuZxTWSuR9nWy76lKuvvLKcX0WZTIZWy69hFXLl3H8+HHkcjmL7rgCFxcXXn39H+zNbycwMkb//faeflgFxlKQ/CNxZ/1W2SnodGj7uiacajY3NyciIoLMzEz9TTojJ5+ATeuNvs/MwhILF18KCgowMzMz2shA/81eLJ9ftGjRqNFnZ1cXcsvBN0mFlS0dXcOftY0XW1tbgoKCyMrK0mclrKysuOPWP/LH391AV1cX9vb2+qOEocYYDcTa2hpPT88Rv0csdhmuQGU8EyKcnZ1pb28nOzubqKgoo8/ZRATL09OTjo4O8vPzCQ8Pn/B1DAkMDDSJ2X1zRvgAI3GZCUYqOBGFzcfHhxMnTky4CtKUMJz+Ld78tFotKSkpeHt7o1QqcXd3HzT9eyLPU1NfT5SrcfGMXGGGhYMbdXV1ExY+6C8d33TuOWw69xzMFfBz9mFUkauwtbUFmUBDyn6cXN1ZMC8YdacLJwoP88TTz3L4RCeBlz6EmYUVPW1NvP7Jv7GxsWHzBGZCent709raSmlpKSdOnMDZ2Znmtg6sHdyMX7NMjrOnHxX5acSu36z/uZbnpBDu6zEut4+B2Nvb4+rqSlFREaGhoTjaq+hubUTu7IFGrUat1tCn7qOxqoyqqiqcnJxQKpX4+vqOOhtwOM46fS37n3oZYcFafaWkoNPRm/8LG+7aNuHXMhRubm60tLQM8q8Up3+IGI4xGukIZayiJRa7DLVBG6/Q+Pv7k5mZSXl5uVFl5kRHLIWEhJCWlqZ3nZms8JkKc0r4xAKX6a4smkzbgLOzM7W1tdTW1k56UvLJxLAYob293ajgRKlU4uLiot/tZWRkoFKpcHNzG+WqY0Mul+Pr5UlrTRkOnr+l8bQaDX3NQ6dWR2KkzdGdt/yR+seeIGX/Lhq1Nig0XbjbKznrxjuxtlNRmJSHl70New4eZt6VjyI3649orFROeK28hDfffZeLL7xwXELf2dnJQzv+zPHiaizdg9C01uKs+Adnrl7OL0cLYF6s0ffb9DYR5G5L7jfvYOHohaa9AXcrHX+8c2IG7yLiGLDMzEy6urqIDgvky++/IGjNZiytbTA3M6fpRDbhXg5cfPHFU7JxO+uss3jrnffJ/PIFbOefgUwmpyPjW8LdrDn33HMnff2BhISEkJqaikqlGrEqV4z8RCEY6rWOVWzEApVjx45ha2trlAYcr2CJlZlisYv4GiYqWAOLZ6ZC+ExhQz+nhE80qp5K4Rut+XsibQMhISEkJyfj4uIyKa/A6UA0YzYUuZ6eHn0xgmHByXCvVfQbdHR0nLIKr6s2n88Tr76DLHAp3ZgjF7TIqjLYtGThmAW2sLCQN9/5gJSMLGxtbNi0fi1bLrvU6GxOpVLx0vPPkpubyz/fepvydghbcQ46rYbitF9QdlWyYs1y9hzJhgE3CDs3H8oam1Cr1eNK9/zzzbfJbjUn7KI79Z+vqswj/HwsGWVXFycSv8VrwQoQBCqPH8ZJ08Brr/6N6upqqqurcXJyIjIyclw30OHOXG1tbXF1daW6uppt116DpfUnfPPT+1i4+KHtbMHNGh594E9TdnMzMzPjvX+/yYcffcR/v9yLTqfjoq0buWLLlmlJmYmWbSkpKSxatGjE1LDhJIehIr/xiJZYoCJOchBf20Ram+Ryub4yU2whmGjEJ65NbElwdXWdlKenKYgegGwU77VTZ0YOcO655/Lss89OyP5oqLYBQ6b6LK6yspL29vZRG1OnE61WO+jmZ9g3pVQqsbOzMypGGCu1tbXU19ePu7dsOEpLS7n2D7dT1a5BYaVE29eJ0gxuueZSrrvm6lEff+LECW574DGsI9fhHrKAvq52KpIPEONmweOPPDjk69NqtRw+fJjvjiTR1d1D/IJIzlh7Gmq1mo2XXYv3BXejcnBCfGRbzQnUiR/z5SfvjfnnpdVqOevCy/A//04sbH7zZRQEgfxPd/LcQ3ew79vvOPTTLwCcvmo5N267Zlz+nL29vfr3uKWlheLiYuRyOaGhoTg4OOjfZ8MbcGNjI6WlpcTGxlJXV0dxcTH29vbMmzdvVp9Di7S0tFBQUDCqw4p4fCKXywcJVEVFBVqtdlzFRHV1dZSVlek9bouLi/VnhRN5DTk5/efNx44dY/ny5aM/aASam5tJT08nPDx8wkcHMpnsZJ7xDftLZlrhxDQz1ib2icyMm2q8vLxITk6mra1t2iugBEGgt7fXSOQGWjx5enqOu+BkJNzc3KipqaGhoWHUfrux8MGnn+O94kIWR8bR29mOhZUNOq2W3V+9zvmbzsXJyWnEx3/y+ZeYhyzD6//ThtYqR0JO20zyl69QUFBAWFjYoMcoFApWr17NypUr6ejowMbGRv/z2bBuJft++hTF6ouxs3emq6mGmp8/4b4bR/Z5HIhGo6FPrcHcytbo6zKZDHOb/grIh+77Ew/eK+i/PhwD20Pa29vp7e3VGzJXVFTwz3c+oktmCciwNzvIn2753ZBzD52dnWlubtabPU/m7NAUcXBwwN3dfVSHFcPzvoEb3olEa4ZuLOHh4ZOK1BwcHPDz8yMzM3NK5nw6OjpiZ2dHWVkZHh4eJhO9TYQ5LXyjTRuYybYBmUzGvHnzyM7OJiEhYco+ZDqdbtDNz9CNXqlU4ubmNumCk9EQX19KSgoODg6TFtSU49n4nvM75HKFvrJTYQ4WLn4UFhYSExNDamoqAIsWLRq06zyeW4Bz7IXGa5TLsXALpLi4eEjhEwSBbw8d4r1PdtPa1YelmYwLN5zO5osu5MF778b6pVf4+KMnsVI6YGMm445rr+S8TZvG9bosLS2JCA2ioSQL1+Dfpjl0tzZBR6N+cO7A92q4aF00ZBZvimL1cG1tLX9/832cl1yI9/8bgjdVlvDoc3/n9Z1PDpkuDgoKIiUlBUdHR70d1amEr68vGRkZo1YFG5b6iz210P+7NpHPdXBwMKmpqVRXV0/6TM3b21vfUjMV2NjY0NfXR1FRESEhIVNyzZlgTgkf9KdoBg6iNNW2ATs7OxwdHSc84UAsOBFvgB0dHQiCoC84cXZ2JiAgYMbKiy0tLfH396egoGBMvoUjobSzo7ezDQsr42Z1TXc76enp3Pv4TnosVAiCgLWmne13/I7TTz9d/30eLs5UtzZg62R8g9d1NA17U//20CFefv8r/FZchJezO93tLbz77R6KS0q5/tqt3P+nu7jmyi0cP36cdevWTfg88+YbruWuR56mr6sDR99QOhqqaEr7lluv3YK1tbWRk41YWCSXy/WFEmOJ1g9+9z0K7ygcDaZgOHkH0loRyaHvfuDyywxaIwSBlJQUKioq9MYE8fHxJuHIMZWIze3Jycn6zMdI3zuw2GWi0ZrocSkWu0z2XhQeHk5FRQV1dXWTLijTarUEBgZSUFBATU3NuAvwTCVKnFPC19vby759+zj99NNNSuBGIigoiGPHjuHm5jZsA7RYcGK4w+/u7sbMzEwfxU2mpHw0WltbKSoqQqPV4u/nN660l6enJ7W1tTQ1NY2ajhyJjetP4z8Hvids3Wbk/++RWFucjVl3I//8bzHuq7fg59Vfot5UUchDz71GQEAAwcH97iYXbTyLR196mx4saGyoRyYI2JprUWpbiY2NHfR8giDwwX+/wm/FeSid3dHpdFTUNtKkCubtLz4io7iKJQvCufHaqwkODqaiosKoRH48LFiwgFef3cG7H35Cxk+/4OzowOWXn0ugvz9HjhwxcrIJCAiY0M2yqqYeK4fB75ulgxtVdb/NPayuruaCSy6norYBS1d/OsuyWbliGffdfeeUZiZMBTMzsyGb24fCsMdPLpdPaoK76Oxy9OjRCXvcGq7L2tqawsJCvSHERNFqtfpCHHEm4Ew3o0+EOSV8b731FkuXLqW2tnbIcwtTRKFQEBISQn5+PvPnzzdKYYn/FUeOiCLn5eU1oYKTiZB+PIM93x9BYe+FTKGg78hxlkcHc/ra08b0/KIryGQnNp+3cSPFpeX8vGcXFq4BaLtasNO24e/pTrVNMI5ev4mOk08ILX4xfPr5l9z3pzuBfnHRNlZw6OV7kVmpQCZH6GrmussuGPJm19XVRVNbBx7O/TvesvIKatt6UPnOo9srDM+EDRyvKubf733I766/lqSkJFxcXMY8OFQ03TaM5DacuY6LzjtX/z4PdDmZDPNCAvnlYDqEGY8aqs9NpNrdkiNHjrBs2TIuu3IrtfaROJ11KTKZDBtNH4l7n+eNt/+Nh4fHlMzNMzVsbW0JDAw0am4fCsMUp/jfyWys7ezssLOzo6ioCGdn5wm/z2LKNSoqiuPHj0/KL1OMaM3NzVm4cCFpaWnD2qSZMnNK+CwtLXniiSd46KGHePvtt01+dypW23V0dNDc3Mzhw4cxNzfXpyoHTv8+2TQ3N7PnuyP4xq7F0ro/xajVRPJL8g8E+hfro6nRsLKywsfHh8LCQr1DxHgxMzPjT3fcyqVlZRQVFaFSqVi4cCG33fsQVg6D0ztWDm5U1pbo//7Fl1+SW16L51k3YeMbhVxhRm9DGe/veYmLLzoyyBrLysoKW2sLOlsbsVY6UlFdg9bWjYbGRpprKygoryY8dAFHf/2CLW1tREVFkZmZOeSUA8OUtDgdvKenB39/f0pLSykqOYGnhztnn7V+Uo34I3Haaav5aPdeyo8fxityMeqeHr568R7aW1so8Annw/334mypo6amFtfr7//NBN3MAtvV2/jyo3u5buvVODg4zMoIYDTc3d2HbG4fiKH4TVb4oP+eZWFhMWyB1VgQxUqpVBIUFKSf4TeR+5/hmaOtrS3h4eHD2qQNhancc+eU8EF/S8Nrr73G4cOHWbly5egPOAkYFpyIu3zDajs7OzuioqLIz88nPj7eZNKzhUVFmDl660UPQGFmhr13MMdz8scsfNB/CJ+SkkJLSwsODg4TWk9FRQUff/4lKRk52NlYs+H0CiKC/EhNLBrU5N1ZU0TUqt8O5z/89Aus/BbiELZE/zUzrzBso8/gX2/9e5DwKRQKNp+7nv/s/x8+S86htaMLSxsZnQVHcQ+OwtojiNzSCpT023b5+vri4eFBbm4urq6u+ve6q6tLn5Lu6urilX++zYm6FmTmlhRmJOHgG47LvCVoiop488Pf89RDf2LVqvFNWBgLSqWSZx9/mDf+8x6//vd5sjPS6XMOJXTzdn2FbNlXr6Cx7B40a87M3o2erk5CQ0PJzs4mPj7e5PpPp4LQ0FBSUlKwt7cfMS0vk8no6+uju7t70jd6nU5HYGAg2dnZEzpTE68h3jM8PDxob283siEbDwOLbVxcXOjs7CQrK4vo6GiTEbbROPU+naMgk8l4/vnn2bp1K99+++1J/wVVq9VGKazOzk59Y7BYzOLn5zdk6sDT05PS0lJ9Jd9Mo9FoQDH456cwM6dvQAHRaIgpTzEVM96UZ3V1NQ888SyKgHh8z7qe3q4O3v/+R6JdzLBqKqQs5Qe8ovtd4SszfsG6tYRLLr5P//jOri7MnAbfCMydvKhpODLkc563aSNd3T28+9/XaC5vwNzKFrfACIJWnIvc3JIOmRnVJwppbm6mrq6Ovr4+ent70el0uLm54enpqZ8PqNFouPamW+jwXETY6mXkHPovipAV6MKWofL3RaVS0l4XzyNP7+TruLgRJ04MRUNDAx9+8infHz6GlZUV5529jgvOO8+osMnT05OH7/sTnZ2dRCSswmftVfpiIblCgdfaK8n42zY0nS2Y2TroH9ddmoZfYAjOzs709fWRm5s7Zf2ZpoRcLmf+/PlGze1iK5BhSrqrqwtzc3Ps7OywsbGZlE2iKDTiQFcx2zORa4iI7jQTmYwu9iwa4ufnR1ZW1qjRsCkx54QP+odFrlq1in//+99cf/310/IchtO/h9rd29nZ4ePjM6gxeCT8/PxITEzEw8Nj3De+6cDfz4++xG/QaSOMBrU2V5Vy2rLxV2na2Njg5eVFcXHxuOfZ7f76f+CzEN/oxUD/pIDwNReR8dU/efTum3n/092kfHQIgLjIUO5/8Rn9rl2n07EsdiHv/JSNsOhMfderTquht7qAJUuHHlukUCi4cstluDo78vd3d1Pb1oND0AI6u3vpqa2i8fj3RLg5631JLSws6OnpITU1lXnz5hmlqFNSUqjXWBDy/zPqKnKScVizDSxsqKtvQKVSonTzpdHBh+Tk5HFFfc3NzfzhjnvocJqH27Ir0PT18I+vviM59Th/+fOOQTfl7u5uNDqwUBlbdlmonLF28qT5iydRrr0BC1d/ukvT6fz+n7z4yotAv3g2NjZO6KZq6giCgEajwc3NTV9xadgKpFKpjIYdi4UukxksLUZrYrHLRM7oBgqf4WT0kWb4jRXD0U52dnYjmieYSkQ4J4UP4JFHHmHlypVcdNFFk+5B0mq1g8yY1Wq13uHEzs7OaHdvSHNzM319fTg7O48afcrlcsLDw8nNzTUJE2svLy/iw3xISv4RR59gFGbmNFWV4q+STdhxxtfXl6SkpDE37mdmZvLZ1/v5ZPfXOC/bjGV1NZ4eHiCTIZPLMXcLQKvV8o+X/0pfXx+APsoRBIEff/qJj7/cR11TKz0njlP9839xXLAGZDI6i9Owrc9i2zUP6r/fsHWgra2NA4cOsffgz9T1yJDptFR//x52Ds7YOzoRExbIOUvCjTwfraysCAwMJDc3l/nzf+vLq6+vx8z+t6pKnVaDTGGOwtyCnu6W316w3Kw/0h4HX371Fe2qAIKWbdB/TeV2Nclfvkx6ejoxMTFG3+/k5ITK1oqOinxUfr+9j+rONizMzbhl68W8/d6rVNVUER4ZzaO7Xubss8/Wf9+8efNISkrC3t6+38h7FiIarLe1tenfb0PXIvE9jYuLG/YaYtX4UM3t41mH+Dg7OzuCg4PHfUY3lPCOZYbfeJDL5cTExJCYmIi1tfWkKkdPBnNW+Ozt7bnjjjt46qmneO6558b8uKF6pmQymb6c3M3NbUwFJ62trew98B0Vje3IzS2x0PVyxsoEIkYRDAcHBywtLamrqzMJt4wN688gpKCA9Ow81F1aViwOIzIiYsIFN+LuMSsra9TzzIyMDJ5/40Nc5p+Gx7waesytyS+vo69Pjb9/f3WhrqtVL6AD+xV//OknXv/kG/yWnIOnsxvuCWez7x9PU5f7HTa2toQH+nHXX3bQ3d1NSkqK0bmrUqkkMzubX/Nrid3yJ4qr6ujWKugoTsHfwYLYFWtozktk+eL4Qev28PCgvr7e6MzG398fdcMX+rSYZ9hCGvKPYhu+DKVNv7dsT1szmvriIdsrRuLX5AwcA42Hf8rkcsy95pGZlT1I+ORyObduu5K//PPfCKuuQOkXSVd9GfWHP+bs05bxwAP388AD9w/7fGZmZnh4eLB//37OPvvsaTeFnyyiL6kocmK/60i/04IgTKq5fawMTC26u7vT1tZGYWHhmLMiw7VV2NjYEB4ervcHnWztgIWFhT4qNfQbNUXmrPABXHvttbzxxhtkZ2cTGRlp9G/D2TtZWFjob3zOzs4T6pnSarX896tv6FX5ELx4MTKZjK7Odvb8eAyVUjlq305oaCjJycljihKnG5lMRlhY2IQrzoZCNEIe6TxTEAQ+2r0Xt5h1OHv5Ex6zhCM/fY/r0gsoq6rCy8uTlqoSbHubWbhw4aDH63Q6Pv5yH35LzsHW0ZXe3j7sXLzZcOvTFOz9Bzdfczn+/v7693rguasgCOzZ/z0+i8/FRuVIuLUdZRWV6LzDyPj+faI8bLn+0k3Dbk7EqMjBwQErKyuioqKI8HYm76fP8U04i9DlG6j81xP01JbgvHgtFakldOX9zJ2/2zbu1JSDvZKajtbBP4PudlTKoXfmN998MxYWFvz9X+9QuK8ZOxsbrj7vLJ7Y8diIz9Xc3MyOp57h15QMdGZW/OVvr3HPrb9n88XjH8U0HYgbV1HkRGs+8X0e6/HDwMntI0U4Y5nkMB5CQkJISUmhtrZ2TJvfkdxfXFxc6OjoICcnh6ioqBGvMxbbM3HW4kjDdU2BOS18CoWCnTt3ct9993HLLbeQlpbGypUrMTc310//Fu2dRpr+PV4qKipo6lMQ7PvbQbCNrRJ771CSj2eNKnwWFhb4+flRVFQ04fJ/U8ff35+kpCTc3NyGvKloNBrKqmqJTOiP7HzCFxDZWEv29+/QJzMjqyYFLzsF2++5zWjn2dfXR319Pd9++y2/Hj2Gn8Idj+BIrG2VmJtb4ODkhJtfKPPnzycgIGDY9Wk0GhqbW/Fw6Y/YzM3NCQ4MIMBPS0HDAi4858wRH29ubk54eDjZ2dn6tPWTjz3MP958m327X0Ct1XFa7DzCA/2obijE1c2RC7Y9apQeHY7e3l7eePNNPvj0C9RqNXELo2gsy8MlYJ7e87O9vhKhNpeVK28f9jo33ngjN95445jL8gVB4K77H6ZY50LQ5dtRmFvQUFnKX3a9i5OjA+vWrRv1GlPFwDP2trY2enp6jDaurq6u2NjYTKrJPDo6Wt+mMtImdLRJDuNh4BndaGnF0d4/f39/MjIyKCsrG7EPc6yfAzc3Nzo6OsjNzR0UUMz08YzInBS+7777jp9//pn09HSKioqora1l165drFixAh8fH3x8fKZ12GJHRwdmNoMrs2xVDjRWlI/pGifTxHomkMvlREREkJOTM+TO0czMDKWtNd3trdioHJDJZESvWE/Q/MWkfvYat1y+gZq6Bp7+2+uo+/qYF+RLXMwCdDod/3znQ7ptPFD7xFJ6opzKvDTWbr4epdIJrVqNtqt11JYKMzMzXJ0caauvRuVqkOoSdAhdLWOyhnJycqK+vl4/NNTOzo67bruF2/74e/0Z8XjRarVcesXVZFR3YrvoAmRm5nyTeRB5TRayT3di4xuJoO6B5nIev+/OMbnljFUYcnJyyCuvI+iSbfobnLOXPz0LN7Dr7femTfgEQTA6Yxf9Z0VTB5VKNW2mDnZ2dgQEBJCVlcWCBQtGvL5oYwYjT24fC+bm5syfP39MxS6j+X0azvCzs7Mb9jMxHt/QwMBAjh8/PqqYzhRzUviqq6uJiIjg8ssvJzg4mJqaGjZs2MAf//jHk3Ie4eTkhKY9Y9DXW5vqCHUffvilIaLJc05ODvHx8Sazk5pKlEolTk5OlJWVDYqeZDIZG9au5NOfvyds+TloEejt7aMk/VcCvVz45Mu9dCr98YndgIWlJSUlOXT8moxWo8YibBXB85dgX1FBaU0zuo5Gju3/L2s230DRsW9ZFTd/VOGTyWRcfuE5vPTB1wSsOA9bBxf6erooObqfNYtjxmW/9uRzL1DT0IKjgz2bzlzD+edtmvDn8MCBA2SUVON66RP6fjsb73Dqv3qBjWuWsXDhAiwtLYmPj5/ywpOqqiosnH0GDV529g+h4PB7U9LQPdB8u7293Sg7MxP+sx4eHrS2tnLixIkRo/yxTm4fK2NtSB+LYCkUCmJiYoxm+E3kOiIymYzo6GiSkpKMBuKaCnNS+K644gqjv3t5eXH55Zfz0ksvce+9907783t4eBDgYkdJThreQRGYW1jQWFtFX20psSs2DPs4tVpNVk4OOYUnEICIYD9sbW0nbGI9GwgMDCQxMRFXV1dsbW2NJr37ensRYJ3Krx+9iLWTJ/R2EOnnztKV6/js13ziV5yjv46TsxuZ331OUXoSS647DwAfbx+0Wh0V6m4q8rPI+OxlzliRwLVXXzHccow4bfVqenv7+OCLTynXyFBo+9iwZjlXGBg6j0RJSQnbn/07VqHLcY32xNbSnPe/P0RZZRV33nrziI8VBIG0tDR9Wf369etxc3Pj0Pc/Ig9aOqjJ3DJsBT8fS+aee/40prVNBH9/f3rqTiDodMgMBK6r9gRBAf4UFxePy9Hf8L0Wi04MC8k8PT0JDQ2d8XNuMG5uH6lK3LDYZbTzvrGcqXl4eIxa7KLVasdUtWllZUVkZCTp6elD9tKOd1KEQqHQD8RdtGjRtE98GQ8z/4kxEe666y6WLFnClVdeOWlT2NGQyWRs2nAmR44mkpr2PWqtFn8PVzZuOmPY2XRarZa9+w9R0SXD3be/8vNIwQlczXrwdHEY0cR6NmJ4RqNUKklMTMTS0tJo0ntQUBCPPHQ/LS0t1NTU4ODggKenJ+98+Al2boOHf9q4+NDV+5NeFGQyCPD3w9vLk8zSH3n6obvG1T8ok8k4a/2ZJMTHkZeXh6en54g7/oF88sUerEOX4RMZS0d7BzpBIGztRfy05x9cUlExrJ+sRqPhznvu48eUbMx8Y5D1dvDcy7t49vHt2KuU0HNi0GO0XW04qMbX+DxeQkNDiYsI5PgPn+C9bBPmVja01ZTSeHQ3Dz5yN21tbTQ2Ng65+zdsAm9razPqeRWLi+zs7Ey2WEKc3J6amjrq5HYx5Tma+I218V0sdhtu+sJ4Im1HR0d8fHzIzMwclLqdyIgkKysroqOj9WJqKpWekvD9P5aWluzYsYPt27fzxhtvTPvOxNLSkjWrV7J65XK0Wu2o5f/l5eWUt2kIXvBbeXxgxHxKMlPw/38vv9nqliH2TBnu7tVqNVZWVqhUKv1mwNbWdshp1k5OTkapRUeVHb2VTYOfSNDi6aikvjQPt8Df2kbaak4Q6Ok87vliOp2Ojz79jK8O/YxC6Yqms5XIAA9u/d31Y6q8zMwtxPW0K2mqLKUo5WcaG2rxDgrHXOnKiRMnhhW+Tz79lJ9yqvC/dLt+EkVXwzru3f4E/3r5BV5/cyt9C8/Ewr7/Jqjt6aD3+P/Y+re/jOv1TYRnn3yc5/76N7755EkEmRlOKhse/9MfWLlyJb29vSQnJxMZGakXura2tkFtIidjHuR0YGVlRXh4OJmZmcTGxo4oNoYN7sOJyVgFSyx2SUpKGnL6wngFy8fHh7a2tkFOLBqNZkJnk/b29gQEBJCens7SpUtN4n2VhM+A8847j9dee41ff/11kDfjdGE4yX0kKqprsHMevJuzc3ans6cbCwvNpEf7nAw0Gs2g9JXYMyW6PgQGBg7aGTo7O5OYmIibm9uo519LEuL56ruX6GgJw86hP7robG1CXZ3HPbf+jr+98R6dTTWo3H1pr6tEfSKVx/50y7h/IQ8cPMiXR7IIPWsb5pZWCDodJcd/4W+v/ZPt99096vVcnB3JP/ItuWlHsQlfjpm/P3nFWfQV/YrmotOGfdxHn+9BueAMGpqa6ezqRi6X42ivwtw7krKyMh576F4efeIBrIMXIyjM6S06yrartgxbXFJYWEhLSwuRkZGTdgSytbXlsYcf5N67+luBrKys6OzsJC8vT38el5aWhq+vL/b29vj4+ExZtbQp4OTkRFtbGwUFBSNWXA+c5DCUoIxHsCwsLPTFLosXLzZK/04kUps3b55+DqHoxDIZBxpPT0/6+vpQq9UmkZmShM8AmUzGCy+8wLZt2zhw4MC0VnaOF2srK/oaOgZ9vbenBxuVpb4RdfHixSaRDhrKw7C7u1vvEj9eyzaFQqEv5hnNtcbNzY0/Xr2Zf73/X6qtnRAAs64mbtpyAXGxsYSEhLB3/wGKSzNYEuDF2Tdtn9CYqt37DuEbey7mlv1pLZlcjv/CFeTufZPKyspRr7l+9TI+vns7vpsfwtLBFQQwc/KhWafh8K/HOO20ocWvpbWNuuZWzMxdUVjbo9FpqW1spq+rj56eHq6/7jrOOfts9u7dS19fH2ee+dCQ0WxpaSlbt91I0YlyLOwcUbfW8dAD9/GH39007p+FVqsdtKERPWjF1oGgoCDMzc0pKChALpePaG01m/H39+f48eOjmkqPVuwy3mIgpVKpr6Y0/B2ZSFGRXC43mrlna2s76Wnw/v7+JiF6IAnfICIjI1m6dCnvvPMO11577UwvR09wYAC/Zuynp9sbq/+fhtDb001vYwUhy07H2toaDw+PGTGxHq6c3NDD0N3dXZ++0mq1ZGVlkZaejr1KRWxs7JgqDB0cHLCxsaGqqmrUc9iFCxeyc948ioqKEASBkJAQ/S+dp6cn11+zddKvuam5FQ9H4zNZmUyGuZ0jLS0twwpfdXU1O194kX2HfkRn40R7cwNarRZBp8XSTEbYsvXs/+4NHrzvniEfb2tpRmdpBi6Bv9llyQSBptyjeHpe/dtrHMGHVqPRcN5Fl9AZuBr30x9AJlfQ11TFUy/8BT8fb84999xhH6tWq42svMSJ72LRiZeXF0qlctibZHBwMMnJyTg6Ok54EocpM3By+0Sb2yciNJ6enrS1tVFUVKTf7ExUsMQoUtxQTzTVKWJKUb0kfEPw2GOPsWrVKi644AKT+cV0cHDgrBVxfHv4GIJ1f9+a0NnMmcsW6YsFToaJ9XA7e7Gc3MnJacSdXVdXFzv//hrFTT1YuPih7Srhvc/3cs/NN4xpjFFoaCiJiYk4OzuPWEAA/eeoAxtopwqZTEZwgC8NlSW4+v62bo26D01r7bAGzdXV1Zx5/iX0uEZiHr4OyvPpQ4FFdzsR4eFY21jT3dpIk7qPvr6+IYsBunrUaKvzaP75A6yD49F2t9N5fD9KZ3dqa2vHtP6DBw/SjhVO8efpv2bh5IX1si288NKrnHvuuUZRuyh03d3dRgVGE5n4LhaCiO4eMzVPcjoxNzefdHP7RNs/wsLCjIpdJhOpqVQqgoKCSE9Px9nZ2aSyYJNBEr4hcHBw4LbbbuPpp5/mmWeemenl6AkJDsbH25vq6moAvRO8iFwuJywsjLy8PGJiYia9wxrO3mmsO/uh+PLr/1Haa0P4uk369TVWneClf/6b5598dEz9RqGhoeTk5EzJa5wMl1+4kade/Xd/r5p3IF1tzZxIOshZK+KHPWv9y7M76XWPxvf0reg0fbQWJKFQmNHR00NvXy/W1lbUpP/IuWesJTs7e8iJ345OjoREnUNHbRnNKV9iZmlD0LJz6ClLH3NvXnl5OXJn40IhQQALtwCKj/yH5ORkfdSuUqn07QNDGa1PBGtra/2cudEav2crdnZ2+Pv7k52dzfz588fd3D7RMzXDYhdbW9tJ90+KLRN1dXUTOhIwRSThG4Zt27axcuVKcnJyiIgY/4id6UJ09x8OR0dHqqqqxmViLfqSio3BYqXdVNo7iRw6fAy/1ZcZ3QScvfzJz7EY8zgiZ2dnamtrqampmbaJ5GNh/vz5PPCHrXzw2R4yj+zGQWXHFevXcs7ZZw37mB8S07Bf2p+OlJtZ4LnyYqp+fheFsz8FrSU4C234WGu45Y/PUFlZOWRad8tF5/HUG//Fb9OtBKy+GICOmhK6kwpZs2bNqOvW6XT4+fnRW56FVqsDGSD0t3d0l2UROS+C6OjoaT+PcXd3p6mp6ZTuQ/X09KS1tZWysrIhK5JFDKM8cZLDcObSY8HCwkIfVSsUiklHaqGhofz000+0t7dP+BqmtLmRhG8YFAoFzz33HA888ACfffaZSRSMjJWRTKzFEUptbW16oTMctzLdlXZqtRoz88HpO5m5JepxDK8NDQ0lKSkJJyenGT0wX7BgAQsWLNDfpEb7mVlZmNPb26X/u8o/Gmtnb0p2/x0/ywbuuO1mVq1ahYWFBba2tvrXaBjZX3TRRRxLSWP/B49h4bcQetvRVefx1788Psi+bqgqWui/Ifu5O1L13T+xX3EFCitbOotT6Tn6Ids/eOek/UzDwsL0Zt3jHbA6WxBTjyqVaszN7WIEOJn7jkqlIiAggOzs7Enfv2QyGS4uLtTW1uLp6TnrbRJlo7gDjG4dYEL09PSwevVqent70Wg0bN68mR07dkz4eoIgcNVVV3HBBReMeNhvilRWVtLa2oqHh8ewzhfin5PpfPHqP98ks90Gv6jfCjO62luo/vlTXnjiYQ7/8gsHfzqCTiewdvlizjzzjGFvwvX19VRXV7NgwYKTtfxJs3PnTl7+8gj+m25F/v8bgJ7Gaqq+eIZDX3ww6JyzpaWFwsJC4uLiBolqTk6OfjrA2rVrsba2Nio6GTh5QKymFXf/LS0t/On+B/lqz1foBAFfPz/+8sRjnHnmmSfnh/H/dHZ2kpGRMaRbyKmCOIA4NjZ21E2FGPU1NDTQ2dk5prPvkTh06BD+/v6Tvk5WVhaOjo6UlJRMqBldLpef7PPcYXehp5TwidWFdnZ2qNVqVq5cyd/+9jeWLl06+oOHoaKigo0bN3Lo0KFRiylmioFO9GIRQk9PDy4uLjg7O6NUKic0Qmmqqamp4YmdL6F2CsLBK4Cu1ibai9O47sL1HPrxMKlVHbjMW4xMJqchL4l5jvDkY9sxMzOjoKCAuro6PD09CQoKQiaTkZmZiaurq0nMJhwLfX19XLzlKo6XNWEVGIuut5Pe4iTuvvFKbr/ttiEfU1hYiEKh0Ke4Dd9vUegGTh4Q3+/Ozk7S0tIAiImJGbLCsKenh56eHuzt7WcsHVVVVUVzc/Ooo3FmM01NTZSUlLBo0aIRfw8FQUAQBGpra+nr65t0lfbhw4exsLAgICBgUi0kx48fJzAwkJ6eHkpLS8c9dsiUhO+USnWK0Qz0p9TUavWkf5F9fHzYvHkzr7zyCnffffdULHNSDOdyIqYqxaITKysr/WgQT09Pk8mve3h48OeH/sT3P/5EdkEmIU4OrLv1OpqamkgrayJ8w7X6tTr5BJG7/z2+/fZbDv38KwU1rVg4uNPXVEV0oAcP3H2HPo3k6OhoMnZII2FhYcHuTz7km2++4dtvD2Jr68IVz/x72In1Op0ONzc3jh8/TkdHB729vUauNiqVCm9v7yEnD+w/8C3PvvQ6cic/kIGu8UXuv/0PnHH66UbfZ2VlNeObOk9PTxobG0cd7DqbcXJyorW1lcLCwmHnV4rn7W1tbdTU1ExJr6NMJhvUkzcRxOpQV1fXYccOjbYOU+GUivig/82Ji4ujsLCQm2++eUqqMnt6eliyZAmfffbZSf2lVKvVRk704vmM2BQs/hlpF5Wfn4+NjY3JV2O98vo/+anODN/oxUZfr85LoylxN8qIVQTEr9P3PRX9so9VAbbcfvPvqa2tpb6+ftZatomMNHnAysqK+vr6YZ3zB1JSUsK22+/H56wbsXHsv3l2NtVRtf9fvPXSsyMWWswUGo2GpKQk5s+fP+WTI0wFQRBIT0/H09MTV1dXvciJ0bt43q5SqbCzs9Nv6CYjGr/88gvLly+nra2NrKwsEhISJnS8kZSURHR0NFZWVgiCwPHjx3F2dh7zvUWhUJxsQ/G5EfFB/w83LS2NlpYWLrzwQjIzMyd9Q7SysuKxxx7jkUce4R//+MeU71yGcjkZaNLr6+uLra3tuM9AgoODOXbsGK6uribjmjAUNtaWaHsHO9N0dbRTXddI1BWn6X/uMpkM//i1fLfnFW68bivu7u7U1NRQX18/a9xARpo8oFKphpw8YGVlRVlZ2ZiGD+/b/y1WQfF60QOwdXLDIiie/31zgN/fdMOE1y4IAiUlJfT09BAeHj5l53JmZmZERkaSlZVFfHz8jKflpxKdTkdHRwdtbW2Ym5uTmZlplKUxdLYxZCont6tUKv3Q2Ym0Amm1Wv3nURw7JA7DHaloxxQ55YRPxMHBgbVr17Jv374piQTOP/98Xn/9dY4dO8aSJUsmfB2dTkdXV9ewQzOVSqW+P28qBFahUBAcHGzyJtarV67gv/ufoXdeLJa2/dV96p4u2gsTcXb3QjFgp2huaQUKc3p6erCxsWHevHmkpKTg4OBgcg3Rvb29g4pODDc1/v7+Yzp/9fX1JTU1ddgJB4bUN7VgrhzcS2hh50hDU/OEX0t2djZ/uO1OKmobkZtbYC3XsfOpx1m/fv2Er2mISqXCw8NjVK9LU8bQ5EGsngb0RWXe3t54e3uTl5dHZGTkqENiJzO5fWBGz8vLi9bWVkpKSsZ9djiwEV4cO5SSkkJcXNyo6XJTSnWeUsJXX1+Pubk5Dg4OdHd3c+DAAe67774pubZcLueFF17ghhtuYP/+/WPa5Wo0mkGpSkP/wpM1NNPNzY2qqiqTNrEODg7mhks38sZH/8LMMxxkMtTVeVxz/hl8ffBHOprqsHP6zaS7paYcJztrvbOOpaUlAQEBFBQUTJtby2gIgkB3d7dR0cnAyQOG1m3jRbTCSk1NJT4+fkSBj18YxY+fHIKIeKOvd1XkEHfZGeN+boC2tjY2X3E1itjNeJ29EplcTmdFHn+48152f+Q1ZRsrX19f0tPThx2zY0qI7SLi+93R0aE3eVCpVCP60fr6+pKdnU10dPS0TW4fyrUlPDyc5ORkfaQ5Hgau09ramsjISNLS0mZVVe4pJXzV1dVcc801aLVadDodl156KRs3bpyy60dFRZGQkMB7773H1q3GXo9DpSoNS8m9vb3HbMg8HZiaifVQXHD+eSxZnEBKSgqCILBo0Ra8vb3x9HDnb//5L64x61C5edNSfYLG9EPc/4etRq/Fw8ODmpqakyLwhkVG4k1Po9HordscHBzw9fWd8n5IKysrgoKCyM3NZf78+cN+39q1a3n/092U/LQb9/krAIHajF/wVHSOqcl9KHbv3o3gGoZj9Gr912x9wumZfzb/fPPf/O2F5yZ03YEYel2qVKoZL7wRGcrJSDRdF9OI46mcFqOv8vJy/Pz8hv2+sUxyGI6hegENDahtbW0nbW/o6OiIt7c3WVlZozrUmAqnXHHLdNPQ0MDq1avZtm0bOTk5xMfHExYWZrSrVyqVJjlPrLS0FJ1Od9JNrKeC5ORkPv5iL2WVVQT6+XDZhRtZuHDhoO/r6emZ8t3nQH/S9vZ2/SilsRYZTTWZmZm4uLiM6P7f0tLCO+9/wP7vDwOwfs0Krr5iy4T9Zx97/Ak+zOrAbcl5Rl/vOJGJZ9m3fP35JxO67nA0NzdTVFQ06my76cAwPd3W1kZ3d7dRu4hKpZqS33GdTkdycjKhoaGjvi9im4OZmdmYn7e7u5ucnBxiY2MH/VtrayvZ2dljLnYRi2SGIysrC1tb22GHMZubm5/s93Fu9PFNF3v37uXrr78mPT2drq4uZDIZHh4eXHLJJZx22mm4ubmZnMgNhU6nIzExkfnz50+bibUpUFFRQWdn54TOiMRd/VCTB0TPypmM3EXUajVJSUmjTvueSj7++GMeee1D3DYZt/XUH/6E8+cp+cuTT0z5cxYXF6PT6cY9JHisiD2RhmewPT09Rh6lYtQ5Xb/j3d3dpKenExsbO+qxh9jcPtZil46ODoqKiobcJEK/0UV9ff2QnrADGU34dDodSUlJBAUF6YdHGyIJ3yzj559/1hu/KpVKNBoNK1eu5LXXXpt1B/DNzc2UlpbOuMHzdCIIAikpKQQHBw+7ixYraQ1veAMnD4i7elNNDTc1NVFaWjrqfMKpoquri5Vrz6AnYCVOsRuQKRS05h+j6+d32P/V59OSSRDfy8DAwEmnr8UeOcN0ZV9fn766UhS6mRiM29jYqH8vR/u8ienLsRS7tLW1ceLEiRHT4jk5OaN6AAuCwJEjR0YUPuiPlJOSkoiJiRnUkiIJ3ynADz/8wF/+8hc+/fTTWScgWVlZuLi4zBq3k4kg7qITEhKQy+VG/VJDVdJO965+usjLy8Pa2nrEM6KppLy8nD898DBHfj2KTK4gNDiIZ/78GHFxcaM/eIKIdl9xcXFjLgQbaPQg9siJZ7CiyJmS6UFxcTFarXZUo3Yx5SmXy0fNPDQ3N1NdXT1iwddokRr0F/EkJyePqaK9tbWVrKysQZPgJeE7BRAEgS1btnDppZdy9tlnz/RyxkVfXx/JyckTbmQ1ZQz7paqrq/VnM+INT/xjyj2N40Gr1ZKUlERUVNSIA0+nmra2NjQazUmrEm5oaKC8vHzITIXhe27Y+G9ra6sXuJN9BjsRBEEgLS0Nb2/vUatZRfFTKBQjiklDQwONjY2jZqbESG3RokVDHoP09vaSkZFBfHz8EI8eTHV1NdXV1UbZiMk24k8ASfimg/Lycs477zwOHTo0626klZWVdHR0zLpUrSHDTR4Qi07s7OwoKioiLCwMe3v7GV7t9NHe3q4vtDLVtOxUkJ+fj5mZGU5OTsP2yIlCN9NnsBNFrVaTnJw8JveasRS71NXV0draOqZxX2Kxy+LFiwf9/Lq6usjLy2PRokVjfi15eXnI5XL9c0vCdwrx+OOPo1AouPPOO2d6KeNCEASSk5MJDw/Xj4NRq9V8//33HPjhFwBOX7WUdevWmcROeah2EbGU3HDywMAbf2dnJ5mZmfqU56lKSUkJWq122opAZgJDd5u2tjY6Ozvp6urC2dkZZ2dnva3Xqfa+tre3k52dTXx8/KgCPlqxi5j1GOv5a0VFBY2NjYOGA7e3t1NSUjKuSSji+ay3tzceHh6S8J1KdHd3s3TpUj7//PMRS8tNkfb2dnJzc4mPj0en0/HIE0+RUt6KU/gSkEFT7jFivO144pGHTlpKdLgqu8mUkpeWlqLVaic9lsWUETcyISEhE25XmElG65ETp0309PRw/Phx4uLiTGJDNl1UVlbqp1WM9Dk3PO8bqtiloqICrVY7Lm/W7OxsrK2tjYpdWlpaqKysHPf0DLVara8kH+78cBqRhG86+fTTT9m9ezevv/76rCuOEE2sy8vLefz1Dwk753pk/7+DFgSB/L1v8vCNm1mxYsWUP7do32YocgMnTahUqklX2YmH9xEREafssFP4raAnPj7eZM9uDX1pxffdsEdOFLmRNjY1NTXU1dXNmmbpiZKdna336R2JkYpdysrKkMvl4zKpF39fgoOD9dZ4jY2N1NfXDztFZCQ6OjpIT09n1apVJjOWSBK+KUCn07F+/Xq2b99OQkLCTC9nXGg0GhITE/n510SOttjiE21ctVWZnUSCqo177xx6VtxYMZw8IJ7NGNq3iX+mq8purpyDVVVV0dLSMmO2bYYMjN7b2tr0Fm6T7ZHLzs7WW4Kdqmi1Wv1xxGhn1MOd95WUlGBpaYmXl9e4nlssdhGngYznrHAouru7Z2Le49yZzjATyOVy/vrXv/L73/+effv2zaqDdTMzM4KDg9n7zQG06sEfB01fD9aW4xOjoSYPiE3g4rzAk12AIHqjlpWVDesscSrg6elJfX39SZ9UYThHzrBlRByxY29vr7dwmwrCw8NJSkrCwcHhpFaznkwUCgXz588fU3O7OK5r4CSHobw6x4KlpSVRUVGkpaWxePHiCV9HxNScrCThmyLmz59PTEwMH3zwAVddddVML2dcuLq6ErtwPr+89wXqiHjMrfrLmdW93XQXJ7N2y+1DPm60cUpj9S/s6ekhMzMTjUZDVFTUtKUjAwMDSUxMxNXVdVrnvZWUlPDjjz8CsHr16hEbg6camUxGREQEycnJ2NvbT0sEPZRPqWGP3MkwX1coFERFRelHGM2mzeZ4sLa2JiQkhMzMzFGNCuRy+aBJDlqtdsIZDtFvNjMzEycnp1PqZyylOqeQxsZG1qxZw4EDB1CpVDO9nHHR3d3NM889z5GMQix9oxGAvoosrth0OluvuhJg0Dglw8kDYupqrOOUOjs7KSwspKSkhH9/vBut0g2Z3Bxdczk3X3cFZ5911rS8zra2NvLz84mLi5uWHegrr+/ijff/i4V/f9l3b2kKN119KX+46cYpf66RqK+vp7KyckxWVCMxXIp6Jn1KDamoqKCtrc0kUrvTSVFREYIgjFq1KwgCOp0OhUKBQqEgJycHd3f3SfVbZmVl0dfXh6ur64RTy3K5fCY+I9IZ31CUl5ezdetWamtrkclk3HTTTdx++9DRzVh59dVXKSoq4oknpt63cLopKSmhsrKSyqoqenp6CQ/rN84Vd/RDFZ1MhI8//S+7/v0+WmsnivJzUDq5sfKyW7Bzdqe7rZkTB97mr4/dM209hgUFBVhaWk6520lqaio33PUwfhfdg7l1f/pN3dXOic+f482/PkVMTMyUPt9oZGdnY29vj7e395i+XxyjZThiBzDyKTW1HjlBEMjIyMDNzW3WVVWPB7G53cfHZ9QUtmFze3Z2Nr6+vpPqY9XpdPz00094eXlN+IzP1IRvTqc6zczMeP7554mNjaW9vZ24uDjOPPPMSe0eb7rpJlasWEFBQcGEPyQnE8OZgV1dXfT29uLr42N0owsODp6yD+3PP//My+99QcA5t9DSrcYtrBNNfSmHP36FM296BGuVIzahi/nft4emTfiCgoJITEzExcVlSs26v9y7D6vwlXrRAzC3UWIdtoKv9+0/6cIXFhZGUlISjo6Og16neA4ripxoxi2+576+vrOiR84wtSu2uZyKiBPPk5OTRx0lZHi+N9mzOegXLXGmp4+PD9bW1pO6nikwp4XP09MTT09PoL/4ISIigsrKykkJn5mZGX/5y1948MEH+fjjj03qQNdw8kBbW9ugmYE+Pj54eHhQVlY2av/QRHn/v7txXnQWVion1G2VyM3MsQ9bTG3ZceqKs/EIXYC10omGxoIpf24RhULBvHnz9ONapup1dnV3Y2Y5eDK6wtKGjs7WKXmO8WBmZsa8efPIzMwkMDBQH811d3cbTYAPDAw0aTPu0TA3NyciIoKsrCzi4uJm7esYDXNzcyIjI8nMzCQuLm7Uye09PT309PRMyc9DJpMREBCg9781pah/Isxp4TOktLSU1NTUMZmwjsaaNWt47bXXOHDgAOvXr5+C1Y0PsYzcUOR6enr0kwdUKhVBQUHD3uzEPqnpMLGuqqlDGdK/2bCzs6GupQ5B6YBC5UpPezMAbRV5xK6Z3jMbsRqwsrJyykri165czqFX3kWIWvZbL6ROR3dxEmtvu2ZKnmMkBk6cEN93rVZLaWkpvr6+uLm5mVyF3VRgb2+Pq6srhYWFhIWFzfRypg2VSoWXlxc5OTn6zelw77ulpSUODg6Ym5sjCMKkz3rt7e2Ry+UTGjhrap83Sfjob7C8+OKLefHFF6ekKEUmk/Hcc89x4YUXsmbNmmmtbhMEwciFfuDkAfEXZTy9UqGhoSQnJ+Ps7DzljdDR4SGklOXh4+CCvcoea7N6uppq6akuwHLhAkoSD6DqquKM02+d0ucdipCQEH3Kcypm2p1++ul8+N8vyNq7C4eoVQC0ZP1EtJeKtWvXTvr6hgiCQHd3t1G6sre3d8j3XRAEkpKSsLGxmdZq1pnG39+ftLS0k97KcTIRBAFnZ2dqampITU0FMOqNHPj7Lp73iQUvE0VMmfr6+tLa2kpZWdm43GBMjTld3AL9Zx0bN27krLPO4q677prSaz/66KNYW1tz222Ta/4WMaywM3Shn45RKwOHuVZVVfHpF3tIycjGXqVk0/q1rFu7dtxplLy8PH7/p4exj9uIS/B8ujvayN7/IX3lxwkPn8faFYvZcunmk2Zv1NTUxIkTJ6ZsPmFvby9ffPEFX35zCIDzzz6d888/f1L9a4abG1Hk1Gq10cSJ0YqNRM/SU7n0H36bPHIyB/ROF8MZAFhZWWFnZ0ddXR1hYWG4uLiMydZsPJPbB5Kamkp4eDg2Njb6gdahoaFjrhZVKBQz4SYkVXUOhSAIXHPNNTg5OfHiiy9O+fW7urpYunQpu3fvHnfa0HDygGjQCwwqI5+uD5MYJcybN4/29nbu3v5nZH6xuAZG0tvRSnX6D5yTEMbvbtg27msfP36cl//5Npk5eVhZWnD+hjO56frrZuzQPDs7G0dHR/1570wi9sgZNoKLPXKGBUcT2dyUl5fT1dU1qydyjIWmpiZKSkqm9Px2uhkYwbe1tekzN2Ikp1QqjcS8q6uL48ePj2lyu3ifH+vk9oEkJSUxf/58/eaqp6eH5ORkvbPLaJiZmc3EhksSvqH4+eefWbVqFfPnz9dHLk899RTnnHPOlD3HRx99xP/+9z9eeeWVYT9whk3gYvHBWCYPTDeiifXho0kcazTHf+Fvfp0adR8FX73Oa395ZMKCoVarJ7ULnSpE27bY2NiTOl5KjOAN2wfEHjlDkZuqzY0gCKSmpuLv76/3YDxVKSoqAjBJY3JR5AwjOUOXm/G0C9XX11NeXj5qczuMPslhJI4ePUpcXJzRZ7G5uZm8vLwxFbuYmvDN6TO+lStXMorwT5pLLrmEXbt2kZKSwqJFiygtLcXW1nbQ5AHxA29KxQdKpRJ7e3sOJ6bgfYZxZGdmboGlawAFBQUTFj5Tcdc3MzMjNDSU3NxcFi5cOC3PMTCC7+joQCaT6cXN29sbOzu7ab05yGQyIiMjSU1NJT4+3mR+/tNBYGAgKSkpNDc34+joOGPrGGjl1tbWZpSmdnR0xN/ff8LHE66urrS2tlJcXDyqyItnfhM57xuqLcLR0REvLy+ys7OJjo42iXvWWJnTwjedaDQacnJySE1NxcvLiyuuuAJLS0tCQkJ47rnnsLe3x8fHZ9KTB6aboKAg5IJAV1sz1koHo3/TdrefMj6JLi4u1NTUUFtbO+lqVrVabZSq7OzsNIrgx2LjNl1YWVkRFBREbm4u8+fPP+nPf7KQy+VER0eTmppKXFzctBaYiRiKnPj+azQafSQ3XVZuwcHBpKamjlrUI95ndDodwLjFb6j7lK+vL21tbZSXl49oCGFq9zhJ+KaB/Px8tmzZQkREBIsWLWLbtm1YWFiwfPlyrrjiiple3rgwMzPjoo1n8a89+3A4/wYUZv1RQkNZAbaa1lPq5ikaHzs6Oo755jRwxE5XV5dR24iLiwu2trYm9Yvv7u5OfX09NTU1p7TbiZWVFSEhIWRlZU1Z8ZKIWHBkGMkZ+pW6uLgQFBR0UqJqsbk9JSUFOzu7Ec/cDMVvqPl9E3nuiIgIEhMT9RHsbGBOn/GdTBoaGli7di3ffvvtrJsJp9FoePSJJ0nLL8fWOwxdbzu2fS08cu8dJnmGMhnq6uqoq6sjOjra6OvD9UoZpqlVKtWYvUpnGrVaTVJS0ilR/TgaeXl5WFtbT9iiznBu5EBTbsPCk5lOHbe2tpKXlzdqczuM/7zvl19+Yfny5cP+u1jsEhcXN+TnydzcfCYyHFJxiynw0ksvUVZWxo4dO2Z6KeOmu7tbL9oqlYr58+fP+C/6dJGenq7vYTQsPhAr7KZqQO5M09TURGlp6ZgKI2Yz4mDVefPmjdqnO7Cqtq2tzWhupKmI3HBUVFTQ2to6pknpYtQ3lshvNOGD/s9TQUEBCQkJg0ROEr45jEajYdmyZbz55puzMlIqKSkBOKljdqaboXrk+vr66Onpwc/PDwcHB5RK5Umt9jyZ5OfnY2VlNeWG3aaGWPpvOJ3eUOTE995w8oQocqY6zX4oBEEgKysLR0fHUc3JB05yGOn7jhw5MqrwAZw4cYL29vZBlocWFhYzsbmShM9UOHjwIC+++CIffvjhrNtli42rCxYsmJVGtTqdbtCIHa1WO+Ruvrq6mubm5lN+3I1WqyUpKYmoqKhTplBpKHQ6HaWlpdTX16NUKqe9dWQmESe3jyXCNZzkMFxEptFoSE5OHpOdoyAIZGZmYm9vb7SZkoRvjiMIAps3b+baa6/l9NNPn+nljJvm5mZKS0unvFhgqhnYI9fe3o4gCGO+0QmCQHp6Or6+vqd8z1t7ezs5OTnEx8efEgbPOp3OqHWkvb0d6Dd/6OzsxNHRkaCgoFPawUaMcOPi4kZNy47m7NLb20tGRgbx8fFjem6tVktiYiLh4eH6YhdJ+CQoLi7mkksu4eDBgyelzHqqyczMxM3NDTc3t5leCjB0j5xcLjcSuYn0yPX09JCWlmaUHjtVKS0tRaPRjDro1NQw3OCI7z0MP0NQvClHR0ef0hEu9De3V1RUjGmTOlKxS1dXF3l5eSxatGjMz93d3U1KSoq+2GWGjgok4TM1tm/fjkql4uabb57ppYwb0Q8xISHhpAvCUKOVDHvkVCrVlPbIVVZW0t7ezrx586bkeqaKIAgkJycTEhKCg4PDTC9nSLRa7aANDmD03o9lg9Pe3k52dvYp71sKUFhYiEwmG7WmQIz6hip2aW9vp6SkhAULFozruQ2LXWboaEQSPlOjs7OTZcuW8eWXX5pM5DQeKioq6OrqmtYRMIbtA0P1yCmVymnvkRNtvgIDA2dNj9JE6e7uJj093SQiXFHkDCM5wyheFLmJbnDKy8vp6OggIiJiilduWoifXz8/v1GN3w3Fz3BD0NLSQmVl5ZgqRQdSWlpKZ2cnsbGx437sFCAJnynywQcfcODAAV5++eWZXsqotLS0sOfrvaRm5ODm4simDWfR1dXFvHnzJt2XaDg/UBQ6cZ6YocjNVI+cKAinwgDO0aiqqqKlpeWkFvWIqWpDz1LDAcmTFbmhEASB48eP4+HhMS1zJ02Jvr4+UlJSWLhw4aiR11DnfY2NjdTX108o6yEIAmVlZQQHB5uUV6ckfDOITqdj3bp1PPnkk+PKn59samtr+f0d99DlEITSJ5yetiY68w7zx6suJsDfj/j4+DELkmjrZJiymg09cuXl5XR3d5/SQ07hN0Hw8vKalpl2Go3GqEeus7NTL3JiJHey7NzUajXJycljEoTZznia2wdOcqirq6O1tZXQ0NAJPbdMJpupWgZJ+EyV1NRUbr/9dvbu3WuyFXVPP/c8P1aBf8IZ+q91tzVRufdVdj7+EC4uLkNOMTd0vDC0dRroQj8bCnzEM7DQ0FDs7e1nejnTiniGO1mPS7VabRTJGXqWGqaqZ/Jz39LSQkFBAXFxcSb7+zdVlJeX097ePqZo3rC5vbq6mp6eHoKCgib0vHK5fKYa/qXpDFPBtm3b+Oqrr3BzcyMzM3NKrrlo0SIiIiL45JNPuOyyy6bkmlPN94eP4r3ReJiutcoJhZMvHR0ddHd34+LiQl9fn9GZnKHjhaur60nzLpwORE/CzMzMIZ0pTiUsLCwICQkhOzubhQsXjinyNjTmHlh0pFKpCAwMNDnPUgAHBwdcXFwoKiqacEQzW/Dx8SEzM5Oqqiq8vLxG/F6ZTKY3s57s9HZTZM5EfL29vfqdi5+fHzY2NuO+xo8//oidnR1bt26dMuGD/rLj008/nW+//dYkS6zP3XwFLmfciLWqf9qyIIBGq6Ho61384aLT8fHxQa1W4+rqanQmM9MFEtNBaWkpWq12VjrvjJecnBxUKtUgBxCxslYsPOnq6sLMzMzIt9IURW445tKcQtGwIDIyctSzefG8r6KiAnNz8yGzOmNBivhmkH/961988skn2NnZ4eXlxT/+8Y9xX2P16tWUlpZO+dpcXV3Ztm0bzz//PI8++uiUX38yqNVq1ixP4Ouj+/Faci46nRZkMrobq1B0NrBu3TocHR3JyMjAy8vrlK989Pf3JykpCTc3t1lnNj5ewsLCOHr0KIA+mh9YWWtK8yMnikwmIyoqipSUlJM+jPhko1AoiI6OJiMjY0zN7eL0kcmc95riZ+OUjvg0Gg1mZmbk5OSwfv16Pv74Y5YtW8aKFSt45JFHOOuss8Z9zdLSUjZu3DilER/0C8yyZcv4z3/+Q0BAwJRee6wY7uTF9gHRyuiFl1+nqscca595aDua0VVl8/T2e/RuDt3d3Rw/fvyUTwMCdHR06PvATqXXKoqb+Ke7uxu5XE5vby+hoaGoVKpZL3Ij0djYyIkTJ055027on0JSWVlp1Nwutg+Jfwwrq/39/bGysprQz0WhUMxU9mfuFbekpaVhb29PYGAg3d3drFixgqeeeoqzzz6bDz74gH/+858cOnRo3NedLuED2L9/P6+++irvvffetPemDZwj193djYWFhVEJueFNTq1Wc+TIETKyc3BxdmLdmjWDdoGnoon1cBQXFyOTyWbtazW8yRm+/0ONWCoqKkIul8/a1zoeCgsLUSgUp/xr7evrIycnB41Gg0KhoLu7G0tLS6N0taHQjWeSw0BMUfhO2VTnK6+8wvz587ntttvo7e1l3bp1FBcXIwgCF154IS4uLvT29ppUWuPMM8/ktdde4/vvv2ft2rVTck2xR86w8EB83eIH3MvLa9TdnLm5OatXr2b16tXDfo+/vz+JiYl4eHic8uXhAQEBJCUl4erqapLnsiIDNzkDd/IqlWrU9z8wMJCkpCScnZ1HNT2e7QQFBZGcnIyjo6PJOtiMl6EieXGT29jYiL+/P56eniP+/hsWu5wKhS6nXMQn7kx++ukn7r77bo4dOwbAGWecwcqVK3nsscfo6+tDJpNN6MB1OiM+gKKiIi699FIOHTo07vUZ9sgZjtixtrY2KiGfzh65pqYmTpw4YfIm1lNBW1sbeXl54+pjnE4Mh+UONAIwjOQmkrLq7OwkMzNzTth8dXd36z1aZ1sVslhda1h4ZG5uro/kBg5L7u3tJTU1lZiYmFEHEo9lksNQmJmZzdRnZm6lOgVBQCaTcf3119PT08PKlSv54YcfuPbaazn77LP1/w6/CeVY2LJlC99//z0NDQ24u7uzY8cOrr/++ilf/4MPPoiTkxN//OMfh/0ecZaYociJU6ENRW4meuRMzcR6OiksLMTc3Bx/f/+T+rwD3W7ESN7KysqoGXwqNznl5eV0dXURHh4+JdczZerq6qiurmbBggUmsakZioEtJJ2dnXqRG+q4YjjG08s42iSHoZCE7yQhillzczMffvghpaWl+Pr6smrVKj777DPq6+uJi4ubFtGaCjo6Oli+fDlfffUVLi4uqNVquru7h+yRM3SgN5Xd6UyaWJ9sxPLw+fPnT6hFZiwYpqvFz4AocoZnMtPtdjOXyv4BcnNzsbW1xdfXd6aXone8MYzkFAqFUSQ3mcKjsrIyOjs7x+RdOtIkh6GQhO8kYhjVQf8A2J07d3Lw4EE+/fRTDh48iLe3N/fee++4or7pprOzk/T0dHbt2kVWVhYdHR24u7uzc+fOIcesmConw8TaVGhpaaGoqIjY2NhJC48gCEabHNHSzTBdLUZyM0FPTw+pqamzMg04XsSBrhERESe1dcXQu1SM5AzNAERbt6nc5IgDZF1cXPD09Bz1e4eb5DAU5ubmM3V/nXvCZ4harWbDhg3ce++9VFdXk5SUxEMPPcSmTZs4dOiQSfRj/fDDD9x5551YWloSExNDTEwM77zzDn/+859ZvHjxTC9v3AiCQFJS0pSYWM8G8vLysLW1HVeT70hnsgMjOVOitraWuro65s+fP9NLmXY6Ozv1Q1inI3sx3BSKgZHcyRAOcdL6eJrbB05yGApJ+GaIwsJCbrjhBr7//nsATjvtNFavXo2lpSUPP/ywSUR8PT09KBQKo110YmIi99xzD1999dWMr28itLe3k5ubazLFH9OJOOB0OMNjUeQMIzm1Wo2Njc2s8y0F9NGBh4fHTC9l2qmqqqKpqYno6OhJXWc4kZsJg+7hEIV+Kia3i0jCN4NcfPHFrFmzhltvvZXvvvuO2267jT//+c+cf/75NDU14eTkZBICaIggCNx4442sXr2azZs3z/RyRkT8HA38BZhIJDRbEStaFy5caBTJtbW1odFo9CI3k4VHU4U42WAs1YCnApmZmTg5OY3qcSky1GR4mUw2ZfMEp5Pa2lqqq6vH5NM6cJLDUFhYWMzUxnfuCp9Wq0WhUJCZmcnNN9/M888/T3x8PIIgUF1dzT333IO9vT2vvvrqTC91SOrq6jjjjDM4ePAgtra2M72cQXR3d/P6P//Ff7/cS1dXNwmxMdz+x5v0DvAajYbExMRJO/2bKoIg0NnZqb/B1dbWIpfLsbe3N4rkTsXzsKamJkpLS+eE04lGo9EXMQ38PdTpdIMiOcAokjNVkRuO/Px8zM3Nx9TIP1pzuyR8M4T4xmRkZBAeHk5HRwfPPPMMR44cASA4OJjNmzdz7rnn6oXSlHjhhReor69n+/btM70UIwRB4A+33UVGo4Dnko1Y2Kqoz0umM/Ur3tn1d72Rc21tLQ0NDROa4GxKjNRCYtgjlZaWdsp7Pork5+djZWWFn5/fTC9l2mlvbyc7O1t/DxE/A8CgSM7U7iHjRafTkZqaSmBgIE5OTiN+72jnfZLwmQDvvPMOb731Fs7Ozmzfvh0XFxcqKyu55JJLKCoqMskPbF9fH8uWLePdd9896f1iI5Gens7vH/gzQRffh8xgN1uZcpAVzj38+bF+oRYEgbS0NAICAmaNibUocoYtBANbSFQq1ZAFDw0NDVRWVpp0D9hUIbZzREVFmbSDzUQY6jMgnsX7+vrqPwemeM+YCnp7e0lJSWHRokWTam43ReE7tZusDEhJSeH222/H3d0dd3d3rr76aiIjIzEzM8PV1ZX4+Hh9Y7qpYWFhwZNPPslDDz3EO++8YzI30/z8fMzcQ41ED8DBP5LUX9/V/10mkzFv3jyTNbHW6XRGO3hR5Ozs7FAqlXh4eBAaGjrmqj4XFxdqa2upra095Ys/FAoFkZGRs96021DkxIhe/AyoVCo8PT0JCwtDLpdz/PhxrKysThlLs+GwtLRk3rx5+mKXkd5b8Z6k1WqRyWRG9yhTuV8ZMmeEz93dnSuuuIJLL72Uu+66i56eHszMzEhNTeUPf/gDQUFBJu00ctZZZ/Hqq6/y448/ctppp830coD+G7zQXj/o693NtXi7GRtYW1tb4+bmxokTJ2bUANhQ5MTzmKFucJPdxYeFhZGUlISTk9MpebZpiFKpxM3NjeLiYkJCQmZ6OaMy8FxWTFmL0by7uzshISHDbnQiIyNJSkpCqVSe8p60jo6OuLm5kZ+fz7x580b8XplMhiAI+uOigQJoSsypVKdarcbc3JyDBw/ywAMP4OjoSHFxMbfccgu33347arWa1NRUk+2bKygoYMuWLRw8eNAkiiX6+vrYePHlEHk2ruFxyGQyejtaKNv7Gs/ef8sggdbpdBw7dmzYkv+pxrCyTozkAH0kN92pqrq6Ompra+dEv5sgCCQnJxMSEmJSkZBhG4n4Z+C57HAp65FoaWmhsLCQ2NjYWRvljhVBEMjIyMDV1XXU5nYwLnaRy+UzufGTzvgGUldXR0FBAQkJCRQVFeHp6YlcLufqq6/mxhtvZOPGjSZZ6HLffffh4eHB7373u5leCtBvqn3n/Q/T0KVDYa1E3VTBH667kmuuvnrI729qaqKsrIyYmJgpXcdQ5eMw80UHx48fx8PDw6SzCVNFd3c36enp09bsPRpDiZxGo9FHcuJGZ6o2jcXFxeh0ulkR5U4Wsbl9LGe5hsUuZmZmkvCZGjqdjgcffJD09HQcHBw444wzWLx4MTfddJO+4tPUaG9vZ/ny5ezdu9dk/BJ1Oh3Z2dl6r7/RRtdM1sTasBG4vb19UPm4Uqk0mco60bd0Llh8QX+zd0tLi76dZbowtHYT/xgaAoh/pvNnLggCKSkpBAQEmMzv4nTS0dGhn9Ax2sZGFD9zc/OZrG6WhG8oSktLufXWW3n55Zdpb2/nqquu4o033uDVV1/l0UcfNdkS7f/85z8cPnyYv/71rzO9lAkxHhNr0bdQFDrR7WI2NAKL1NTU0NjYOOvbOcaCIAgcP34cLy+vQYOKJ3NNQ5Nu0b/UFFxvxMrHudK+UlNTQ21t7YgVy4Ig0NbWRkpKCqmpqdx2220zNcdRquocioCAADIzMyksLOT000/nkksu4dJLL2Xjxo160autrUWn040ptz0c+/bt4/bbb0er1XLDDTdw//33T2rdV111Ff/617/IyMiYledHFhYW+Pr6UlxcbGRibWjOK0ZyhpZO/v7+M27pNBHc3d314neqRwYymYyIiAiSk5Oxt7cftxgNJ3Kif6mjoyP+/v4mUzBkaWlJaGgoWVlZc6KR38PDg8TERL7++msefPBBfXo5PT2d5ORkUlNTyc3NxcbGhtjYWJOd3zhnIz6NRoOZmRmHDx/m008/JTExkaqqKrZt28Y999yDmZkZSUlJfPDBBzQ1NfGf//xnQs+j1WoJCwvjwIED+Pj4kJCQwAcffDDpVNCxY8e4//77+fLLL2edEEB/oVFiYiLOzs709fXR2dmpN+cVhW42itxwiFMN5sKoJoD6+noqKytHtL0aODjXcKagYbpyNkRSBQUFmJubExAQMNNLmVZ6enpISUnhtttuIyQkhIqKCszNzYmJiSEuLo7FixcTFRVlKml9KeIbiHjzWbFiBa+++ipKpZJvvvmG0NBQysrK+Pjjj/nxxx+JjIzk8ccfn/DzHDt2jJCQEIKCggC4/PLL2b1796SFLyEhAX9/f3bv3s2FF144qWtNN2q12sjSSZwlplQqaWhoYMGCBaeUyA2F6G5SWFg4aln4qYCrqysNDQ1UVVXh7e0NoI/kxM9CT0+Pkcj5+PjMWt/P4OBgkpOTcXBwMKmq1smgVqvJyckhJSWFlJQUjh8/jk6nIyoqiiuuuIJ///vffPPNN/p722xizgof/ObjuWvXLuzs7Oju7ub9999n9+7dWFpacu+997Jy5Upg8Hy/sVJZWWk0yNLHx4ejR49Oeu0ymYynn36a9evXc9ZZZ03bENTxIk6FFm9wnZ2dmJmZ6SO5oKAgo1lieXl5tLW1zYnRRV5eXqSmptLc3DxrHGwmSm9vL05OTuTm5lJdXY1arcbS0lIvct7e3tM+OPdkIpfLiY6OJj09fUyTDUwNrVZLfn6+/lwuLS2N7u5uIiIiiIuLY+vWrcTGxhr97q5cuZJt27Zx4MCBWfd657TwiblnOzs78vPzeeSRR1Cr1Vx44YVcddVV+u8zxbYG6M+3X3311bz44os8+OCDJ/35+/r6BkVy5ubm+lSlm5vbqFOhg4ODSUxMxNXV1WTObaYL8fwrPT2dhIQEk/xMTYS+vj6jdGV3dzcWFhaoVCr8/Pyoq6tjyZIlp3RED/0mDYGBgWRnZ5u0XZ1Op6O0tFQfyaWmptLS0kJoaChxcXFcfPHFPPXUU9jb24/4GtasWcOWLVsoLS0lNDT0JL6CyTOnhc8Qf39/4uLiuPHGG/WpCo1Gg0KhMLpBjTfy8/b2pry8XP/3iooKfepnKrj99ttZunQpV199tVFkOdUY3tza29v1IidGcmMRuaEwMzMjKCiIgoKCOVH1aG1tjbe3N0VFRbNyOv1Qmx1R5ETnGysrK6PPgU6nm3HHnpOFu7s7TU1NVFZWmsQoLp1Opx++nZKSQlpaGrW1tQQEBBAXF8f69et56KGHcHFxmZBQm0o/8XiZs8Uthgycwzfw72lpaeTl5XHZZZeN+9oajYawsDAOHjyIt7c3CQkJvP/++1N6k//666956623+Pe//z0lu0yx4EC8wYk7eMMxK9bW1lO2o52NJtaTQXQ5CQ0Nxd7efqaXMyyGaWvDiN6w8GQsnwOdTkdycjLh4eEzVdZ+UhGNu8cyyXwqEQSB+vp6fSSXkpJCeXk5Xl5exMfHk5CQQEJCAl5eXiYbjU4xUh/fWBEjOjG9WVRUxOuvv46Hhwd33333hK65d+9e7rjjDrRaLdu2beOhhx6a8jVv2rSJW2+9lVWrVo3rsQOr6np6eox28KIf4XT/onR1dZGRkWGSJtbTgam93oEFSIZns+KfiUT0Ip2dnfrm51MlxTsSHR0dZGVlTdvrFQSBlpYWUlNT9SJXXFyMs7MzcXFxepELCAgwic/XDCEJ30TZsWMHOp2ORx991KQ/QPn5+Vx55ZUcPHhwyHJ5w9Jxw6o6seBAjOYGpqlOJsXFxcjl8lO+JFzkxIkTqNXqk255pdFoBkVyCoViykRuOMrLy+nq6iI8PHxKr2uqVFZW0tLSMunsjmiqbShyBQUF2NnZERsbqxe50NDQObGpGAeS8I2X9PR07rvvPurr63n88cexsrLi/vvv5+WXX2bJkiUzvbwhueeee/Dx8eH666+nqKgIKysrvdj19vYaVdUplcoZFbmhONkm1jONIAgkJSUxb968aUuJGZoCiJGc2EoifhYMK/WmE0EQSE1Nxd/f/5Rv5If+15uZmYmLi8uYDTDEBv6MjAx9Q3h2djYWFhbExMQQHx/P4sWL9SPVJEZEEr7x0Nvbyw033MBHH33Eo48+SlRUFH/961+prq7mH//4B2vWrJnpJeoRBIETJ06QnJzML7/8wnvvvYeNjQ2enp489thjhIaG6puATUnkhmO6TKxNlY6OjimbZWfoYWpo7zZQ5GYycyE28s8V71KNRkNSUhLBwcFDWrj19fWRnZ2tF7mMjAwA5s+frz+XW7Bgwaxo4jdBJOEbK+IZ3/vvv49Go+Giiy7ir3/9K0ePHuWZZ54hLCyMuro6vL29J9zbN1U0Njaybt06fUVqXFwcBQUF5Ofn8/zzz8/YuibLZE2sZxvFxcXIZLJxVT0ONOpub283OZEbjtraWurq6mal3d5EyMrK4rrrruPgwYOUl5frRS4tLY2+vj4iIyOJj48nPj6eRYsWTUuaeY4iCd9YGShmR44c4b333uOaa64hMzOTzz77jKioKB5//HEsLCz07hOmglarZfXq1bz44ouztj1ANLFevHjxnDiz0Ol0+irAoUa+DDdyyVDkTN2oeyCZmZk4OztPygPXlNHpdBQVFQ2qsFyyZIk+XRkbG4tKpZJEbvqQhG+8CIJAaWkpwcHBxMTEEBgYiFar5aGHHkKpVOLj44NWq+XWW2+dsI/ndHHkyBG2b9/OF198MatuhoZUVFTQ3d096xpjJ0p7ezu5ubnExsYOmg4OMz9XcKpRq9UkJSWxaNEik9o4TgSdTkdlZSVJSUn6ApSGhgaCgoL0/pVxcXFs27aNm266iY0bN870kucKkvBNlP/85z9ERETg5OREcHAw0D9t4ZlnnuGTTz6hvb3d5BpzBUHgmmuuYcOGDZx//vkzvZwJIRZ+REREjDr4crai0+mMIrmGhgYAnJ2dTymRG46mpiZKS0tn1VQDQRCora0lOTlZ73pSVVWFj4+PURuBh4fHoNfU0NDAGWecwVdffWUSze1zAEn4xsvAJnaAnJwcnnjiCYqKirjyyiu57bbbZmh1o1NVVcWGDRs4dOjQrK2QbG9vJy8vj7i4uFlzYxwOnU43KJLT6XRGkZytrS0pKSnMnz/fZLxXp5v8/Hy9gbepIQgCTU1NpKam6s/lSkpKcHNz05/JLV68GF9f3zFnVjIzMwkICDhlN3MmhiR8k6Grq4vt27dz6NAhLr74Yu6//35aW1tRKBTodDqcnJxQq9UmV6X2zDPP0NXVxX333TfTS5kweXl52NnZTanN23Sj0+no6uoy6pUzFDmlUolSqRyyHL21tZXCwkJiY2NnvdiPBZ1OR2JiIlFRUTMqBoIg0N7eTlpaGikpKSQnJ1NYWIhKpTKK5EJCQmbt8cEcRBK+ydDd3c21117Ljh07cHV1Zfv27dTX19PX10dfXx979uwxyZ6a3t5eli5dykcffTRrUysajYbExETi4uJM0sRabC42jOS0Wi22trZGPZPj+Xzk5+djbW09rd6rpkR7ezs5OTlT0tIxFgRBoLu7m/T0dH3hSW5uLtbW1sTExOhFLiIi4pRNM88RJOGbKAMnM9x66614enpy4YUX4uzszGOPPcaiRYu47LLLeOWVV3jgU+Zu+AAAGGVJREFUgQdmcLWD2bNnD++88w5vvfXWrI0gamtraWhomPEqVXHatGEkp9VqsbGxMRK5yUb+Wq2WxMTEOdPID1BaWopGo5kWF5ve3l6ysrL06crMzEzkcjkLFy7UF59ER0eb5MZKYlJIwjcVHDx4kBdffJFXXnkFDw8PLCwsOHDgANu3b8fOzg5/f3927txpUkbLgiBw7rnncuedd7JixYqZXs6EmAkTa0ORE/vl1Go1tra2Rm0E05Xebm5upqSkZFYVfkwG0bg7JCRkUoNcNRoNubm5Rr1yWq2WqKgo/ZncggULTor/7FzlnnvuYc+ePVhYWBAcHMxbb701U8N5JeGbCn755RduueUWUlJSAPjiiy/4/e9/z7Jly7j77rv1Q2tNjdzcXK655hoOHDhgkinZsTCdps5i6sswklOr1YMiuZMdEeTk5GBvb4+Xl9dJfd6ZQkw/xsfHj+lzqtVqKSws1Kcr09LS6OzsJDw8nISEBOLj44mLi8POzk4SuZPI/v37WbduHWZmZvr6gmeeeWYmliIJ31Rx7733Ul1dTWZmJk5OTlxyySX8/ve/591336WtrY1ff/2VO+64g9jY2JleqhF33XUXQUFBbNu2baaXMmGmwsRa9EI0FLm+vj6sra2NTJpNIe0lnm+eCr1uY6WqqoqjR49y4YUXGn1dnOlnaNTc3NxMcHCw3torLi4OR0dHSeRMiM8//5xPP/2U9957byaeXhK+ySK2N9TV1XHJJZcQFRXFww8/jJeXFx0dHZx11lkcOXKEt99+m61bt870cgfR0tLCqlWr2Ldvn0mlYsfDeE2sRZEz9K/s7e3F2traKF1pyj6IjY2NlJeXs3DhwjlxQxcEgY0bN3L++efj6emp75Wrrq7WW/MtXryY+Ph43Nzc5sTPZDazadMmLrvsMq666qqZeHpJ+KYCUfwqKyv15fX33Xcfr732GjfddBOlpaVcdtllXHLJJVP2nJ988gmPPfYYOTk5HDt2jPj4+Alf61//+hdpaWk8++yzU7a+k81wJtaGY5cMRc7KysookjNlkRuOrKwsnJ2d8fDwmOmlTDmCINDQ0KAXuJSUFKqqqigtLeWqq67itNNOY/Hixfj4+EgiZ0KcccYZ1NTUDPr6k08+qTfNePLJJ0lKSuKzzz6bqfdu2CednQc+M4R4tuTt7U1VVRWbNm3CycmJo0ePEhERQXl5Ofv376e1tZW///3vrFixgnXr1k3qOaOjo/nss8/43e9+N+n1X3fddaxatYrs7GwiIyMnfb2ZwMnJiaqqKiorK7GwsDAaoGs4dsnHx2fWTKQYjbCwMJKSknBycjKJFOxEEQSB1tZW0tLS9MUnhYWFODk5ERcXR3x8PNdccw2BgYF88803vP7661x00UWnxHt4qvHtt9+O+O9vv/02X331FQcPHjTJ908SvglSXV2tr+qE/l9qX19f/P39uemmm6itrWXDhg2Tfp6IiIhJX0NEoVDw3HPP8cADD/D555/PmkbcgZFcd3c3dXV1+Pj44ODggJeXl8nNFpxKzM3NCQ4OJjc3lwULFsz0csaEWBUrNoSnpKSQl5eHjY2NfnjqxRdfTHh4+JC9chs2bGDPnj289957M5Umk5gg+/bt49lnn+WHH34wWQciSfgmSFxcHM7Ozuzfv5/169eTkZHBY489xtdff83y5cv5+OOPcXNzQ6PRmFQl5fLly/Hw8ODrr79m06ZNM72cQfT19Q0SOQsLC30kJ4pcRUUFPT09c2Z0kZubm36cjym+ZnF4qihyWVlZmJub64enPvDAA0RFRY2r/WPnzp3odLppXLXEdHDLLbfQ29vLmWeeCcDSpUt5/fXXZ3hVxkhnfBNAbGpvamqis7OTXbt2sWfPHjZs2MBdd93FsWPH9F8bC2PJl69Zs4adO3dO6oxPpLKyknPOOYfvvvtuRqsF+/r6jApPurq6sLCwMCo8Ga7fai6YWA9EHNc000Nc1Wo1OTk5+nRleno6giAQHR1t1Cs3VypRJUwW6YxvKhFTM05OTrz55pscPnyYjz/+mPDwcABCQkIoLCwkLy9P/7WRGC1fPtV4e3tz2WWX8dJLL3HPPfeclOdUq9VGtl6dnZ2Ym5vrBc7NzW1cAzhlMhnh4eHk5uaeEibWY8HCwoLAwEDy8/NPmouNVqslPz/fqCG8p6eHyMhI4uLi2Lp1K7Gxsdja2s6J90Di1ECK+CaIOLDWMJXZ3NzMjz/+yIEDB/D3959SUZnKiA/6U1NLly7l008/nfIGabVabRTJdXZ2YmZmZlRdOVVTpmejifVkEASB9PR0fHx8cHFxmdJr63Q6SkpK9OnK1NRUWltbCQ0NNeqVs7e3l0ROYjYgtTNMF2KLw4EDB0hLS6OiogIvLy8uuuiiKRmi+vnnn3PrrbdSX1+Pg4MDMTExfPPNN1Ow8n7nmY8++oh//etfE76RaTSaQSKnUCiM0pXTGQ2Yuon1dNDb20tKSgoJCQkTPj/W6XRUVVUZzZWrq6sjICDAqFfOxcVFErlpZipbliSMkIRvunnmmWeoqalh/fr1nH766bPiJqzT6TjnnHO49957Wbp06ajfr9VqjbwrOzo6kMvlgyK5k10taiom1ieTqqoqWltbx1T1KwgCdXV1eoFLTk6moqICb29vI5Hz8vKSRG4GyMnJQS6X87vf/W5KszoSkvBNG2LKs7q6GgBPT88ZXtH4yMrK4vrrr+fAgQNGZeVardYokhNFbmAkZwotEYIgkJqaSlBQ0EyZ4Z50BEHghx9+wNzc3Mh8XBAEWlpajNKVxcXFuLi46HvlFi9ejL+/v0m8dxK/MdXHGRJSccu0Ie6QZ5vgiYiu9U899RQWFhYUFhZy/fXXI5fL9YNT/fz8sLOzM9kbpUwmY968edNmYm2KyGQyvL292bx5M8899xw5OTmkpKSQn5+PUqnUi9zll19OaGjonPiZSEiMFSnim4OUlpayb98+kpOTSU9P16cwL7vsMlatWsWKFStm5Y1yKkysTRXRd/T48eP6aC4nJwfoT1nfeuutJCQkEBkZaVJ9oxL9nOyWJQlAivgkDKmqqkIQBG666Sbmz5+PlZUV//jHP8jMzGTVqlUzvbwJExAQwLFjx3B3d5/1A1z7+vrIysrSi1xmZiYACxYsID4+njvvvJMFCxZgZmbGmWeeSVRU1KxxdZmLnOyWJYmRkSI+CaD/TG/lypW88sorzJs3b6aXM2GGM7E2ZTQaDfn5+SQlJZGWlkZaWhp9fX1ERUURFxdHQkICMTExw7aAFBYWcuWVV3LkyJFZGalL9CNFfFOOVNwiMTo//fQTf/7zn2fSTX1KyMjIwMPDA1dX15leyiB0Oh1FRUX6hvDU1FQ6OjoICwsz6pVTKpXjeg/q6+tN8vVKjM50tizNcSThkxgdQRC48sorueiiizjnnHNmejkTpre3l9TUVBISEoY0QD5Z6HQ6KioqjHrlGhoaCAoK0otcfHw8Tk5Os3qjISFhokjCJzE2Kioq2LhxI999992snF0nUl5eTk9Pz5SYCIwFQRCora0lOTlZH81VVVXh6+trJHIeHh6SyElInBwk4ZMYO0888QQymYy77rprppcyYabTxFoQBJqamvSFJykpKZSWluLu7q4/k0tISMDX11c6c5OQmDkk4ZMYO93d3SxdupTPP/98Vk/9bmtrIz8/f1Im1oIg0N7erh+empKSQmFhIfb29kYN4cHBwZLITSP79u3j9ttvR6vVcsMNN3D//ffP9JIkTB9J+CTGx2effcZnn33Grl27ZnVqLi8vD6VSOSYjbkEQ6O7uJj09XZ+uzM3NxdramkWLFulFbt68eTN6djjX0Gq1hIWFceDAAXx8fEhISOCDDz4gMjJyppcmYdpIfXwS4+OCCy7g9ddf59ixYyxZsmSmlzNhgoOD+eabb1AoFLi7uxv9W29vL5mZmUbDU+VyOQsXLiQ+Pp577rmH+fPnz+jsOwk4duwYISEhBAUFAXD55Zeze/duSfgkJowkfBJDIpfLeeGFF7jxxhvZv3//rI1wzMzMqKur48477+SBBx7Qi5zoWBMVFUVCQgJ/+MMfWLhwIVZWVrM6wj0VqaysxNfXV/93Hx8fjh49OoMrkpjtSMInMSziRO333nuPrVu3zvRyxoxWq6WwsNCoV66srIwHH3yQDRs2cNVVV/HCCy9gZ2cniZyExBxEEj6JEdmxYwennXYaF1xwASqVaqaXMwidTseJEyeMKixbWloICQkhLi6OCy64gD//+c80NjZy5ZVXcvvtt0upy1mGt7c35eXl+r+LI5UkJCaKVNxyCnLPPfewZ88eLCwsCA4O5q233prUuJ7XXnuN/Px8nnzyyalb5AQQBIHq6mp9r1xaWho1NTX4+fnpe+USEhJwdXUdMpJ74oknsLa25k9/+tMMrF5iomg0GsLCwjh48CDe3t4kJCTw/vvvz6n5ixITQqrqnEvs37+fdevWYWZmxn333Qf0D8qdKBqNhhUrVrBr1y7CwsKmapkjIggCDQ0NRpFcWVkZnp6e+uGpCQkJeHt7jzld2dvbS2FhoXTDnIXs3buXO+64A61Wy7Zt23jooYdmekkSpo8kfHOVzz//nE8//ZT33ntvUtf5/vvvefbZZ/nkk0+m/FxMEARaW1tJTU3Vi1xRURFOTk5GDeGBgYFSr5yEhMRYkYRvrrJp0yYuu+wyrrrqqkldRxAEtmzZwqWXXsrZZ589qet0dnaSlpamF7n8/HxsbW2JjY3V98qFhYXN2kpSCQkJk0ASvlONsQy2fPLJJ0lKSpqyaQtlZWWcf/75HDp0aMw+nj09PWRkZOgrLLOysrCwsCAmJkafsoyMjJQKTiQkJKYaSfjmGm+//Ta7du3i4MGD2NjYTNl1d+zYgbm5OXfcccegf+vr6yMnJ0cfyR0/fhxBEIiOjtanKxcuXDirza9NnW3btvHVV1/h5uamH14rITFHkYRvLrFv3z7uuusufvjhhymf0Sb6eH722Wc0NzcbNYT39PQQGRlJfHw88fHxxMbGDjs8VWJ6+PHHH7Gzs2Pr1q2S8NHv13r8+HH9Z1FiTiEJ31wiJCSE3t5enJ2dAVi6dCmvv/76lF3/2Wef5YUXXmDVqlX6dGVsbCz29vaSyJkApaWlbNy4cU4KX19fH8XFxTg6OuLu7s7x48fZtm0b+/fvx8nJCeg/Z5Y+p3MCSfgkpg5BENDpdFLxiYkyl4RPq9Uik8n01b4ajYann36aDRs2EB8fT09PD5s2beKKK65g9+7dZGVlsWPHDi699FLMzCT/jlMcyaRaYuqQyWSS6EmcVMTNllwuN4rWDD+HLS0tKBQKvvrqKz766CN6e3v58ssv8fDwYM+ePdx+++04Oztz55134u3tzWmnnTYTL0XCBJCET0JCwqQYKhU51Garq6uL1157jYMHDxIQEEBDQwN33XUX8fHxdHR08PDDDxMaGkpoaCj5+fksWbIEGxsbwsLC+O677yThm8NI3cASEhIzgk6nIzMzk76+PqBf8IBBotfb20tycjI7duzg8ccf58SJEwB88cUXHDlyhCeeeAJPT09KS0uxt7dny5YtODk56YVS7AnVaDQAJCQkkJycfLJepoQJIgmfhMQpxJYtW1i2bBl5eXn4+PjwxhtvzPSSAFCr1ajVaqOvlZaWcscdd3Ds2DGgX/C6u7v57rvv+OmnnwAoKCjglltu4f7770elUpGTk8MTTzwBQGZmJn5+fsTFxXHVVVexevVqfv31V/z9/amrq9P3ucbFxVFbW0tLS4v+7ykpKSfplUuYIpLwSUicQnzwwQdUV1ejVqupqKjg+uuvn5F1CIKgj+Bqa2u55ppr2Lt3L9BfkALg4OBAdHQ0ubm5APzwww+cf/75vPbaa3z66afs3LmT0NBQ1Go1Xl5e3Hnnndx///3U1taSmZmJt7c3jo6OALi4uGBnZ0dJSQleXl4oFAoaGxvp7e3Fx8eHmpoaSktLAZg3bx4rVqygo6PjJP9UJEwFSfgkJCSmDMN0pZiytLa2xt/fn8LCQqPvtbW1xdfXl7y8PKC/B/Hee+/lrbfewtnZmZ07d1JZWUlsbCyenp4ABAUFIZPJaG1tJTQ0lH379gGgVCr59ddf9eOLrrjiCp566ilCQ0MpLS3lrrvuIiAgAABLS0s+/vhj7Ozspv3nIWGaSMUtEhIS40an0xmJm4j49+zsbOrq6lizZg3W1tYEBQUZpTQBLCwsCAoKIjExEYA9e/bwySefYG9vT2RkJK+99hpubm64u7uTk5NDe3s7SqUSe3t7SktLufLKK3nnnXc499xzaW5uJiAgAKVSSXV1NWeffTYLFy7UC2ZERITROoc7T5SYG0jCJzHnKC8vZ+vWrdTW1iKTybjpppu4/fbbZ3pZJonYRmDYKwcY/X9VVRVeXl4IgsB3333H3XffjaurK35+fmRlZXHzzTcTGBjInj17jB4rk8nw9fWltbUVjUaDnZ0djz/++CATdG9vb7q6uigoKCA2NhYnJycKCwvR6XTs2rWL9PR0YmNjeeutt8jPz8fNzQ1AL3pilahhtagkeHMbSfgk5hxmZmY8//zzxMbG0t7eTlxcHGeeeSaRkZEzvbQZo6WlhUOHDvHrr7/S1dXFAw88oJ91OLCNQKPRsGvXLnQ6nb545oYbbuCWW24hOjqaI0eOoNPpeP/993n++ee57LLLCAgIQKfTUVxcTFBQkL4nz9XVFaVSSXFxMddddx1vvfUWbW1tWFlZ8cEHH3DTTTcRHh5OQECA/mxwx44dqFQq5HI5HR0dHD16lKeffprq6mq2b9+OhYWF0XolsZMYiCR8EnMOT09PfTSgVCqJiIigsrJyzgpfX18fDzzwADU1NcTFxbF8+XL9v2k0Gnbu3Mkvv/xCb28vzzzzDDExMXz22WfI5XL27t2Lg4MDMTExLFy4kFWrVvHII4+we/du4uPjAThy5AgbNmzA0dGR7OxsgoKC9KlGlUqFSqXSD5p1dnbmjTfeQKFQsGzZMsLCwvDy8mLHjh1Af/Tm4OCgX5+1tTU+Pj7cf//9xMXFYWlpKVmSSYyKJHwSc5rS0lJSU1NZsmTJTC9lxigpKSE1NZVff/1V/zVRmA4ePIiVlRUvvfQSZWVlPProo7z44ousW7eO6upqVCoVNjY2rF69msTERGxtbSkrK+Obb77Bw8ODyy+/nJSUFDZt2oSHhwfHjh1j48aN+uvb2tpy3nnnYWVlBcA555zDOeecM+Q6hxI0pVLJ5s2bjb4miZ7EaEjCJzFn6ejo4OKLL+bFF19EpVLN9HJmFA8PD15++WXc3d0JDg4mMjISKysr3n//ferr6ykvL+fw4cN0d3fT1dWFn58fFRUV9PT0YGdnR2RkJJmZmaxfv56GhgbKyspITEyksrJSL5gLFiygtrYWQO+TaWFhwXnnnTdoPWJa09CiTBI0ialCEj6JOYlarebiiy/myiuv5KKLLprp5cwo4eHhrF+/njfeeIOAgACampqwt7fniSeeYP78+XzwwQfcd999/PGPfyQ4OBjoT4/+/e9/54svvmDbtm3U1dURHR1NdHQ0l19+OVu2bCEhIYFHHnlEP2T46quvHnYN4pmfiOQFKzGdSNMZJOYcgiBwzTXX4OTkxIsvvjjTyzEpysvLKSgo4ODBgzQ2NrJ161buvPNOjh49CkBTUxO5ubnEx8ezbt06oqOjOXbsGN7e3rz77rvY29uPeMY2UOAkJKaRYVME0idQYs5x+PBh3nnnHQ4dOkRMTAwxMTF6V5G5jq+vL+vWrcPR0REbGxuWL19OTEwMmzdvZtWqVaxYsYLs7GwsLCzw8PBg7dq1pKSksGfPHuzt7YHfUpJarVbfDiEiiZ6EKSClOiXmHCtXrmSUTMeco7S0lJycHMrLyzl+/Djp6en64cW7du3i0KFDeHh4GFW+KpVKcnNz0Wq1yOVyBEEYMl0pnc1JmBpSqlNCQoLm5mZ27NhBc3MzS5cuZc2aNYPcTkQ0Gg1mZmbk5+djY2ODj4/PSV6thMSYkCawS0hITA6pP05iliFNYJeQkBidodoIRCTRkzhVkCI+CQkJCYlTEamqU0JCQkJCAiThk5CQkJCYY0jCJyEhISExp5CET0JCQkJiTiEJn4SEhITEnEISPgkJCQmJOYUkfBISEhIScwpJ+CQkJCQk5hSS8ElISEhIzCkk4ZOQkJCQmFNIwichISEhMaeQhE9CQkJCYk4hCZ+EhISExJxitLFE0hwSCQkJCYlTCinik5CQkJCYU0jCJyEhISExp5CET0JCQkJiTiEJn4SEhITEnEISPgkJCQmJOYUkfBISEhISc4r/AwyGF+BdmiUZAAAAAElFTkSuQmCC",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "\n",
    "# form the design matrix and target vector\n",
    "X, y = iris.data, iris.target\n",
    "X = (X - np.mean(X, axis=0)) / np.std(X, axis=0)\n",
    "\n",
    "# display data in 3D\n",
    "fig = plt.figure(1, figsize=(8, 6))\n",
    "ax = Axes3D(fig, elev=-150, azim=110)\n",
    "ax.set_xlabel(\"Sepal length\")\n",
    "ax.set_ylabel(\"Sepal width\")\n",
    "ax.set_zlabel(\"Petal length\")\n",
    "p1 = ax.scatter(X[:, 0], X[:, 1], X[:, 2], edgecolor='k', s=40)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "We can implement PCA using a small number of `numpy` operations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "def pca_project(X, p=2):\n",
    "    Sigma = X.T.dot(X) / X.shape[0] # form covariance matrix\n",
    "    L, Q = np.linalg.eig(Sigma) # perform eigendecomposition\n",
    "    W = Q[:,:p] # get top p eigenvectors\n",
    "    Z = X.dot(W) # project on these eigenvectors\n",
    "    return Z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Visualizing the data, we obtain the following structure."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x12d072828>"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 864x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "Z = pca_project(X, p=2)\n",
    "plt.scatter(Z[:,0], Z[:,1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "We can also add labels. The classes are well-separated."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x12cbc9f60>"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 864x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(Z[:,0], Z[:,1], c=y, cmap=plt.cm.Paired)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "The separation is better than if we just chose the first two dimensions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x12c8c2cf8>"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 864x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(X[:,0], X[:,1], c=y, cmap=plt.cm.Paired)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "We can train two classifiers on this data and compare their accuracy.\n",
    "\n",
    "PCA dimensions result in better accuracy that just choosing the first two dimensions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Accuracy on first two dimensions: 0.83\n",
      "Accuracy on two PCA dimensions: 0.92\n"
     ]
    }
   ],
   "source": [
    "from sklearn.linear_model import LogisticRegression\n",
    "\n",
    "# train softmax on non-PCA data\n",
    "logreg1 = LogisticRegression(C=1e5, multi_class='multinomial')\n",
    "logreg1.fit(X[:,:2], y)\n",
    "print('Accuracy on first two dimensions: %.2f' % logreg1.score(X[:,:2],y))\n",
    "\n",
    "# train softmax on PCA data\n",
    "logreg2 = LogisticRegression(C=1e5, multi_class='multinomial')\n",
    "logreg2.fit(Z, y)\n",
    "print('Accuracy on two PCA dimensions: %.2f' % logreg2.score(Z,y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Pros and Cons of PCA\n",
    "\n",
    "PCA is perhaps the most widely used dimensionality reduction algorithm.\n",
    "* It is both highly intuitive and effective\n",
    "* It is also fast and easy to implement\n",
    "\n",
    "Its limitations include:\n",
    "* Linear projections may be too limited in some applications\n",
    "* Choosing the right dimension $p$ can be somewhat of an art"
   ]
  }
 ],
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   "collapsed_sections": [],
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  "kernelspec": {
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   "language": "python",
   "name": "aml"
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  "language_info": {
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