problem_sets/problem_set_4.ipynb

{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# ECON526: Assignment 4"
      ],
      "id": "c920325d-ca77-4ece-bbfb-3418a8d7413f"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Student Name/Number: (doubleclick to edit)"
      ],
      "id": "7a9f18c0-68fa-47eb-9cfa-f78a78c83ec1"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "### Instructions\n",
        "\n",
        "-   Ensure you modify the field above with your **name and student\n",
        "    number above immediately**\n",
        "-   Modify directly and save as the `.ipynb`, and submit directly. Do\n",
        "    not export to PDF or HTML, and leave the filename as is. Canvas will\n",
        "    automatically append your name to the filename.\n",
        "-   Submit directly to canvas as a `.ipynb` file.\n",
        "\n",
        "## Setup\n",
        "\n",
        "Feel free to use the following packages (and we have added a few\n",
        "convenience imports)"
      ],
      "id": "22c638f9-594c-47fc-8cdc-62484f2c454d"
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {},
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "import scipy\n",
        "import seaborn as sns\n",
        "import pandas as pd\n",
        "from sklearn.decomposition import PCA\n",
        "from sklearn.cluster import KMeans\n",
        "from numpy.linalg import cond, matrix_rank, norm\n",
        "from scipy.linalg import inv, solve, det, eig, lu, eigvals\n",
        "from sklearn.metrics import confusion_matrix"
      ],
      "id": "59b516a5"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Q1\n",
        "\n",
        "Take the following code from our PCA examples in class:"
      ],
      "id": "3b1f29ef-48e9-4123-af57-202eea4042ae"
    },
    {
      "cell_type": "code",
      "execution_count": 2,
      "metadata": {},
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Correlation of Z_1 to Z_hat_1 = -0.9993288763718938\n",
            "Correlation of Z_2 to Z_hat_2 = 0.3969207044136635"
          ]
        }
      ],
      "source": [
        "N = 50 # number of observations\n",
        "L, M = 2, 3 # number of latent and observed factors\n",
        "Z = np.random.randn(N, L) # latent factors\n",
        "F = np.array([[1.0, 0.05], # X_1 = Z_1 + 0.05 Z_2\n",
        "              [2.0, 0.0], # X_2 = 2 Z_1\n",
        "              [3.0, 0.1]]) # X_3 = 3 Z_1 + 0.1 Z_2\n",
        "X = Z @ F.T + 0.1 * np.random.randn(N, M) # added noise\n",
        "pca = PCA(n_components=2) # one less, and correctly specified\n",
        "Z_hat = pca.fit_transform(X) # transformed by dropping last factor\n",
        "# Scale and sign may not match due to indeterminacy\n",
        "print(f\"Correlation of Z_1 to Z_hat_1 = {np.corrcoef(Z.T, Z_hat.T)[0,2]}\")\n",
        "print(f\"Correlation of Z_2 to Z_hat_2 = {np.corrcoef(Z.T, Z_hat.T)[1,3]}\")"
      ],
      "id": "90d16ce1"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Q1.1\n",
        "\n",
        "We said that it was doing a terrible job at separating out the 2nd from\n",
        "the 3rd latent factor. Can you show code to demonstrate this and then\n",
        "provide a description of why you think it is happening?"
      ],
      "id": "d8efaf4b-10d7-40af-b6ef-bb99a80d6acd"
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "metadata": {},
      "outputs": [],
      "source": [
        "# Your code here"
      ],
      "id": "5a13b51a"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "**(double click to add an explanation here)**"
      ],
      "id": "106fb0f5-12b9-492a-adbc-59738c37c1c4"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Q1.2\n",
        "\n",
        "Change the $F$ matrix in our previous code - leave the noise and other\n",
        "parameters intact - to make it so that it is more clearly a data\n",
        "generating process with 2 latent factors as is specified. Hint: get\n",
        "better separation between 2nd and 3rd singular values"
      ],
      "id": "f2f9c862-e384-40d2-bd46-ea2f93298b94"
    },
    {
      "cell_type": "code",
      "execution_count": 5,
      "metadata": {},
      "outputs": [],
      "source": [
        "# Your code and explanation here"
      ],
      "id": "190cdec3"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Q1.3\n",
        "\n",
        "This example contains the same $F$ as the original."
      ],
      "id": "0413c3b1-4e4d-45a6-92f8-60f1e0481bd1"
    },
    {
      "cell_type": "code",
      "execution_count": 7,
      "metadata": {},
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "[3.09820150e+01 3.51428933e-01 1.56859240e-15]\n",
            "[9.99871353e-01 1.28647051e-04 2.56297883e-33]"
          ]
        }
      ],
      "source": [
        "N = 50 # number of observations\n",
        "L, M = 2, 3 # number of latent and observed factors\n",
        "Z = np.random.randn(N, L) # latent factors\n",
        "F = np.array([[1.0, 0.05], # X_1 = Z_1 + 0.05 Z_2\n",
        "              [2.0, 0.0], # X_2 = 2 Z_1\n",
        "              [3.0, 0.1]]) # X_3 = 3 Z_1 + 0.1 Z_2\n",
        "X_new = Z @ F.T + 1e-16 * np.random.randn(N, M) \n",
        "print(PCA(n_components=3).fit(X_new).singular_values_)\n",
        "print(PCA(n_components=3).fit(X_new).explained_variance_ratio_)"
      ],
      "id": "c14479bc"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Give an explanation for the key differences of these results compared to\n",
        "our baseline"
      ],
      "id": "e8c1b422-7846-4c10-9225-27890eb6c3c2"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "**(double click to add an explanation here)**"
      ],
      "id": "c8306950-c8e6-48ad-8355-a0c374998b88"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Q2.1\n",
        "\n",
        "The clustering example we gave in class worked quite well. Modify the\n",
        "example to make it perform worse and briefly explain why"
      ],
      "id": "fc2d895a-e5ba-415b-8f99-4cbd768796f4"
    },
    {
      "cell_type": "code",
      "execution_count": 8,
      "metadata": {},
      "outputs": [
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "C:\\Users\\jesse\\anaconda3\\lib\\site-packages\\sklearn\\cluster\\_kmeans.py:1412: FutureWarning:\n",
            "\n",
            "The default value of `n_init` will change from 10 to 'auto' in 1.4. Set the value of `n_init` explicitly to suppress the warning\n",
            "\n",
            "C:\\Users\\jesse\\anaconda3\\lib\\site-packages\\sklearn\\cluster\\_kmeans.py:1436: UserWarning:\n",
            "\n",
            "KMeans is known to have a memory leak on Windows with MKL, when there are less chunks than available threads. You can avoid it by setting the environment variable OMP_NUM_THREADS=1.\n"
          ]
        },
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
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Nf7SHnM5yiomsLCMpPi5Sna69Un2feFuXRVXR1DF3yZWXr5fSvvB3+LhYAjMn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\n"
          }
        }
      ],
      "source": [
        "# Modify this directly for your example\n",
        "mu_1 = np.array([0.0, 0.0]) # mean of k=1\n",
        "mu_2 = np.array([1.0, 1.0]) # mean of k=2\n",
        "sigma = np.array([[0.2, 0], [0, 0.2]]) # use same variance\n",
        "N = 100 # observations\n",
        "X_1 = np.random.multivariate_normal(mu_1, sigma, N)\n",
        "X_2 = np.random.multivariate_normal(mu_2, sigma, N)\n",
        "df_1 = pd.DataFrame({\"f1\": X_1[:, 0], \"f2\": X_1[:, 1], \"k\": 1})\n",
        "df_2 = pd.DataFrame({\"f1\": X_2[:, 0], \"f2\": X_2[:, 1], \"k\": 2})\n",
        "df = pd.concat([df_1, df_2], ignore_index=True)\n",
        "kmeans = KMeans(n_clusters=2, random_state=0)\n",
        "df[\"k_hat\"] = kmeans.fit_predict(df[[\"f1\", \"f2\"]]) + 1\n",
        "cm = confusion_matrix(df[\"k\"], df[\"k_hat\"])\n",
        "sns.heatmap(cm, annot=True, cmap='Blues')\n",
        "plt.xlabel('Predicted k')\n",
        "plt.ylabel('True k')\n",
        "plt.title('Confusion Matrix for K-Means Clustering')\n",
        "plt.show()"
      ],
      "id": "1146e4b8"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Q2.2\n",
        "\n",
        "Repeat the same thing as Q2.1, but change a different parameter to make\n",
        "it perform worse in a different way. Briefly explain why"
      ],
      "id": "8a4ff2a2-7a25-449e-84df-1f65348f47a3"
    },
    {
      "cell_type": "code",
      "execution_count": 10,
      "metadata": {},
      "outputs": [
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "C:\\Users\\jesse\\anaconda3\\lib\\site-packages\\sklearn\\cluster\\_kmeans.py:1412: FutureWarning:\n",
            "\n",
            "The default value of `n_init` will change from 10 to 'auto' in 1.4. Set the value of `n_init` explicitly to suppress the warning\n",
            "\n",
            "C:\\Users\\jesse\\anaconda3\\lib\\site-packages\\sklearn\\cluster\\_kmeans.py:1436: UserWarning:\n",
            "\n",
            "KMeans is known to have a memory leak on Windows with MKL, when there are less chunks than available threads. You can avoid it by setting the environment variable OMP_NUM_THREADS=1.\n"
          ]
        },
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "image/png": 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BZRQTWV6S9NTs5VqbuVe5uXlKjL9eH05/UL0emaM1m/f6efS4YgL0f51+C0qG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\nAIAlEJQAAABL+H/FpT6PI+WWjgAAAABJRU5ErkJggg==\n"
          }
        }
      ],
      "source": [
        "# Modify this directly for your example\n",
        "mu_1 = np.array([0.0, 0.0]) # mean of k=1\n",
        "mu_2 = np.array([1.0, 1.0]) # mean of k=2\n",
        "sigma = np.array([[0.2, 0], [0, 0.2]]) # use same variance\n",
        "N = 100 # observations\n",
        "X_1 = np.random.multivariate_normal(mu_1, sigma, N)\n",
        "X_2 = np.random.multivariate_normal(mu_2, sigma, N)\n",
        "df_1 = pd.DataFrame({\"f1\": X_1[:, 0], \"f2\": X_1[:, 1], \"k\": 1})\n",
        "df_2 = pd.DataFrame({\"f1\": X_2[:, 0], \"f2\": X_2[:, 1], \"k\": 2})\n",
        "df = pd.concat([df_1, df_2], ignore_index=True)\n",
        "kmeans = KMeans(n_clusters=2, random_state=0)\n",
        "df[\"k_hat\"] = kmeans.fit_predict(df[[\"f1\", \"f2\"]]) + 1\n",
        "cm = confusion_matrix(df[\"k\"], df[\"k_hat\"])\n",
        "sns.heatmap(cm, annot=True, cmap='Blues')\n",
        "plt.xlabel('Predicted k')\n",
        "plt.ylabel('True k')\n",
        "plt.title('Confusion Matrix for K-Means Clustering')\n",
        "plt.show()"
      ],
      "id": "b0ee8c71"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Q3.2\n",
        "\n",
        "Take the following example of generated data and latents"
      ],
      "id": "5f34e910-b74d-4cea-990a-1dea69fb5e01"
    },
    {
      "cell_type": "code",
      "execution_count": 12,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "image/png": 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6iXUSga1l0CBJl8snEm7+VqRhLy+qH7QfK9YluXJiDRS72OUkNQpyxEFqWJRK\nETRMWyYqVCoUIkmNPqC+eyZJ0sUUCojsDfduEmuQcs5BVF+RpvrsDvfX+YSDR80lQZLqlgwcpIap\npP6DJEkNm1IlRvEu3vEU1kWU6L542hFg0INlVV6NmWDKBJRi2sPTp9a7LF0eOVUhSZIk1TzfFnDb\nz45VYlUaGPsfCOkkdlElH4BvJsP7veGDXvDDDEg/UV89lipJjjhIDU9+GuQng6UQ9EFiG6Uc1pSk\nxkWpEjuhZm0Q2zatxSJfincYqHWQGQ/zry4bkbDb4eRfsHCcSB7lF12//ZfckoGD1LCkH4cfboeU\nQ+JrD08Y+Qz0mCrXOUhSY6NQiJTWrkrBH/zJ9TRGforI5yIDhwZLTlVIDUduEiyeVBY0gBh1+PMp\nOFfBQitJkhoXS7GoH+POue1115cm4KeffmLUqFH4+vqiUCiwWCy1+noycJAajpxzooqlK+tfF4uo\nJElq/FRqCOvmvr2itqJ8yIgXlXPzUmq+bzXJlCVGUU1ZtfoyRqORkSNH8sQTLtJ/1wI5VSE1HEYX\nSWRK5J5zn9pWkqTGRaGA7reI+jHWYsc2T19RP8aVrNPw+5NwbKVYExHYGq55V2wBVXvWfr8rK+Mk\nrH4W4laKZHMKJXQYD6NfEH2uYdOmTQNg/fr1Nf7crjSKEYeqDsPk5+czc+ZMfHx8CAwM5KGHHqr1\noRupBlQ0pxnVD7Rym5YkNRn+rWD6csfEbiEdRWp4v5bO5+cmwdc3QtyKsloXGSfhq+showHtxMg4\nCZ+NgGO/i6ABxH/jVonjGSfrt381oFEEDlUdhpkzZw5bt25l9erV/PDDDyxZsoQXX3yxlnspXTbv\nMGh3tfNxpQcMfwK09VuOVpKkGuShFiMLd/wB922G2Vvg9l8hvJvronNZp1wHCDYr/PMOmE213uVK\nWfMcFBeA7aIPqzaLOL7mufrpVw1qFFMVVRmGycrKYvHixaxatYr+/UXBpJdffpnHHnuM5557DpXM\njd5w6QPh2vdhx+ew/WNRSrtFf7j6NQhqV9+9kySpNniHicelpB9335ayX9yU1bqa61d1mLLg6Iqy\nkYaL2SxwdKU4rxGnvG8UgUNV7Nq1C7vdzvDhw0uPjRo1ioyMDE6cOEH79u1dXmc2mx2mM0ymBhK9\nNgf5qeLTRMImMIRAt8nQewbYLSLznNyGKUlSRWsDgjuVZaKsTwXp7oOGEnarOE8GDg1Hamoqfn5+\nqNVl5V6Dg4NL29wFDq+88govvPBCnfRRKicvCZbNhvi1ZcdUarjlO4gZJoYzJUmSAmLFeoiLd14p\nlDDkIVHG+1JyEyHlICT8I9ZYtBoq1lOoauhWaAgS/akoeFCoHLNpNkKNYo1DVdhLFs2Uo3A1X3aR\np556CqPRWPrIyMioje5JFzu6yjFoALCaYck0yDtfP32SJKnh8YmA25ZBq2Flx3yjYOr3ENT20tdn\nn4WvJsDim2DTu/DbPPjfQDi7TaS/rgk6f7F7QukmEFF6QIdxNT7akJmZyd69ezlxQqwB2bdvH3v3\n7iU/30WCrRrQ5EYcQkNDyc7Oxmw2l446pKamAhASEuL2OrVa7TBKIdWBgjTY9j/XbWaTWH1cE6W0\nJUlqGgJiYfIi8d5hLRY3YJ+IS19nKYatH0Ha0YuOF8H3t8E9G11nt6yO0S+IcuIXL5BUeojy46Nr\nfmT7l19+YebMmaVf9+nTB4B169Y5TNvXlCY34tCrVy8UCgUbNmwoPbZ27VoCAwNp06ZNPfZMcmKz\nisyQ7si8DZIkXUznJ0YYQjtXLmgAEWjsXuS6zZghamnUlMDWcPc6aD9WTFuAmJ5oP1Ycr4U8DjNm\nzMButzs9aiNogEYy4pCZmcmZM2cchmFUKhVt2rQhJyeHUaNGsWjRIvr160dAQABTp05l7ty5LFiw\ngIKCAp5++mlmz54td1Q0NPog6DIR/nnbuU2hlDspJEmqIXawmStottbsywW2hilfi90TBeliTUMj\nXgx5sUYROFQ0DBMTE0NcXBxGo7G0/aOPPuL+++9n9OjReHh4MH36dJ599tk677d0CSoPsXti//eQ\ne9F6hpFPV26LliRJ0qXoAqDjDXBgiXObxktU7ayV1/VvUgFDCYXd1WpCCZPJhF6vx2g0otPV897g\npi7rtNj7fGQ56INh4H0Q0qlJ/sFJklRPMk7C/DFi2qKEQgET50OHa0t3cBUVFREfH09sbCxarbae\nOlt7Kvr+KnvfaxQjDlI9s9vFPCCIJE0V7VIxZoK5UGR59PSt3PP7t4QB90HP28RWzIaUc16SpKYh\nsDXctUYkYDqxGvxioM9M8PQRH1qMGdCiH+hqafShCZGBg1SxnPNw5FfYtUB83XsmdLwWfCMdzzNl\nw/ndsPYlkcwptAuMehZCOlcuVbRCAZ7eNd59SZKkUv4xMHA29LkDUMKZTfD5KMdCW12nQYf7XG7t\nbwpq4vtqcrsqpBqUmwjf3Qq/Py62MaUdFf//3a2irYTVAkd/g68nQOJusSAo4W+Yf5XYIy1JktSQ\nqD0hPxG+u8WpOqf60BIwGx3WzTUlZrNYJOrhUf1xAzniILmXtA+S9rg4vgeS9pdthcpLgj+ecj7P\nbodVj8KMVeAdWrt9lSRJqor0Yy4LYyltZgJO/UaKZgoAer2+UkkEGwObzUZqaioGgwGlsvrjBjJw\nkFyz2wEFTPxcJC45vxt2LYSiXNF+6Cdof6GSpSkTCrNdP0/GSXGNDBwkSWpIigvcNoWc/RU6X0/K\nqcOg0oKHJyhVFa/vaiRUKhXR0dGXFQzJwEFyVpwPJ9fDiodEASqAloNh4mfw831iAaRXuUCgoop0\nHlpQaWq1u5IkSVUW0sltk6LtVYRue5ngU39j1gaAhw4mfAwRPRt18KBQKFCr1Zc9giIDB8lZ6lFY\ncqvjsdObxJTEiKdgxSOigmUJQwjEjoD4dc7P1WOaY5AhSZLUEHiHwxUPOSeg8w4Xxa+2f4LSbkdr\nShHHl94GszY4LwxvhmTgIDkqyocNb7huy4wHrQ9c94HYylRC5wfXvisKUyXvLzseOwKGPCK3V0qS\n1PB4+sDAB0SQsPl9MZLa6ToI7gArHr4wXVtOQRrkJ8vAARk4SBczG8WiIXdsFpEm+uIStv4tYdpS\nsdvCmA5e4SLzoyGwdvsrSZJUXYZAaD0SovqDrVjUx3mnq9NOi1K2Gk5N3UjJwEFypDFAWHfnmvcl\nAtu6r3vvFSIekiRJDYExEwpzRO0br1D3o59aA2CAYiN0vB4O/uDiHG+ZBv8CmcdBcqQxwNBHyqq6\nlRfWDfyi675PkiRJVWG1QOJe+PZmeK8HfNAHVj0G2Wcrvk6jhxFPiKJU5SkUcO17YiRVkrUq3GnW\ntSrMJji3A357CDJOiG1IHW+AK5+XgYMkSQ1f2jH4ZAhYCh2Ph3YVU6qX2h6elQBHfhOpqf1bQd87\nwT+2cllwG7HK3vdk4OBGsw4cSuSngClH1I/wChGjEZIkSQ2ZpQh+fxJ2fuG6febv0HLgpZ/HbheB\nh1IDKlXN9rGBkkWupMvnFSq3UkqS1LgU5cPZre7bUw5WLnBQKCrOUdOMyTUOkiRJUtOh1kFArPt2\n/1Z115cmSgYOkiRJUtOh0YvETq4YgiG4fd32pwmSgYMkSZLUtAS1h0kLQOdfdiy4A9z+K/i1qL9+\nNRFyjUNzUGyEvGRRiKpkL7LWu757JUmSVDu0XtDpeojqKxLSqTQiNb5XcH33rEmQgUNTl58icrFv\n/0xkfVQoRIKTMa/K1KmSJDVdSpUYXZAjDDVOBg5NmdUKu7+Crf8rO2a3w+GfQaGC694TkbkkSZLk\nzGYDpZzRv5j8iTRleYmw+T3XbYeXiekLSWpGrDY7ZqutvrshNWTGLDi3C355EH6cAXGrIDepvnvV\noMgRh6bMUijytLtit0Fxvuu2wlyRPdLTR+5jlpqEbGMx8WkFfLXtNDlGMxN6RtInxp9wX/n7LZVT\nmAPbP4X1r5YdO7wcWgyAmxaAT0T99a0BkYFDU6YxiF/03ETnNg9PUQ67PFM2JO2Djf+BnHMQPQgG\nz4XANqCSvypSzUrJLSQ9rwizzU6Ql4YwH088VDU/CJpXaOarLad5a3VZ1de1R1PpFO7N57f3JcJP\nBg/SBbmJjkFDibNb4dQG6H5L3fepAZJ3g6bMJwKufAmW3uncNuhBx4ItliI48AOsfLTsWFYCHFoK\nd66G8O613l2p+TiSlMu9X+/idIYRAG+tBy9P6MKoDqF4edbs21JKbqFD0FDicFIea46kMH1gTI2+\nntSInd7kvm3nAmg/XozENnNyjUNT12aU2M/se2FlsT4Qrn4d+s0CtbbsvLwkWP2M8/WWIvjjaVGz\nQpJqwLksI7d8trU0aADIK7Iw97u9nExzM312GfacyXbbtmTHWbKNxTX+mlIjpaigJoVCKXalSXLE\nocnT+UOXGyF6IFhMoNKCd7jzSuH8NLGuwZWEjVCUAzrf2u+v1OQdTcoj22h22fbpxnjemtwdT3XN\nFRWqaPrDQ6lAIW8GUomWg9y39btb5r+5QI44NBc+4SJ/u2+k6+1FHlrnYyW03hVH4pJUBWezjG7b\nTmcWkFdo5lR6PvvOZhOflk9BkeWyXq97C1+UbmKDGYNj8NWpL+v5pSbEO0KMyF6s9Six5ksC5IiD\nVMIrVCyCzDjh3NZvlqySKdWYzhHu54j7xwSwfE8ir646gs0OSgXc0COSx8Z2IMzHs1qvF+bjyRsT\nu/GvH/c7HL+ibRCDWgdV6zmlJsrTG3rcCjFXwMGfxC6LzjdAUDuRcVcCZOAglfAOhSmL4avrHfM7\ntBoOfe+SuyqkGhMTaKBTuDeHk/IcjuvUKsZ2DWfyJ1uw2cUxmx1+2nMevUbF09d0qtYUhl7jwbiu\n4XRv4cfqQylkGYu5snMosUFeBHtXMNImNU+ePhDWVTwklxR2u91e351oiEwmE3q9HqPRiE7XxLZr\nGTMg4yTs/wGwQ9ebxGiDIVBsw8w8JYKHoLbgGwUG+alMqlnns018vP4E3+88R5HFxhVtgnhibAee\nWraffedync7XqJSsfngoLQMN9dBbSaoDxUZRIsBSJIKXesgZUdn7nvwY2dwUZMC6l2Hn/LJjOz6D\nnrfB6OdFoOAbVW/dk5qHSD8dT4/vxL3D2mCz2/HVqUnNK3QZNAAUW20Yi6019vr5hRYSc0xsiEuj\n0GJleLsQovx1+Bs0NfYaknRJOech+zRovERpgIM/gNUsFrBf/Qa0HtEgt3/KwKG5yTzhGDSU2PMV\n9LxVji5Il5SUbWL/+Rw2HksjJtDAqI7ipqvxcD+NUGi2klkgtj0GGDR4qlVo1Soi/cs+1eQVWvDT\nq13uuPDx9MC7hvI75Bda+HH3WZ7/5XDpsbf+PMbMwTHMGhqLzQ4eCgUhPlq540KqPWlx8NUEkVTq\n3HY4tbGsLS8JfpgO05dD7PB666I7jWpXxeuvv05ERAR6vZ7rrruO5GT3tRaGDx+OQqFweLzzzjt1\n19mGat/37tv2LK67fkiNUnKOiT1nszEVW+nXKoDdZ7K4+t2NbDuVifXCwgSbzXH283RGAU8tO8Dw\nN9cz/M31PPPzQc5kFDg9d5ivJ/+6qr3L1507ui3h1VwcebHEHJND0AAwIDaAAbGBzPtuL0PeWMs1\n7//D/E2nSM0rrJHXlCQH+anw/W2Qex4iejgGDeWteV5MLTcwjWbEYcGCBbz88sssWrSI2NhY5s2b\nx5QpU9iwYYPba+bNm8fjjz9e+rWPT8Mb8qlzKg9Rf8JVzga7LP4juZeaW8iXm0+zcHMCJrMVrYeS\nm3pH8dw1nXngmz38cO9Afth1jlPpBYxsH8KQtkGggCmfbCU5t+wG/MOuc2w6kc4P9w4k0l9felyl\nVDCuazg6jYr//B5Hcm4hId5aHrmqPVd2CkFVjXTUhWYr57JMrD6cTGJOISPbh6DXqjBoVBRcmPrw\n1npw5xWtuO/r3VguBD1p+UW89NsRjiTm8uy1nfGRWzalmpSfIkYcFAoodg6iS6UdBXPDC14bTeDw\n/vvvM3fuXG688UYA5s+fT+vWrdm7dy89evRweY3BYCAsrHJbaMxmMxZL2X5xk8lNMqTGqtgoFjy2\nGipWC6t1sO0TOLut7Jwet9Zf/6QGy2azY7fb+WHnWf634WTp8SKLja+3neHW/tEMahPIkaRcPt0Y\nD8Dqwyn0ifFncp8WDkFDicScQnacznIIHAD8DRpu7BXF4NZBFFqsaD1UKBR2NhxLY1t8Jh3DfRjW\nLpgWAXpU7pIzXFBssbLxWBr3fr2rdJfGV1tO0zHMm/9M6s6cb3YDcEPPSBZtOV0aNJT34+7z3Du8\njQwcpJplKRL/tdtBrXd/XmBbUVeogWkUUxVFRUXs27ePkSNHlh6LjY0lJiaGbdu2ub3u008/JSgo\niB49evDWW29htbpfXPXKK6+g1+tLH4GBgTX6PdSrwlzY8zV82Be+mwrL54hHz9sgdoQ4p+sUsVdZ\nki44k2nk662nuefrXfz3zzjah/swuI3z38WPu85xdWfnAD1Ar2FjXJrb599QQVuoryctAw0Yiy1c\n894mHlqyj+92nOW5Xw4x9t2/2X8uu/Rcq82Gq81hyblFPPjdHi6OB44k57HnbBb9WwUA0C7Um70V\npKVOKRf42Gx2ErNNHDiXw5GkXNLyihzOzSgoIjW3kGJLzS3klJogQ3BZwJB6SGT2dWX0c2K3WwPT\nKEYcMjIysNlshISEOBwPDg4mNTXV5TXTpk0jNjaW4OBgtm7dyuOPP052djYvvfSSy/Ofeuoph2kN\nk8nUdIKHjOOw6l+Ox8wm+G0u3PYzDH0UgtqDV3C9dE9qeOLT8rnp4y1kXFjQuBrw+PsUb0zsRmZB\nMUfK5WAosogpLutFN+iMgmJ6tPBz+xptgr0q7EN+oZlXVhwhLd/x5mwyW5n73V6+vrMfR5LzWLr7\nHD6eaqb2i6ZVsAF/vdgZkZpbSJ+WAeSYzBw471hrZcX+JGYMimHbqUyyjMWE+HiS56ZOhs+FRZmm\nYgv/nMjg8aX7Sxd6tg724oOpPQnx1rLhWBqfbownr9DClZ1CmTE4hhi5fVRyxSccrnoJVjwCm96F\nCZ+CPgDiVokpY30gXPUKRPat75661CgCh+qkmrjrrrtK/79r166oVCrmzp3Liy++6HKltFqtRq1u\ngsORVgts/8x1m80KqUeh/6y67ZPUoBUUWfjP73GlQUMJi83OqyuP8K+r2/PE0gOlxz3VSqL89by8\n4ojjE9nhuu4RfLk5wWkawEOp4OouFU8jZhYUszbO9QeDM5lGjibncc9Xu0qP/bjrHA+MbMPdQ2Ip\nKLKQlFNImK8nXSN9eWBkGxZsTmDLSbHQzGqz06ulPyM7hHAms4BZQ1vxeLnvqURMoJ7QC4sy49ML\nmPXVTsq/HZ1My+eWz7by/s09efj7faXHF25O4Jd9iSybPUjmnpCcqTTQZZKYitj4H/jrRRhwL4x4\nSkxf6PxE+mtX5QEagEYROAQFBaFUKp1GF9LS0pxGIdzp3bs3+fn5pKenExzcjD5Z28xiIY47eUl1\n1xepwTEVW0jNK8JYbMVL60GYrydZxmL+POx6x1JGQTH6i7I3zhgUw67Tmew+k1V6rHOED7cNbMnb\nq+N4fWJXXl5xpHSbpb9ezXu39KRFQMWJ1exARZ8ZCs3O0wEfrjvBDT0juevLnZxKL1t05qFU8PrE\nruQXWjhwPocxncNYuOkUfno1SoWSgiIrMwbF8PXWsrUObUK8eGNiNzQeSpJzTJxMzadtiBfHUhxH\nJrKNZuLTCwj21jpMXWQWFPPt9jM8clV71NVY2Ck1cTo/iB0mdlWYC0HrA5rGkWywUQQOWq2W7t27\ns27dOkaNGgXAqVOnSEhIoH///pV6jn379mEwGAgKamZ5CtQ66HgdnFzrur31iLrtj9RgpOQW8t6a\n43y/6yxmqx2DRsX9I9owvls4SoUCsOOl9SC/yFK6TiDKX0e4nydD2gZx4Fw2Mwa1Ymr/aI6n5qNW\nKTBfmK+4a0gsz/x8kLwiC2n5xTw1riNaD3HzDPX1pHcLfzw8Kr6Z+hs0XNEmiH9OpDu1RfrpyHSR\n72FEhxC+3JzgEDSAGC158bfDPH9tZ95Zc5whbYO45+tdDoHJtd3CWXrfIOLTC9ColGTkF1FQZOad\nNcf581AyBq0HE3pG4qdX88zyQ6XbT0Fkwgz20jqteVh9OIW7h8QS6CVTW0tuePqKRyPSKAIHgPvv\nv5+5c+fSu3dvYmNjeeihhxgyZAg9evTg/PnzjBo1ikWLFtGvXz9OnjzJd999x9ixY/H392fbtm08\n+uijzJkzp3kmdGkzCvyiIfuM4/EW/eWCyGbKWGThrT/i+H7XudJjBcVW3vgjDr3Wg4V39CXXZCEj\nv5gAg4a0vCJ8dB7kFZr54u8EYgINPHpVe2KC9PjqNPjp1fwxbyhrj6aSkFZAoEFD3oWqlocScx0K\nTEX4erJszuDSKQB3fDzVPHdtJyZ9vIUcU1mQoFEpefqajvzn9zina0a2D+GN34+6fL5ck4UQby2v\n3tiFB77Z4zSacTQ5j80n03njwvN+Nr03j3y/32GNxX/+iGNE+xDmjGjNe3+VFYRrE+LFkh1nnV7T\nT6+Row1Sk9NoAoc77riDlJQUZs+eTXZ2NqNHj+azz8TcvdlsJi4uDqNRlOvVaDT88ccf/Pe//6Ww\nsJCYmBgeeeQRHn744fr8FuqPXzTc/hvsWgD7vxfza33vgi43yopvzVRKbiFL95x32fbeX8d57tpO\nPPjd3tJj82f05bWVRzieWjZM/9XW07x+Y1eu7xmJTq0iNtiL2AsLHje7GCUoUWixYavkuqW2od78\ncv9g1h1NZfPJDDqG+3BdjwjOZxmdRhUAFAqF0y6K8nQaMR0T4acjq9yIRbCXlsfHduDJn8Q6h4Gx\ngWyNz3RamAmwLi6VSb2j0KiUFFtttAoSaxjKBzcl7h0WK7dySs7ykiEzHhL3gn8MhHUR79ONRKMJ\nHACefPJJnnzySafjMTExDgsoW7RowcaNbjJxNVf+LWHE09D/XpF0xBAi/is1acUWG2qVwmmkLa/I\n4jDUXl5GQbHDzbdvjD9bTmY4BA0lnv75IANiA4kJclwAGOGnc5i6KG9c1zACq1ATomWggRmDWzF9\nYAzKC7kbvLQeDG8fzPqLtnQGe2m4rnsE32w/4/Q8Bo2KEG8tkX46FszoW7rAskWAjkg/HTMW7Cid\nauga5cs/x90HP8dTxXXtw7y5pV80hWZr6bROibuuaEWvaP9Kf59SM5F9VmyLTy5X5l3nD7f/2mgq\ncjaqwEG6TCoPOcLQDNhsds5kGlmxP4ntCZl0DPNmUp8oogP0pfUkfDzVeCgVLpMehXhryS0s+/Q8\ntF0wP+w853QeiLUDJ9LynQKHUF9PnrmmE88uP+RwPNhby91DYiusa+GOslzCp1AfT96c1J34tHxW\nHUzCR6dmbJdwovx1tA/3YV1cKkk5ZfkXFAr47+TuhF2YHgnx8STEx5M+MSKXQ46pmL4xAZzLEqMw\nBUWiboY70QF6HrmqPasPp3DXlztpHezFT7MHkZZXRF6hmTYh3oR4a+Vog+TIXASb3nYMGgBMWfD9\ndJj5O3iH1k/fqkAGDpLUyOUXmvFQKfG8sNshLiWPmz7eUvrpd8OxNL7YdIqv7+pPhK+O1NxCdBoV\nH0ztyUNL9mG6aHfCHVe0cggUbHaoKEmjqzadWsUNPSLpEunL/H9OkZJbyJWdwhjbJYwWARVkyquC\nYG8twd5a+sc65lvx9lTzw70D2XwygzWHU4gO0HNTnxZEB+hQu1mQ6avT8OTYDnQI8+aTjfH8fjCZ\np6/pyOaTznUCPNVKNB5KZi8WmSc1KiWxwQZUCujRwg+DVr6tSm4UpLqvCZQZL2pYNILAQWGvTpKE\nZqCydcklqb6czzKxLi6V5XvP46NTc8fgVrQONvDAt3vYkVC2NdLH04PHru6AQaPibJYJP70aX52a\nNYdTmDm4FXcs3EG2yUyAQcNDo9uRayrmzT+PlV7fs4UfA1sH8tH6k059UKsUrHl4WIW5CootNoot\nVgxajzpfnGyz2R1GKipzfmpeEVabHaUCFm87w4frT5QupDRoVLx6Y1fm/3OKfedy6BXtz+wRrfl1\nXyIHz+fQMkDPnJFtaRfqhbenHG2QLpJ1Gt7t5r79rrUQ1bvu+nORyt73ZGgsSY3Q+SwjUz/fxukM\nY+mxv46k8s1d/R2CBrVKwX9v6s5//ojjRLk1Cn56NW9O6s6SnWdY8cAVGC1WDBoPwnw8WX9R0qU9\nZ7OZOTiGLhE+HEzMLT2uUMB/byob/ndH4yE+odeHqgQNJeeH+ZZ9P/cOi+XGXpGcSM3HoPUg1MeT\n11YeZt+5HIK8NNw9tBWzv95NsdWGj6cHQ9oGczw1D7DTPcoPj4t2VKTlFZFfaEbtoSTEW1utKRup\nEdP5iTT/8euc2zx9G032Xhk4SFIjY7PZ+W1/kkPQUCItvwiFoixx0tWdw/htf5JD0AAiadFrK48w\npW8Liqw22oZ4l7b1jw1k6X0D+b/VxziWnE/rEAORfjo+ua0Px1Pz+P1gMmE+nozrJtYUaNVN9+bn\n5anGy1NdulsE4OUJXZmZmo/NDv/3ZxzFVhsDYgO4Y3ArFm5O4NvtZwjx0TJ7WBuu7BRKkLeWgiIL\nu05n8fwvh4hPL0CnVjFtQDR3XhHrEKhITZynL4x5Bb640rkq5vi3RLbIRkBOVbhRo1MVeclgzASF\nEgxB4lFVxgwouDDfaggUucylZimzoJipn23laHKeU9u0AS05l2lk/TGx2+D/JnfniaUHKLa6Lpn+\nwdSexAZ50SnCueR8rsmMyWxBp/aQi/xcOJ1RwLA31+Oj8+C/k7oze/Fup8WmdwyO4f4RbUjKKeR4\naj4aDyXr49L4cddZbHa4slMob07qhp++8rtMpEbOZoXMU7BrPiRsAv9WMPgBUS9IW3H9ltompyoa\nAosZEnfBz/eJhS8AYd1gwscQ2rlyz2G3Q+phWH4/JIrFWIR3h+s+gNAuDTaXuVR7VEpF6ULIiy3b\nfY4l9wzkSHIuKblFqJQKt0EDiF8vXzdBgY9OLQOGCqiUCrQeSib1imKBi3ocIGpWjOoYyrQvtmG/\nsMj02u4RvHZjN574aT+rD6eQmlckA4fmRKmCoDYw+gUozhdls9WNax1dpe86NpuNl19+mT59+nDV\nVVfxww8/OLSnp6cTGxtb4x1s1LJOwZfXlgUNILbhLBwvFslURvZZWDCuLGgASNoHC8ZCdkKNdldq\nHHwvLIR0pchiw1en5qf7BvHh1J54KBWl5aMv5qdXE+Wvk0Pl1RTspWVi7yjah/mwq9y6kvJsdkjN\nKyqdOrLZYfneRI4m5fLNXf0Z3TGEjPxil9dKTZxKLfI3NLKgAaoQOLz66qu8//77TJo0iW7dunH3\n3XczZ86c0nar1crp05W8GTYHNhvs+RqsLt4UTFlwdpvr66wWyIiHnQvg93+L865+1Xlqojgfjq6s\n+X5LjcKA2ACn6pJdI31YdiGXQEpuEd1b+DGmcxjPXdcZ3UUjFAoFvHBdZ7GNsIoLCCVBqxa1PZQK\nCPFxX4tC4yLl9Lc7zpCSW0RssBfB3rKOhdS4VHqq4ssvv2T+/PmMHz8egNmzZzN+/Hhuv/12FixY\nUGsdbLSsRc5JPspL3g/dJjsfT9wFi64Hs6nsmH8ruP4DkSDEWi6t7dltYJ8jM0A2QyE+nrw6oQv3\nDWvN5pMZtA424KFUcOsX28g1ifwN3loP3p7Sg0FtAlnx4BV8u/0s209l0CrIwMzBrWgT4iVzDlym\nCD8dozqGUGi28sxFya4AWgcbSMoxOR0vNNuwA59ujKd1sBdtQup3bluSqqLSIw7nz5+nU6dOpV/H\nxsayYcMG9u7dy0033YTZ7JynvVnz8IQW/dy3R/ZxPpZ7Hr6/3TFoADHlcXCpqHJZXou+MmhoxgIM\nWrq38OO+4a1pHezFrK92lQYNINJKz/pqJ+eyTMQGe/H41e1ZdGc//jOpO91loqIaE2DQMrZLOLcP\nbOnw5xgbZODp8Z34eINz/osAg6a0LPg7a46RklvodI4kNVSVDhwiIyM5duyYw7GQkBDWrl1LQkIC\nN910U413rlFTKMSIgtpFljyvUIjs5Xw8PxXyklw/3+FfoO2VZV9rDND+mprpq9To/X4w2eXiPJsd\nll8oZuWhUuKr09RbToWmLMhby6Nj2rPm4WEsuWcAvz5wBd/c3Z/EbBPpLtYw3D2kFT/sFNU0k3IK\nsVSwgFWSGppKv4OMGTPG5ZREYGAgf/31F1ar1cVVzZx/K5i5CsJ7lB2LHQEzVriuhGar4M3DZhHb\nOUHsppixUlRVkyTgVIZzpcjKtEk1x9tTTetgL/q3CqRrpC9hvjrGdg3jjYndCL2wBiLC15Nnr+lE\nZoGZ3WeyAWgX6uV2l4wkNUSVHqt86aWXSExMdNnm5+fHX3/9xe7du122N1tKFUT0gGk/galcHgdP\nX9fne4WIVbYmFyu0O4yHqL4we1v1c0FITdaI9sH8uMt1IapRHULquDdSiQCDlsl9ohjcJpBT6QWc\nSM3n+51nOZJUloPj6fGdCPSSCyQlxOL4vESwFIHGC3zC67tHLlU6cPD398ff332JWG9vb4YNG1Yj\nnWpyDIHicSk+ETDhE/j2ZrCXG33QB8Co5yBAbneVXOsR7c+YzqEoFQrOZZk4cD4HEJ9wLy4CJdUt\nhUJBlL8ejUrJroQszmWKNUwtA/U8e00nekb71W8HpYYh+wzs/gq2fQxFueAbJXI9tBktUlU3IDJz\npBv1VuTKbBJZxXZ/CRnHofUoaD8OAlzv25ck64Uy2iv3J7H/fDZtQ7zp3sKXA+eymdQnmugaqkYp\nXT6L1UZqbhHFVht6jQq1SokdOwEGOeLQrOWnwtqXxfv+xW76EjrfUCfdqOx9TwYObtR7dUy7XWy9\n9JAZ5aSK7T+XzZRPtjqUx/bWerDkngF0inAzLSbVq/NZRn7em8hPu8+j8VAwfWAMI9qHyGRczVXy\nAfh0mEhHfTH/GLjjzzopt13Z+55cXt1QKRQyaJAuKS2vkIe/3+cQNIDYivnvZQfIKpBZCRua81km\npn2xnTf/iONkWj5HkvJ48qcDPPL9XtLy5LbMZikvxXXQAJCVIBL+NSByI7ckNWBZBcWk54uhbX+9\nhjAfT4dS0VkFZqfKlyX2ns0hy1iMv0EGoA3J5pPpnEp33umy6WQGJ1ILCPaWow7Njsbgvs0rBFQN\nayqrWoHD9u3b+fzzz4mPj+err74iPDycH374gZYtW9KvXwVJjyRJuqQsYzEZ+UUUmW08/fNB9pzN\nBiDQoOHp8R3p3dKfCD8dHiolyPxfjUqh2cov+1zvTgNYF5dK62ADBq0HhWYrGQXFYIcALw1BcudF\n02UIgnZj4Ngfzm1XPAK+kXXfpwpUeapi6dKljBw5EoVCwT///IPJJFYIp6am8vzzz9d0/ySpWUnJ\nKeSxH/exIyGLe77eVRo0AGQUFPPQ9/s4mJjLgXM52O12Ag0aOoR5u3yuPi39CZCjDQ2KSqnAX+++\n4qhOrWTVwWSeWX6QnQlZjH33b656ZyNTPtnKvrPZMlFUU+UVCkMfhw7XlOXrUetg8ENiK34DyxBc\n5cDhhRde4PPPP+eTTz5BrS77AxgyZAi7du2q0c5JUnNitdr4ZvsZNp3IQOOh5FyWc40DgO93nuW3\n/Ukk5xYS6KXlrcnd8boofbSfXs28K9tyOtNIQaHF5fNIdU+tUjK5TwuXbUoF9GjhT26hmZ92n2fF\ngSQm9BSfNE+m5TPl0y2czTLWZXeluuLpI3bOjXoObl8Bty6FO9fAwPvBz/XvS32qcuBw4sQJBgwY\n4HRcp9ORm5tbI52SpOYoNb+IBZtO4adTk1pB7YLTGUa8PD1IyysCoFO4DysevIIXruvMDT0ieWxM\ne16d0JWnlx3k+g82sTYuFauLdNRS/Yjw03HvsFiHD5FqlYLnr+tMQkYBOy+U6F5xIInh7YNLzyk0\n21i5P7muuyvVhbwU2Pw+fDYS/nwKtnwI8etB0TD/bqu8xqFVq1bs3r2bmJgYh+MrVqxwKIIlVZPV\nDHnJ4r9ab/AKvvQ1UpNgtdnJLbRQZLER4ed+K1SXCF+MRZbSYEChUODt6cH2UxnkF1lZuvs8J9PK\nFkw+s/wgvaL9ifSvh23FkpNgby1tQ7z4ZFpvzmWZUKuUBHppWLE/kRt6RrHxeBogfh8sVscbx95z\n2dhsdocFslIjZy6ELR/A5vfE14kXMjDHrxX/7X8fqBpWSvIqBw7PPPMM9913H8nJydhsNv78809O\nnjzJhx9+yLffflsbfWw+cpNg60ew43MwGyG4PYx7EyL7gkYm8WnqvLQe9G7px67T4ubQKdybw+VS\nEwN4KBVM6BVJtrGYYO+yxXK5JgsrDrj+NJptNJNZUCwDhwbC21PNiA4hHDify46ETLKMxXSN9OO6\nHpE88/NBSjLreKqVTlPbfVr6y6ChqclLgu2fuG7b+CZ0ur7BTVdUeari5ptv5ttvv2Xp0qUYDAYe\nfvhh/vnnH5YsWcL1119fG31sHoxZsOJhEXWaL8xjpsXBoush5VD99k2qE356Dc9e0xkPpYJsk5lH\nx7RnfNdwPC7cKDqEefPOlB589nc8od6ehPmUbdvzUCnQqNz/OWs85M2mIQkwaBnWLpg3J3Vj7qi2\n7D2bxT1f7SK53BTV1H7R/La/rFqul9aDMZ3D6qO7Um0qyhO1KVwpzBbZhBuYKo04mM1m3n//fW6+\n+Wb++uuv2upT85SfDHErnY/b7bDmebh5cYPLVy7VvI7h3vzywBWcyTDyrx/2MaRtCP+9qTsAZ7OM\nvLryCIk5hdw/oo3YjnlBkJeWCT0jWXKhVHN5nSN8HEYnpNpjs9lJzDFxLsuEsdhCTKCBEG8tXp6u\nd1J4earpGO7DlL4tOJ1hJDWvCD+9mvuGtSbIS8P8TQkAdIvy5dGr2vPP8TRsdjvRAXqHf3+pEdN6\ng4fWdfDg6Sd2VzQwVU457eXlxcGDB53WODQ1dZ5y+uR6+MrNiI3WW1TFbGB7eaXak2Ms5oVfD/PT\nnvNObR5KBasfHkarIMekMeezTNzz1U4yC4q5sXcUYT6eZOQXc333CGKCK0gwI9UIq83O3rPZzFy4\nnVyT2MmiUMA9Q1sza2hshVtj7XY7ybmFFJptaD2UhPp4kldoJjHbRHx6AXHJeXy7/Qzp+cVoPZR8\nc3d/ercMqKtvTapN5kJY+5JY53CxK1+GAbPrbI1DraWcvvrqq+VoQ23QV/Am4NcSPGQ2uebEV6/h\n/pFt8NE5Dwo+clV7Qn2cRxAi/XV8Mr03L1zfhXVHU3lj1VH+OZFGWkERBUVyS2ZtS8wxcfv8sqAB\nxIDhxxtOsvdMVoXXKhQKwn11tAoyEOGnQ3Vheuq5Xw5x/zd7eH/tCdLzRfrwIouNh7/fV+HOG6kR\nUXvCoAdh8Lyy0QWtt9ia2f3mBrcwEqqxOLJfv3488cQTbNmyhR49eqDXOy7au+OOO2qsc82KdxiE\ndoWUA85tI5+qXFluqUlpFWTglzlX8O32M6yPSyPYW8N9w9vQOcIHvcb5T9ditfHP8QweX7q/9NiO\nhCxu+ngLC2b2ZUT7kLrsfrNzIiWffDcB2kfrT9KnVQA+bqYsXMkymtmR4DrgOJ1hJLOgmBAf+YGi\nSfAOhRH/hj4zodgoUlD7RICq8r8vdanKgcP//vc/vLy8+Ouvv5xGHhQKhQwcqssrBG7+Gn6eDac3\niWMaLxj1LLRwzpshNX0KhYKYIANzR7Vlav9oFIBeq8JUbMViLSLoonULKbmFvLLysMvnem75ITrf\n6yNvNLUos8DNAjdEGnGzpWpZHy+5nLXcCTabnZS8QixWOzqNSqanbkhM2VCQDthBH+h+dNlDKyph\nNgJVDhxOnTpVG/2olNdff5333nuP7OxsRo8ezaeffkpYmOtVxvn5+TzwwAMsXboUtVrN9OnTefPN\nN/HwaMB1vfxjYMrXoja72SR+wRpw1CnVvvNZRv634SQ/7DxHsdXG0LbB3D4ohi83nWJCr0iGtA0m\n8MJNIttkdhgmL+9MppG8QgshPnXZ++alY7j7EuajOoS6nHaqSIBBw6whregS6YvNLnbOJOUU8vnf\n8Xhp1QQaxL97Wl4RP+89z0frTpBlNNM2xItnr+1Ezxb+eHk24Pe75iAtDn6dB2c2i68jesG170Jo\nF1A23sWtjea3asGCBbz88sssWrSI2NhY5s2bx5QpU9iwYYPL8+fMmcP27dtZvXo1BQUFTJs2DW9v\nb1588cU67nkV6QMqXu9wMasFsk/DybWQfgxiroDI3uAbVXt9lOpESm4hMxfu4FhKWTKnDcfS2H06\ni7dv7sFdX+5kzog2zBnRGr3GA08P93OhWg8lGo/G+0bVGIT7enJNt3CHLZQAvjo1UwdEo67GXHXL\nQANP/HQAY7EoudwqyMBrE7oS4acj2FuLscjCB2uP8+WW06XXHE/N57YvtrP4rv4MbhN0ed+UVH1Z\np2HBWDBmlB1L3C2O3fM3BMbWX98uU5V3VUyfPr3C9kWLFl1Wh9zp1asXY8eO5ZVXXgEgPj6e1q1b\ns2fPHnr06OFwblZWFsHBwaxatYorr7wSgPnz5/PYY4+RkpKCqhJ/wHW+q6K6zmwTuzHK7/X1i4bp\nv4jc51KDlZxj4kymkbOZJloFGYgK0BFSrqTy5hPpTP18m8trZw6O4WhSHjsSMlnz8DBiggzkmsw8\n+O0e1h9Lczp/2oBonhnfCa264S20akpScwtZF5fKpxtPkVdo5spOodw1JJaYQD2KCgoVFZqtZBaI\nxY/+ejU6jQfb4jOY8ulWp3ODvbT8PGcQkf56TqXnM/r/NrpMKd4hzJvFd/UvHZGS6ti+72DZPa7b\nRj0PQx6q0+5URmXve1Uecbj4pms2mzlw4AAJCQnceOONVe9pJRQVFbFv3z7efPPN0mOxsbHExMSw\nbds2p8Bh165d2O12hg8fXnps1KhRZGRkcOLECdq3b+/0GmazGYulbJi3pOpng5abCD/OcE4Qkn0G\n1r0G174jM042UKfSC5g+fxtnM8v+7TqFe/PZ9D5E+ot/s90VrMTffy6HjuE+bInPICO/iJggAz46\nNS9P6MLsr3ex/3xZ3ZhhbYOZM6KNDBrqQIiPJ1P6RjOqYygWqx1/vfqSP/ezmUY+WHuCn/eex2a3\nc023cB4Y2ZbF2864PD8tv4iEjAIi/fVkFZjd1iE5mpyHyWy97O9JqqazOypo2wLWB0B10S3YbAKl\npkHupCivyoHDggULXB5/6qmnqOLgRaVlZGRgs9kICXFcFR4cHExqaqrT+ampqfj5+TlU7wwODi5t\ncxU4vPLKK7zwwgs13PNalp8qggdXDi2FUU+DJrpu+yRdUraxmMd/3OcQNAAcTsrjP3/E8dqNXdFr\nPGgZ6D73QoSvZ+liPF25HRZR/nrmz+xHam4hWUYzQV5aQry1+Mvy2nWqsosTz2UZueWzrQ6VUJft\nSeSfExm8OqELv+xz/fedlCO2Yla0hiHYW4taJomqPxE9KmjrJYKGogKwmKAoF47/CXG/g28L6HcX\nBLQGrVeddbcqauy3aubMmXz88cc19XQOqhqQuDq/omFCEIGP0WgsfWRkZFR4foNQ0c/FXrUV3FLd\nSc8vYruLbXbdo3wJ9/UkPV8EBN1b+DqVywaRVOia7hH8dSSV9qHehFy0uyLIS0unCF8GtwmifZi3\nDBoasF0JWS7Lp6flFZGQYaRtiOsbR9sQb0BMW/SL8Xd5zoMj2zj9bkh1KOYK8HSxYNbDE7pOhnM7\nYdndkLQP5l8Nqx6H+HWwZxF8MlRkErYU132/K6HGAoc1a9Y45XSoKUFBQSiVSqfRhbS0NKdRCIDQ\n0FCys7Mxm82lx0qudXU+gFqtRqfTOTwaPK9gsb3HlfbjQCczyzVExRdty/PRefDBLT0Z2i6Yv4+n\nM++7vfy6LxFPDxXf3N2fCN+ydQ+h3loWzOiLp1rJuzf35LPpvZ22ZUqNQ46pmI0nnNejlNhxKpOX\nb+jCR7f2Yvbw1hg0Yvi6b4w/kReqp/obNPzflB4MbF32PqBWKbhvWCxju4Zf8gOTVIv8Y2DGCrGD\nokRQW7jrL0g9DJ+PApUW9n0L+SnO1//6oCiA1QBVeapiyJAhDr+Mdrud5ORk4uPjefvtt2u0cyW0\nWi3du3dn3bp1jBo1ChDbQhMSEujfv7/T+b169UKhULBhwwZGjx4NwNq1awkMDKRNmza10sd64R0B\nN34K30wGW7m5TH0AjH6uwQ5zNXd+eg3BXlrSLowsvHhdF95ec4yTaQWl5+w+s4erO4fyyoSu/DR7\nMOn5RdjtdjQeSt74PY51cSIQHtImiGeu7VT6CVRqPHJNFkK93efVCPP15J01x9kSn0G/VgG8e0tP\n9p7O5taB0Q7BYpS/nv/d2ovUvCKMRVb89GrCfD3xlGta6pdCAWFdYfpyKEgTI8SGQLET7usJ4pxW\nQ+HPp11fbzaJwMG/Zd31uZKqHDiMGjXKIXBQKpUEBwczZMgQOnfuXKOdK+/+++9n7ty59O7dm9jY\nWB566CGGDBlCjx49OH/+PKNGjWLRokX069ePgIAApk6dyty5c1mwYAEFBQU8/fTTzJ49u1I7KhoN\npRJaXgH3bYY9iyE9DlqPhLZj5I6KBizc15OXbujCvV/vol2oF+ezTQ5BQ4nfD6Vwz7DW9Iz2J8zX\nk3NZRiZ8uLk04ADYeDydmz7ewi9zBhNdwZoIqeGx2u30jPZDo1JSbHUchVIpFVzRJoivt4ptlttP\nZWK12fhkWm+CXAQbfnoNfno5JdUgGYLEo0TyQbE+DcBurThPj7Jh3q+qHDg8//zztdCNS7vjjjtI\nSUlh9uzZpQmgPvvsM0DsiIiLi8NoNJae/9FHH3H//fczevRoPDw8mD59Os8++2y99L1WqT0huANc\n9RJYzTJZVCOgUCgY2jaIn+cMYvfpLP487GKY8oK/j6fTM1rMYe9IyHQIGrpF+TKqQwhKpYLDSbky\ncGhkvLUeeGk9+P6egaw8mMTiracpKLbiq1Pz73EdWbztDOU3TOw6nU16QbHLwEFqRFTlAry4ldBl\nIuz43Pk8nT94Ncwy6tXajpmUlOS0ViAjI4OQkBCs1trb/vPkk0/y5JNPOh2PiYlxWhDp5eXFwoUL\nWbhwYa31p8GRQUOjodd60KOFP7FBBraczHR7nr++7N9064XzPNVKXruxG/Fp+Szfm4jFZmdc1zC6\nRPoS5S+33zYG8Wn5vLryCH8dFZ88h7UN5ttZAyi22EjIKODLzac5cD7H6TpZrKwJ8AoWI8Wn/xGJ\n+25aBKc3i3UPJVQamLxIZA5ugKocONjtdpcLbvLz8/H0lJGwJFWFj07DzCtiWH3EedRBqYBBrcuG\nODuEi3UMT4ztyMJNp9h3ruzG8vGGeP44lMLXd/UvXTgnNUxnM41M/mRLabVLgPXH0jhwPodvZw1g\nwaYEDiXmOl2nVIC/nI5o/HT+cP378PVEyIyH5bPhyhfBQwepR8Q0c6uh4Bvd+KcqSopXKRQKHnzw\nQYddB1arlT179tC3b9+a76EkNXEdwny4Z2gsn2yMLz2mUip4Z0oPIvzL/s5GtA/hkw0nMWhUDkFD\niVPpBWyPz2BCL5luvCHbFp/hEDSUyCgoZlt8Bi9d35kb/7fFqX1K3xZye2VTERArdlxkJYiSAQGx\n4N8KetxS3z2rlEoHDiVbG+12OxaLxWGro1qtZuLEidx9990138PmzJQtfqkOLRMrbDtPgMDWYAiu\n755JNSjAoGHOiDZM7B3FnjNZ6NQqukX5EebjuDI+yl/Hojv7s3CT+0Jzv+xL4pruETLxTwO26aT7\nHDFb4zPQaVR8dGsv3l59jOOp+fjq1Nw7rDXX9wgnIcPI0t3nyDYWc133SDqF+xDqK0d6GyWfCPFo\nOai+e1JllQ4cvvrqKwDatGnDo48+isEgF2LVKlM2bP0YNrxWdmzbx9D5Rhj7hijDLTUZPjo1Pjo1\n7ULdb6v0UClpF+pNiwD3f3sh3lpUcu9+g9Yx3Idle867bGvhr0etUnIqPZ+HRrelU4QvGg8l3p4e\n/LDrHC/+WjYPvmxPIr2j/fjg1l6E+8rpqWbHlA1FeWI6wyu0Tqc1qvyx5LnnnpNBQ13IOesYNJQ4\n9BMk76/7/kj1IinHxJ4zWWw6kU58Wj75RRau7BSC0k1scOuAaJTuGqUG4cpOoWhdVCrVeijpHRPA\n3O/2YrXBVxe2Ykb46UjNK+Kl3w47XbPrTDYbXRQ1k5owSzEk7oXvboV3u8LHg+Gfd+o0WVSVF0fa\nbDY+/fRTfvzxR86ePeswZQGiaqVUA+J+d9+24wuIGQoecqFUU3Y4MYcZC3aQmie2YCoVMGtoa+4Y\nHMMHU3sx77u9pfv/VUoFz13biZggGdQ3dC38dSy+qz9zv9vL+WyRbjrC15Mnx3Xkf+tPAvD5P/E8\nM74TBcViF8XeM9luM8x/vfUMYzqHyTwOzUXGcfhitNh+D2DMhLUvQtJeuPZdkQCwllUrj8MXX3zB\ngw8+yPPPP88TTzzBmTNnWL58ucutklI12SrYdmUzAxfeRUxZoFC6zokuNVqJ2SZu+2I7GQVli+hs\ndvh4w0l6tPBldMcQVj88lPi0Aiw2G21DvAnx0aLXVPlPWqpBGflFpOUVUVBsJchLQ4i31qEIGYgp\npz4xAXx5R18OX9g9kWU08/bqY8Sni0RguSYLrYIM+OkuvcVazkw1I0X5sP71sqChvCO/wPAnGmbg\nsGjRIubPn8+YMWN4+eWXufXWW2nTpg19+/ZlzZo1tdHH5qn91bDhdddtfe+GgnRRTW33IpFBsu8s\niB0K3uF120+pVpzNMjoEDeV9sO4E/VsF0jLQUGEFTaluJWQUMPvr3RxOEsGAqBnRmhmDYwgwOO+G\n0KhU/OvH/RRZnAvSqVUKknMK8b0QOPSI9kOhcF3XbtqAlnK0obkoyoMzzjtuSmWfgdDay+Bcospr\nHNLT0+nQoQMAvr6+ZGaKpDRjxozhjz/+qNneNWd+0dBvlvPx2JEQ1A6+uwV+mweJuy9UWZsFy2ZD\nnvsshFLjkeUmaADIyC/GbJXVTxuStLxC7vpyZ2nQAGC22nlv7Qn+Pp7u8ppQHy03923hsu3a7hGs\nPpLCe2uPYyy2EObjyTPXdHI6r09Lf4a2lbusmg0PLfhEum93V/SwprtR1Qvatm3LyZMnadmyJZ07\nd2bhwoV06NCB7777Dn9/1+VdpWrQB8KwJ8Quil0LwFIIPW+D8B5wZrMoxXqx+LWQcQy8Q+u8u1LN\nah/mzXs398BmB7VKyd6z2Xy99TQms5UhbYJKP4mWdzbTyJb4DHYlZNE50oehbYOJDtDLxZJ1IDG7\nkBOp+S7b3vvrOINaBxF8UQ4GrVrF7OFtyDaa+WV/Ina7mHa4qlMYQ9oG8+gP+9CpVeSYzIT76pjU\nK4r+rQJYtuc8mfnFXNcjgo7hPoT6yO2YzYY+AIY9Lj44XswvuuKgogZVOXB44IEHOHVK7CN/7rnn\nuOaaa/jkk09Qq9V8+umnNd7BZs0QCIaBED0A7Dax3cZmFWVY3Tn8G8QMqbs+SjUus6CIlfuTeX/d\ncQrNYmRhYOtA3r25B3vOZDG+WwTL9pxHr7mQ78HXk3NZJiZ/soXMkpGKnWDQqPhu1kC6Rjmvf8nI\nL8Jss+Pr6eE0By9VXmpuIWsOp2DwdP8zTMwudDtCFOrryewRrRnTJYxiiw2Nh5LNJzP41w/7sNrs\n+OrUpdtrfXRqOut86Rwh1zM1a9H9YeSzYirbeuHvPagtTPkGfBto4FCSQRJg0KBBnDlzhri4OKKj\nowkOlkNmNSYvWWyvMWWDd5godqL3BxSgq2Dxi06O+jR2G4+l8+afcQ7HtpzMYHzXcIzFVq55/5/S\n4yqlgvdv7sm+c9llQcMFBcVW5i3Zy3ezBpR+2k3PK2Lj8TQ+Wn+SrIJihrQN4v6RbWkVZEAlRyaq\nJKugmJd+O8yv+5P45Lbebs/rGe2Hl9b9W623p5rnfjlEWl6RU9tdQ1o5jVRITVReMqQfg1MbxchB\nq6Hg28J595w+EAbOhi43iiqbap3I41CHI83V/qiRmJhIfHw8ffv2pXdv9380UjVkxsO3N0NauZtH\nx+tg3H/E4sfeM2HvYtfXdrqubvoo1Yq0vELeWu0YNET56xjcOghvTw++3HLaoc1qs/PAd3v43629\nXD7fybR8MguKCPbWkmcy8+5fx0vzAwD8vDeRPw6l8OsDg2kT4j75lOQsJbeQX/eLvfObLwR2Kw44\n7qVXKuDxqzvgU8HuiHBfTxbM6Mu0L7aRbSxbLT+mUyjju4a7rA0kNTE55+H76XB+Z9kxlRqm/iBG\nkFUX3arVOlHTIqBV3fbzgiovjszMzGTcuHFERUUxbNgwzp8XGdDuuecennjiiRrvYLNTkAE/3uEY\nNIDYarNjPlgtENgGRjzl2K5QiD28vq4XW0mNg9lq51yW2Ntv0Kh4Y2I37ryiFe3DvPhpt+tsg1ab\nneOp+cQEuq6MWbISPyWvyCFoKGEyW/lw3QlMxbLyYlUcSylb0/DVlgQGtwli9vDWBBjEJ8SeLfz4\n8b5BtA+rOCBTKBR0jvDhtweu4Os7+/HOlB6sfHAIr03sRohcv9D02ayw7zvHoAHElsvvb4O8xPrp\nVwWqPOLw4IMPolKpOHfuHO3bty89PmnSJObOncvrr7vZQihVTn4KJO5x3bbtY+h9O/hGQf97odP1\ncHYbKD0gqi94R4BWbs9rzDzVSjqF+3AoMZdXJnTl4w0nOZqcx20DWpJfQUllk9nqUNeiREygngAv\ncSM7m2l0e/3GY+nkmCxyvUMVBHqVjSLY7PDvZQfo09KfR65sR4iPlq6RvoSVSwWdVVDMuWwTfx9P\nw9NDxZC2QYT76vDy9EChUBDlr5dl0ZujgjTY+bnrtqI8MRrhF123fbqEKr9L/PHHH2zYsIGICMc6\n4W3atOH0aedPM1IVmd2/uVOUW5YYytNHPILbuz9fanQCDFqeHt+RV1YeISGjgKPJeQDsOp3FmM6h\n7Dqd5fK6ke1DWLb7HIPbBKJSKNh7Npsii43/m9yDEG/xqdW7ggV8AQYNapUcEq+KFgEG/PRqh+mF\nnaez2Hk6i8+m93YIGtLzi/jvH3F8t+Osw3O8dH1nJvSMwquCfxupibPbwOK8vqWUzUWyp3pW5akK\ni8X1p57z58/j5eV12R1q9gxB4OFmeDK8B2jkPHRT1zXSj1dv6Mqfh8pychxOyiU22MvldMR13SMI\n9/XkvVt60jrYizBfT96a3J3f5w6hW7kdFRF+OoK9XC+0mz2iNYFu2iRHJrOVw4k5fLkpnrdu6k6g\noWzxmlIBs4e3pndLxwXMR5JynYIGgGeWHyIpx1TrfZYaMH0gdJnouk2laZDTz1UOc8eNG8frr7/O\nwoULATE/l56ezpNPPsl118mFeZfNO1ysX1j9jONxhVJUxTTUTYIPqf54eXrQIkDvlEr4meUHeeWG\nrhxJzmXd0VQ81SruuqIV3Vv48dXW03x0oc4BwPc7zzG2SxgvXt+Z4AsjDhF+Or68oy+3z99BWn7Z\nJ5zbBrRkSBu5I6qytsdnMnPhdmx2WBuXzmNXd0CvUaHXqGgVbCDE29NhF0WxxcaXmxPcPt/6uDTa\nVlAVVWriPLQwcA4c+dW5UNXY/4hddQ2Mwm53VzrFtfT0dCZNmsSxY8dITU2lbdu2nDlzhq5du/L7\n77/j5+dXS12tWyaTCb1ej9FoRKer45K1xgw4sxXWvSqqZEb2hpHPQHBH0Mjyuc3Fkh1neXypcyXU\nni38eHtKdwK8tPh4qjmUmMP49/5x8QywcGZfhrd3LMGemG0iKaeQ3EIzLfz1hHhrK1z1L5VJzink\nxo82kZhT6HBcqYC2IV58fZfY+lpssZJpNKMEvLQezPpqF/+ccJ1B8uHRbXlwdLs66L3UoGWdhmOr\n4OgKsV6t/z0Q2BY86y6orOx9r8ojDkFBQaxfv56///6bAwcOkJ+fT/fu3bnqqqvktqGaog+EDuOh\nxQCRMVLrLdYzSM3KsHZBDIgNYGt8psPxW/pHE+rjWbqQ8a/D7tOMf731NINaB6EpV8Y5wk9HhJ8M\nQKsj21jsFDSAWBwZl5JPjqkYk9nKgk2n+HVfIhqVkjkj2zCxd5TbwGFoOznaIwH+LcWi957TxVZM\nVcMN5isdOAwaNIiVK1eWjiicO3eO22+/HYNBruKvNXJaolkL89Xx3s09OZGaz++Hkgk0aLi6SzhR\n/jqH3Q8VDRlWaThRuiS1yv2yMIVCBBCTP95Ccm5ZcPHUsoMsuqMv3aJ82X8ux+GacV3DyDaZsVht\neFTw3FIzomn4O2sqHThs3bqV4uKyzHT33HMP/fv3JzY2tlY6JkkShPh4EuLjyaA2QW7PGdUxlLfX\nHHfZNrVftMNogzsFRRYSs01sOJZGQZGFoe1EnQu5YNJRkLfW5SgQwOgOoSRmmxyChhKpecXcPjCG\nLGMxfx1NReuh5OouYWTmF/PAt3tYNXeI3IopNRrV3gNUxaURkiTVkkg/HXcMjmH+pgSH40PaBhHo\npaHYYkXj4ZzjoURBkYXf9ify+NIDpcfeXnOca7qF89y1nUoXV9a3zIIiEtKN/H4oGa2HkrFdwojy\n19fp+gxfnZpXJ3Rl+vztpYm6QOTLeHp8R5779ZDL61RKBQ8t2UvLQD0DYgOxWG28+Xtcaen0/EKZ\nfKvBK8wVixfP7RC7HSJ7ibUIjWCEoKbJzcOS1Mj56dWMaB9CjxZ+bDiWhtlqZ0jbILKMZqZ+uo1V\n84bQMtD9lGJybiHPLj9EiLeWHJOZIosoyPTb/iSu7xHJlZ3qP3BIzyvilRVHWLa3LHvm+2tP8OhV\n7Zg+MOayg4e8QjOpeUWczzLhpfUg3M+TcF/HdSCpuYVsO5XJsj3neXJsBzQeSpJyCokJMqAE9p7N\nxs9NP+x2O55qJaczjJzOcMzVolEp0WvcB3ZSA2DMhG2fwobXyo4plHD9R9DxWtA2r1QEVQoc3njj\njdI1DcXFxbz99ttOpbRffPHFmuuddHnsdsg+A+d3QdpRiOgFYV1E5kmpycgoKOb5Xw+TlGOiX6sA\nPJQKXl15hKwLiYlOpuVXGDhkFhTz1k3dScwpJMRbS36RhbdXHyOjoJgFm04xqHUghgqKNNWFw0m5\nDkFDif/+eYzRHUMvK3BIzy/ig7XH+XLL6dL03CHeWhbe0Y9O4WJRcnpeEf9edoA1R1IBWHs0lQdH\ntaHYYueNVUcpKLaiU6v4703d+Xmvc4rgPw8lM21ASz7/+5RT2y39WsjU0g1d+jHHoAFE4qaf74WI\nnhDSQRwzZonsvwVpoPMTxQm9mt7i10q/GwwdOpTdu3eXfj1o0CAOHjzocI7cVdHApByCL68BU7ls\ng37RMP2XeiuOItU8ux0sNhvGYivr49Kc2i1W99OK57NMvPDLIQ4m5pYeaxmo5z+TuvHQkr0UWWzY\n6nlasthi46utCW7b1xxJoUN49XcdbTqRzsLNjllvU/OKuH3+dpbPGUyEn46EjILSoAFAp1bRJtiL\nB7/bW3rMZLay8Xgaj1zVjvf+Oo75ws/doFFxZecwdGoVN/Uxs2z3eSw2Oyqlgsm9o5gzoo3LdOFS\nA2G1wA43KaEBjv0uAofcJFj5KBz9rawtvAdMXiR2TDQhlQ4c1q9fX4vdkGpcfiosvdMxaAAxAvHn\nMzDhE1nXookIMGiY1CuKt1Yfc2pTKqBtaNkwqsVqIzm3kNRckQAqr9CM7aK44HSGkY/Wn2Rq/2hi\ng73w9qz/bWHmCoIfV23JOYUcT8lj37lsYoO96BrpS5S/zunDTUZ+ER+uO+HyedPyikjMNhHhp2Pt\n0VSHtj4x/vztYnvlkh1nGdUxhEV39CPbaKbQYkMBzN90igPnc7iuewTvTOlBmK8ngV4aQr090dfz\naI50CXar8/toeaZMEVxs/cgxaABI2gvL74cpi0Dn7/Lyxkj+xjZVBWliesKVuBWiXQYOTYJKqeDG\nXpH8tOc8p9ILHNqevaZTaa0KY7GFjcfSefSHfaUFs/z0ap67tjPfbj/D9lNlOwV2nc7ivmGxdI4Q\nKatzTWZScgs5lpKPl6eK2CAvwn0962QLocZDyS39ol2OpgCM7hjq8PXZTCPT5293+Fn46Dz47u4B\ndIrwdTjXbLWTWVCMO7mFYrrH96KpEKVCgbuBmL+OpOKn13Dv0FiufvdvrOUis+V7E4lPy2f+jH4E\ne8sdK42Chxa63Qwn1rhubzdWLJp0NyqRsFF8kJOBg1SvCnOgMA+USjGHpnTx5m2zur/ebhNRtNRk\nRPrr+fqu/myLz+CXfYkEe2uZ1r8lrYIMpesTzmYauW/xLocbXrbRzOM/7ueDqT3ZfiqTVkEGpg9s\nSYBBQ4iPJ1q1koz8Ij5ad5IvNpXNz+s1Kr64vS99Y/xrPXiwWG10i/RlcOsgNp10/JR/a/9o/PUe\n7DubzbI954ny9+Tg+VynACrXZGHON3v4/p4BDrtEfHUeDG4TxHIX6xIAogPEivmRHUJ4bVVZIL77\ndBYv3tCFH3edc3ndTb2iCPfV8eHUXjz3y0FScotQKuDKTmE8e01HGTQ0Ni0HQUhnSL1o10ybKyGw\ntahiWVGBworaGiEZODQmVgukHoY1z0H8etAHwID7ocfNosZFeYZAcezi3OcALfqLa6UmJdJPx429\norimWwQqpQKV0nFY/vud51x+Si622jhwPofJfaIY3CaIN/+IK91qOKRtELf2j3YIGgDCfDz5dvtp\nYoP1hPpULQtlscVKXqEFrYcSLxfTIKZiC0UWG3a7nWMp+Szacpois5Wnx3fgTJaJpbvOofFQMrlv\nC1r46/hlXxL/+SMOgP+b3J0VB1z8zgOn0gtIyyt2CBx0Gg/mjmqLArDY7Gw5mVG6RXJqv+jS0ZoI\nPx1vTurG40v3Y7NDXpGFs5lGxncNY8WBZIfXuaZbOG1CvPDy9GBM51B6tPAlx2RG66EiyEvj8nuW\nGjjfSLj1e4hbCbu/Elkd+90DscPA60JK96C2kO4in4rGCzx0YC4EddNYBCsDh8Yk4zh8MbqsBGtB\nOvz1PCTugmvfcwwGfCLhho9g8STH0QeNAcb9t0kNm0mOXCV8slhtTp/Cy0vLK+K6HhHMXLDDYc1A\n62AvvvinLGgI8tLwzDWdSMopZPfpLD7ZEM+Uvi2IDtA7ZLN0xWqzk5BRwJebE9h0IoNwXy33DW9D\nhK8nOSYLvnoPMvKL+ezveM5nm+jTMoAr2gZxLCWPzhG+/HE4hW+2naFfqwAsNjv3L97N/6b1dljb\noVQoKlwPYbXZLvq+CzmXZSpdBPr41R2w2GxoVEqGtQ8p3a1h0Howvls4vVv6s+eMKFneJ8afm3pH\ncWv/lny74wx2O9zSL5p2oV4EXRhRUCgUhPnqHEpsS42UbxT0mwVdJomtmDq/sjavEBj7Jnw9Aafo\nfND98MeTMGA2tB7leoS4kWkUgcOCBQt46aWXSEpKol+/fnz22We0a+e+KMyMGTP48ssvHY7NnTuX\nd955p5Z7WouKC2DDm67rth/5FYY/6TyKED0I7t0EO7+AlMMQPRB63AL+ckdFc+OhUjK6Y4jTIr8S\nV3cJY9WBZKebrl6jIsck5vlVSgWv3diN5385xPnssuRHCzYn8OHUXlzZMRR1BVkqT6blc8OHmzAW\nW0u//udEBvNGt6XIbMNXr+b1ctMBB8/n8uOuc7xzcw/sdjt3L9oFiPwSAP56NcdT8x3WEBxMzKF/\nqwC2nXLO7Bhg0DhkwkzPL+KFXw+XPh/AH4dSuKpTKK9M6OI0naDXeBAb7EVssOOe/XA/HQNiRXp4\npVLuLGvy3I3WRvWDmb/D6mch5SD4tYR+d0NmPJxcC0n74J6NTWI7fIMPfdauXcusWbN48skn2bFj\nB2FhYYwfP94h/bUrEydOJCkpqfTx0ksv1VGPa0lRHpzZ7L4903l/OGpPCOkIY16HqT/AiH9DYBtQ\nyq1fzdGQtsGEuJhbbxVkoF2oF/EuRiT2nctmUGuR7npUhxD+PJTsEDSA+ID16A/7XKZaLpFfaObN\n3+NKg4by/rf+JGO7hvGui7TZ+UUWNh5L41C57aKlr4vYNVLeDzvPcdeQWAwXJVRSKOCNid0IK5cv\nIT6twCFoKPHn4RSOp+S7/V5cUSoVMmho7rQGERRED4CrXoFO18Omd2Hz+6LdmCEWSTYBDX7E4YMP\nPmDy5MncfffdAMyfP5/g4GBWrlzJDTfc4PY6T09PwsIaXh3zalNp3K9ZAFFR0+21HuIhNWstAvT8\ncO9APlx3kuV7z6NSKripTxR3D4klzEdHrxb+bDmZ4XDN5pMZzBgUw2/7ExnaLpj//O56p46x2Epy\nbiEtAlyn380xmVkX5/pNs8hiI8dkxmR2vWB3/7kchrR1rtWRbTQT6uOJh1KB5cKoQ47JzFt/xvHe\nLT3ZGp/B7jPZtA42MGNQDK2CvRxu7sv3OCeUKvHjrnMV1geRJJfsdrEt02p23d5Ech01+BGH7du3\nM3LkyNKvDQYD/fv3Z9u2bRVet3LlSoKDg+ncuTNPPfUUJpOpwvPNZjMmk8nh0aDoA2DYY67bfCLA\nr0Xd9kdqlFoGGnjx+s6se3Q4fz08jKfGdSTKX49SqWBS70h0FyUistvh173n+frO/kT4elaY5E1Z\nQZvZasPL033wWlHVyTOZRga1dh0YL9lxluev6+xw7GhyHu+vOc7dQ1qxcGZfXr6hC50ifJ2+Nw8P\n9/1VV9AmSW7pA8UaCFe8w8AQUrf9qSUNPnBITU0lJMTxhx0cHExqqvshn7Fjx7J48WLWrVvHM888\nw1dffcWsWbMqfJ1XXnkFvV5f+ggMbIAlraP6wchnHeu0B8TCtGVNYt5MqhueahURfjrC/XQOxa+i\nAw18f88AukaKLIwKBYxoH8yDo9tx0ydb2H06ixt6RLp8Tm+tB6E+7rcYmq12JvZy/Tsa4euJSqkg\n3Nf1ivNxXcKITyvguu4RTm1ZxmJigwx8eltvZgyKYULPSD6b3puPp/chxEeHt6fabYEvd98LwOQ+\nMhCXqkGjh2FPgO9Fvz8eWpg4X3zIawIU9noqc3nvvffyySefuG0fNmwY69evR6PRsHTpUq699trS\ntsmTJ+Pt7c0XX3xRqddau3Yto0aNIjU1leBg13nDzWYzFktZhTqTyURgYCBGoxGdrgGtiDabxHRF\nfqrY4uMdKiJZSaohmQVFZBvNKBUKAr007EzIYubCHSgV8PG03ry+6qjDegiVUsHn0/swpG2Q25wO\nR5NyOZqcxzcXJZry06t5c1I3PtsYz51DYvnXD/vILVcpsnuULy9c15k/j6RwXbcIUvOK+GprAkVm\nGzf3jaZXS3/UKgVZRjN6jQo/nQd6beW2O2Ybi/lw3Qk+u6h+xO0DWzJ3dDsCDJqq/NgkqUz2WTi/\nG05tFNs0244Gv5gGP2VsMpnQ6/WXvO/V23fx+uuv8/TTT7tt12rFp5eQkBCn0YW0tDRat25d6dfq\n3bs3AAkJCW4DB7VajVrdCPZXq3VilCEgtuafuyADivNB6SESS6nkIsrmKMCgJcBQNnqQUSB28tjs\n8OiP+3hmfCdMZiv7zmYT6uPJDT0jiQnUV5gIyk+v5ot/4hnTOZzpA1tyOsNIdICeCD9P3vz9KNsT\nslAqTvHVnf1JyjGRmF1I1yhfYgINBHtr6REttg93CIcBsYHY7HaH+g7ld0tUlp9ew+zhbbiueySr\nDiVht9kZ2zWcFv56/GXQIF0Ovxbi0fn6+u5Jrai3wMHPzw8/P79LntevXz/WrVvHnXfeCYDRaGTb\ntm3MnTu30q+1b98+AGJiYqrT1aav2ATJ+2DV4yK3uqef2HPc+3Y5miGVVogEkYHxXz/uJ8zHk3ah\nXnSJ8KVVkMHtGgVTsYVjKfks2BTPo1e157Gl+8kqMBPioyXXZGZou2DemNQdBQq0aiWhPp50b+FX\nYX9c5amoLn+DBn+Dhq5Rvpc+WZIkoBHsqpgzZw5XX301I0aMYMCAAbz00ktEREQwbty40nM6dOjA\na6+9xoQJE8jPz+ell15i4sSJhISEcPDgQebNm8fkyZPdjjY0e2lHYMFYkYoaoDAb1r8KaXFwzVsy\nWVQzF+6r45quYfxWLkNicm4hJrOVF6/vUuHCxu2nspixcDt2O2w7lcUjV7VHgagk2SXClxYBOodM\njpIkNXwNPnAYNWoUn3zyCS+++CLJycn079+fFStWoNGUDSXGxcWRk5MDgEqlYs+ePcyfP5/c3Fxa\ntGjBxIkTeeaZZ+rrW2jYCnNgzfNlQUN5h5aKnRwycGjW/A0anr22M1e0Deazv+PJLbRwZadQ7h4S\nS8tA19svAVJyC3nq5wOlifSScgp57Mf9eGk9aBvixVWdQmXQIEmNUL0tjmzoKrtIpNHLOQ8f9RcJ\nplyZ/ovIxy5JiGyLVpsNP50GrbriNTDHU/K48u2Nbtt/nzuEDuWmQSRJql+Vve81+O2YUi3z0FS8\nldNTzv1KZYK8tIT66C4ZNIDIzeAumaJCAVq1fPuRpMZI/uU2d4ZgUefCleAOzlU3JamSgr21XNXZ\n9eLa4e2CCarGTghJkuqfDBwkiLkCRj0nkpSUCO8Bt3wr8kRIUjUYtB48Pb4jncK9HY63C/Xixeu7\n4C3LS0tSoyTXOLjRbNY4lDAXQV6iKMSi1oNXKBgaYPZMqdFJzSvkfJaJ89kmIv10RPjpCPWRiyKl\nRqogA4zp4v/1gWBoOjVNKnvfk4GDG80ucJAkSZLcs9sh9TAsnwOJe8SxsK5w/YcQ2hWUjX8AXy6O\nlCRJkqSakpUg8t2UBA0AyQdgwTjITqivXtULGThIkiRJ0qXErxd5by5WnA9Hf6vz7tSnBp8ASqpF\ndjsUlJurawJDbZIkSbXi3A73bWe3i/fTCkrLNyXyTtFc5Z6HbZ/AwnGw4GrY/B7knKvvXkmSJDVM\nUX3dt7Xo12yCBpAjDs1TbhJ8dxsk7io7tuY52L8Ebv0RfCPrr2+SJEn1LesMnN0Kp7eIBZCtR0Ds\ncJEQ7+LpCo0XdLimXrpZX2Tg0BylHHQMGkqkHoZzO2XgIElS85V+XCyCLEgrO6bxgjtXw4yV8Mv9\nF+2q+AD8Yuqlq/VFBg7N0dEV7tsO/AAdr5XrHSRJan4Kc2HV445BA4gFkD9Mh5mrYNpPF9aG2Ztc\nHofKkoFDc+QV4r7NO0wGDZIkNU/GdIhf67ot/Tjkp0JoZ9AH1G2/Ghh5h2iOOt3gfiFPr9sq/zxF\n+ZB5Ck6uh2X3wd//J/64LMU10UtJkqS6ZbdDRTkRZb5EQAYOzZNfNNzwMSjLDTgpVTD+LfBvdenr\nTVkQvwGWTIPvboHDP0On62DfN/C/gXBmi/wDkySp8dEHQsvBrtv8Y5rltIQrMuW0G00+5bTZBLmJ\nkHYU7DYI6SgqYWoMFV9XbIQdn8PqZxyPe/rCjZ/B97eB1htmbZSLLCVJanxSDost6uV3T3hoYfov\nED2g/vpVB2StisvU5AOH6so4CR/2A5vFua3T9aDWwb7v4O51ENmr7vsnSZJ0Oex2kV766Ao4/Q+E\ndYOuN4nR2PxkyEsGsxF8IsArDLRe9d3jGlPZ+55cHClVTV6S66AB4MRfcNVLInBAxqOSJDVCCgUE\ntIJB98OA+8Q0LkDSfvjmJhE4ACiUcMUj4pxmVklYrnGQqsajgnLIWm8xBaIPFGW5JUmSGrOSoCH7\nLHx1Q1nQAGKK9+834ey2eulafZKBg1Q13mFgCHbd1v0WOPwr3Pg5eEfUbb8kSZJqS1YCGDNct/39\nJhiz6rQ79U0GDlLV+ETC1CWg9XE83mootL0KrnsPYq6QuSAkSWo6Lk4IVV5+GliL6q4vDYBc4yBV\njUIB4T3h3n8g9YhY8xDeXWzxlFuVJElqioI7uG9rPRI8/eqsKw2BDBykqlMqwb+leEiSJDV13mHQ\n+UY49JPjcY1BLKJUV7D2qwmS48mSJElS82O1VD7LrT4Arn4Nxr0Jfi3FVG3nG+GutRDQunb72QDJ\nPA5uyDwOkiRJTVB+GqQegh1fiLUJvW4XOWe8wyt3fV6K2JKu87t0wrxGRuZxkCRJkqTyCjLgr+dh\nz9dlx479Aa2GwYRPwKcSwYO33GoupyokSZKk5iErwTFoKHFqAyTuqfPuNFYycJAkSZKah7iV7tt2\nL5SVfStJBg6SJElS86DSVtymkLfEypA/JUmSJKl56DDOfVvfu0All/1VhgwcJEmSpObBrwWMetb5\neI9pENKp7vvTSDX4wGHjxo2MGzeO4OBgFAoFJ06cuOQ1FouFhx9+mKCgILy9vbn99tvJz8+vg95K\nkiRJDZanrxhZuHcTDH8KhjwCd6+FK18ELzc1eCQnDX5cpqCggD59+jBhwgRmzZpVqWteeuklvvnm\nG5YsWYK3tzczZ85k9uzZLFq0qJZ72wwV5oqKcenHQOMlytH6RMohP0mSGiZPXwjzhbAu9d2TRqvR\nJIBKSEigVatWHD9+nDZt2rg9z2azERoayquvvsrdd98NwNq1a7nqqqtISUkhMLByddNlAqhKKMiA\nTe/Clveg5NdI6w1Tv4eofjJ4kCRJakQqe99r8FMVVRUfH096ejojR44sPTZs2DAAdu7c6fY6s9mM\nyWRyeEiXcH4nbH63LGgAKMqDrydC7rmqP19BOiTtgwM/wqm/IacazyFJkiTVqib3kTA1NRWAkJCQ\n0mMqlYqAgIDSNldeeeUVXnjhhVrvX5NRmCtGG1wxGyHlMPjHVP75cpNg+Rw4+VfZMa8QuG05BLYG\nY6YormUIERU6JUmSLpaXAsYMUKpAHwSGyo0wS1VTbyMO9957LwqFwu1j+PDh1Xre6s68PPXUUxiN\nxtJHRkZGtZ6n2bAWiz9Qd4zpVXguK+z+0jFoAPDwFGW7/3gSPh0KX4yB7Z+IIEOSJKmExQxntsKC\nq+F/A+HDfrB4IqQeru+eNUn1NuLw+uuv8/TTT7tt12orSNRRgdBQkUc8NTUVb29vAKxWK5mZmQ6j\nEBdTq9Wo1epqvWaz5OkHHcZD2lHX7RE9K/9cBSmw7WPn41e/DstmiSmMEqseh+Or4YaP5SpoSWoO\nrBYxglDRSGP2KfjyWvGBpkTiHlg4Hu5eD/4ta72bzUm9BQ5+fn74+fnV+PPGxsYSFBTEunXraN1a\nlDvduHEjAH369Knx12u2VB7Q8zYxUlBw0ehC5xvBJ6Lyz2W3QWGO47GYK+D0JufnBjixBnLOyMBB\nkpqyrNNwci0k74fYkRDUFrzDRFXK8mw22P21Y9BQwpgJ53bKwKGGNfjFkfn5+ezdu5fDh8WQ05Ej\nR9i7dy+ZmZml53To0IFly5YBoFQque+++3j22WdZu3Yt27dv58EHH2Tq1KmV3lEhVVJAK7hzNQyY\nDb5RENIRrv9IjBToq/Cz1vpA7AjHY+Hd4fRm99ec3VG9PkuS1PBlnoL5Y8BcAP6tYO2L8OV4WPkI\npMWBzVp2rrUIkva6f67k/bXe3eamwS+O3LlzJyNGlN1UrrvuOgAWLFjAjBkzAIiLiyMnp+wT67PP\nPkt+fj433XQTxcXFTJgwgY8++qhO+91sBMTC6Bdg8FxQeoAhqOrP4ekjErCc3gSWQnGsKA/0Ae6v\nkaVtJalpMhfCxreg4zWQtB/2LylrO/AjxK0S0w/B7cQxlRai+ooKl65UZdpUqpRGk8ehrsk8DnXM\naoGME7D5PRFABLWHfnfD4knO56p1cN8WMeIhSVLTknMePugDEz+H76a6Pqf7VBj/Fmj04uv0E/DJ\nFWC+aBu9Vwjc9Rf4Rddun5uIZpvHQWqkVB4Q0kG8GdzxJ0z8DCJ7w6jnxMKoEp6+cOtS8G1Rf32V\nJKl2efpCfor79pNrHHd1+cfAjFUQ1rXsWMxQmLFSBg21oMFPVUjNjFonHiX6z4JO10FWAnjoRJEa\nn0jHYEKSpKbDEAydJ4ostO7og+DEX2AzQ7sxIjiI7Am3/QymTFBcyOOg862zbjcncqrCDTlVIUmS\nVE+yz8K5HbDqMShIc24f8yrs/AIyTooPEjNXVi3hnOSSnKqQJEmSGie/FtByENy8WKxTKKFQQI9b\nxa6KjJPiWO552DkfrOb66WszJEcc3JAjDpIkSQ1AzjnxKBl5OPEX7FrgeI53uNhp4RNW591rSip7\n35NrHKSGpyAdcs6KXA2GIIjoAT4twENm9pSkBslqFqngrcWgMYBPeM09t2+UeCT8IzJBuuKhlTVs\n6pAMHKSGJS8ZVjwCR38rO+ahFaW6W14hS3VLUl0wF4pdDVazyLPi5T5dP3nJsPVj2PEpFBeIHU9j\nXoXYYWJ3RE3xbSGCkuIC57a+d1XcR6lGyTUOUsNyfLVj0ABgKYJvbxFzmbWp2AgZ8XB8DZzaKFLe\nWuS8qdTM5JyDlY+JXAof9BY1IBL+EcHExQpzYc0LsOntsht6zln4/jY4s61m++UdAZMXiURz5bUY\nAF0myhGHOiQ/vkkNR0E6bP3QdZvZCJknRc55qwVyz0HKIbGXO6KnqI1RlTTXFzPlwL5v4c+nwGYR\nx9R6mPKV2A/uoan+c0tSY5GfBj/MhHPby46lHRXBw6wNEN7tovOTYf+3rp/rz6fENGNNjQR4qMXf\n4pxtEL9RVM6NHQaBbWUm2TomAwep4bBZxKd+d4qNImg4tx2+niiCiRKdbxQ1Mqr7BpJ+DH5/3PGY\n2QjfTBFvVIFtqve8ktSY5J53DBpK2G2w/nW48VPQepUdN2WDu/X16cfKUsjXFA+N+FuUf4/1Sk5V\nSA2HPlAke3JFoYDg9uKN7ZvJjkEDwKGfRNXM6rCaXZf1BhHMnPq7es8rSY1N9mn3bUl7oTjf8VhF\nSZp8W4BKjtQ1RTJwkBoOlRr63AVeLkYNrnhUlNTNPCUKYLmy5X3XZbgvxVosFoLFDocJn4hPVTd9\nKWpleHiKIVFJag68K9gNERDrmNUVwCsMWo9yff7Ip8XfrNTkyKkKqWEJiIE7/4QDS+HIL6JC5qAH\nxdyq1tv5E095RXll6xOqQmOAIQ/Dma1iR0dxvhjhaDNaFNrRV6PipyQ1Rr7RENi6LLlSeSP+7bxL\nQu8P170Hqx6HuBVi2kJjgKGPQ5sr66bPUp2TgYPU8PjHwBXzoO8doNSA1lDWFtTW/XUdrgVdBaW4\nK2LMhA1vlH1tt4sdHsUFMOGz6j2nJDU2PmGiiNzPs+HMZnFM5w9j/wOhXVxf4xsFN3wsRu2KC0Dn\nJ/I4yGmKJksGDlLDpFSJN6yLeYfDoLmw+V3H44YgURCrOrsf8lNh3Suu205vhqLcqj+nJDVWAa3g\n5m+gIAXMRWLUzyei4sJynt7iITULMnCQGhdPHxg8F2IGw6b3wJQB7cdDz2niDa86rMWi+qY7dRU4\nFOVDQarIw+/pB17BdfO6UtNlKRIJmiwm0HiJXAjKSixt0/uLhyS5IAMHqfExBIpSutEDxU3f0+/y\nMkp66CCkE6QcdN3uauSjpmUlwB9PQdxKsfUttDNc8w6E95A5JKTqyUuGze+LKpJmkxiVG/U8dBgv\nRhEkqZrkrgqp8fL0EW+Gl5uG2hAIV73kuq3zjTW7MtyYCWlxkHpETJEA5CbCVxNExky7TRxLOSTy\n8mfG19xrS81HUT789TJs+UAEDSB2HP1yP5xcW799kxo9GThIEkBkHzGv69dSfK3Ww8AHRM79msq3\nn3pEJK76sB98NEAEBmd3iBS/rgIEazFs/xQsxTXz+lLzkZcM+xa7blvzvGiXpGqSUxWSBGL0osN4\niOglVoZ7aMQe9ZqaJsg6DQvGgimr7Fj6MfjyGpixQmz/dJWB7/wuMBfI6Qqpagqzy0avLpZz1nXd\nCUmqJBk4SFJ5NVkOuLyz2xyDhhKWQjFF0XKwKCR0seAOYvRDkqrC09d9MOodLirOSlI1yakKSaoL\niXvdtyXvh5COzscVShh0v3yTl6rOKww63ei6beTTtRcgS82CHHGQpLoQ2auCtr7Q81Y4v1tMTQD4\nRMK178piPlIZm03Uask9Jwq++UWLhbuu6kV4esOYl0GjF1VfbRbQ+sCwx6Hd2Lrvu9SkKOx2d6XN\nmjeTyYRer8doNKLT6S59gSRVJOsMfD7CuZaGWgf3/iMCBGOGaLcWi4JfPhH101ep4bHZIHE3fHOT\n2JkDYkTqikdgwH1iZ5ArxQWQlyKKwmm9RUB6ubuQpCarsvc9GTi4IQMHqcalxcGvc+HMFvF1aBe4\n7n0I715xVj5Jyj4D/xvsOhnZ1O9FXhNJukyVve/J0FOS6kpwe7j5WzCmixXv+kCRh0JqOvJTxcNm\nFaMAJZkaLUWgUFX/035anPsMpv+8DdEDam7bsCRdggwcJKkuyVS+TVfKIfhxprjJg1iDMPlryDgB\n+74BjTf0u0sshNW7mVpwx1hBufiCNJnrQ6pTMnCQpOqw2cReeZXa9eI0qXnJPiNycpSsPwC4+nX4\n9UFI2lt27PAyGHg/DH20aqnM3VWmBGh7lRxtkOqU3I4pSVWVfUZU51x0PXwzGY6uhPy0+u6VVJ+S\nDzgGDSP+LVKJlw8aSmz5QLRVhU8EdLrB+binL/S9SyYIk+qUHHGQpKrIPgMLxonseyVOb4beM2H0\nc3VTEEsCqwXyEsGULRJkeYfW78hPxomy//cOEwXY1r7s/vxTf4tCZpWlD4Sxb0DsMFG4qjAH2o2D\nIfMgILba3Zak6pCBgyRVls0GB350DBpK7FoA/e5unIGDKRvyU6A4H3QBIrOg2vPyn7coXywUtBSK\nT8Y+ESKb4eUyZcGhn2HNc+IGqlCIm+jYN8CvxeU/f3n5KRe+hyIwBIvESSoXn+7De5T9f9fJcHa7\n6/NKeFTj5+sdBn3uEKnRbVbxb1UT/06SVEUycJCkyjJlwcGl7tvP7azap8iGIPssLL8fTq0XX3to\nYfA86Derajs+8tPAWiRuiIYgUbjrz2fhyM/iJucVIub821wp6oJcjjNb4Ld5ZV/b7RC3Qvz73Px1\n1RceupN6FL6fBunHxddqPYx5RVRM1fk5nhvUFgLbQsZxMfqxbwn0mg4Jfzs/r0IBMYOq3y+v0Opf\nK0k1oMGvcdi4cSPjxo0jODgYhULBiRMnLnnNjBkzUCgUDo958+bVfmelpk3lAVov9+2Xe0OsS4U5\nYtpl7zdwdmvZcUsRbHgD4lZV7nlMWWKNx5fj4d1u8OW1Ir32T7Pg0FIRNID41P7jHSKJ0eUoSIe/\nXnTddmZzzVV9zDkHX11fFjSASKL020NiPcPFfCJg2lLodjNkxIvRCZvZdX6Fcf8V2zQlqZFq8CMO\nBQUF9OnThwkTJjBr1qxKXzdx4sT/b+/eg6Ku9z6Av7nJXVfYXUhFECgchZS8njMZKFaSHZ6UwiZR\naUqPY4+BTYOGkiYeU8fSGjNBicxHzcmxnkynUbmkzzFRfIARn/LSyiETBsEDst5g4fv88YvVlV34\n7bqwy/J+zezU77a/74evsJ/9/r4XbNmyRb/t7e3dHcWjvsRjgLTUdtWpjsdc+kkTOdm7Ow1AdbmU\nHDReBQaNBpJ2Aic+kprX2xV9CDz+rNQ8bkpbK/DLD8D3/3l/X+3/SQnJv/5p/JpjHwDJTwJefpaV\nX3fX+BLk7e40WPa+D7uhMZ2EHN8g1fXDieLAYODFTVJyc6sW+GIaEJMOPDlLaiVx8waGPQOoR3ae\ngBLZObtPHOLj4xEfH4/KykqzrvPw8EBgYCd/9IgsETQeGJ0MlP3X/X0ubsCs3dJ0vvZM1yw9ajn0\nzv19Df+SWhde/kIaOtg+MuDmH9LU1525eQ04mmm4z9kVaG4yfU3dRaDljmXlB6QputUjgGulxo9b\n6zFFZy0XjVelBAZGWpj6eQH9hkqPZuYdBA6/C/z7irRc+xPPS3M49OffJerd7D5xsNThw4ehUqmg\nVqvx0ksvYcWKFZ1OodnS0gKdTqffvnPnEf64kePyUQPPZUnrA1w9I7VCDIqWmqrtfRXLpmrgyIqO\n+1ubpSGC0XOAf34i7QsY2fVy3vdudlwqvE0nTXRkivIJ6cPfUl7+wLMfADsTOh6LeEHqX2ANysdN\nHxs6UVowqjNuHkDwX4C53wH3mqSEyieQwybJIdh9HwdLxMfHY/fu3SgsLERmZiZ27drV5WOOf/zj\nH/Dy8tK//P2t9M2FHI+XHxAYCYx9HYicCfgNs/+kAZBmGGy5bfxY1SnDjp3Pr+26c6Sbp/E1NmrP\nS8MRjZm6yvLHFO0GPQW8ugcY8OcICld3aUTL9I+sN6plwBAg+OmO+13dgb+myh/N4K2ShksqhjJp\nIIdhs0WuFi5ciOzsbJPHY2JiUFRUpN+urKzEsGHDcOnSJYSHm7fUcEFBAeLi4lBbWwuVSmX0HGMt\nDv7+/lzkihxHzTlgm5EPQwDo5yONejj5ifTfoIldP4e/pwUOpgEV3xjud/MEZh+Qhqi2d5D0DZTe\nNyzOep1Im6qBu03Sh7lPIOBm5eSt8Q/gdA5wZoc0VDX4aWDaWkA1AnB1s+69iOyA3a+O2dDQAK1W\na/K4u7u7wYf8oyQOjY2NUCgUOH36NMaNGyfrGq6OSQ5HWytNi9y+lsKD/poKTFggdfL0Uct/z8ar\nwPeLgd8K7u97Yhrw4seAh0KaB0F3F3CXOY+Drlm6pq0V8PC1Xp8FS7W2/LlolU56LPXwMEwiB2L3\nq2MqFAooFIoeuVd5eTkAICQkpEfuR2SXfNRA0i5g10uGUx4PiwEm/t2yzp0DhgCJuYC2BrjTKH2w\n+gbef2RgzqyGjX8AP38mtVS03AYGjwHiNwCBUbZ7FOTiBgyw806vRD3M7jtHarVaXL58GdeuSX/o\nfvnlF2i1WgwdOhR+ftKz0uHDh+PDDz/EjBkzoNVqkZWVhcTERKjValRUVCAtLQ1JSUkmH1MQ9Rmq\nCOCNY0BDpTRywD9cShgeZXlvL79H77egvQ58+3fDCZP+OAvkTQMW/GR6Yq3bN6Q4btdJyYpvoNSv\ngIi6jd0nDiUlJZg8ebJ+OyFB6k2dl5eHlJQUAMCFCxfQ2NgIAHBxcUFpaSm++OIL3Lx5E0FBQUhM\nTERmZmaH9ybqkwYMtr9v0U3XjM+y2NoCHN8I/McWoN9Dc7HcvAYcXAJc+vH+vsFjgFe+lDojElG3\nsPvEITY2Fl11w3jwuKenJ44cOdLdxSIia6r/zfSxq2ekIY0PJg66e8D/bDZMGgCpleKHJdK8FFxq\nmqhbOORwTCLqZXwfM31MEdxx7gdtDfC/O42ff/mY1MGSiLoFEwcisr2BwffnZXhY7NKOrQe65j9n\nbzThUWanJKJOMXEgItvrPwiY8y0QEHl/Xz8f4G+fAIFG1gBx7296xIbHgN65vDlRL2H3fRyIqI9Q\nPg7M/e/7cz94KaWEwsXIZEu+AdJQzd0vdzw2dRXga2edP4kcCBMHIrIf3kr5Q0OH/gV4/Ufg6PvS\nNNcDhwFTMoGgCYCLkamwicgqbDZzpL3jzJFEvcTtf0sTRrm6P9p8FER9nN3PHElEZBVeAwGwTwNR\nT2HnSCIiIpKNiQMRERHJxsSBiIiIZGPiQERERLIxcSAiIiLZmDgQERGRbEwciIiISDbO42BC+7xY\nd+5wsRwiInJ87Z93Xc0LycTBhLt3pZX3/P39bVwSIiKinnP37l14eXmZPM4pp01oa2tDQ0MDPDw8\n4OTkZOviAJCyQX9/f9TX1/e5abAZe9+Lva/GDTD2vhi7PcQthMDdu3ehUCjg7Gy6JwNbHExwdnaG\nn5+frYthlKenZ5/6hXoQY+97sffVuAHG3hdjt3XcnbU0tGPnSCIiIpKNiQMRERHJxsShF3F1dcXK\nlSvh6tr3njAx9r4Xe1+NG2DsfTH23hQ3O0cSERGRbGxxICIiItmYOBAREZFsTByIiIhINiYORERE\nJBsTBzt3/PhxvPDCC1CpVHBycsLly5e7vCYlJQVOTk4Gr7S0tO4vrBVZErdOp8M777wDpVIJX19f\nzJs3D1qttgdKa315eXkIDQ2Fp6cnYmJicPHixU7P7811vm7dOgwaNAheXl5ISEhATU2NyXO1Wi1e\nf/119O/fH/7+/liyZAl0Ol0PltZ6zIk7Nja2Q/1u3ry55wprRQcOHEBcXBwGDBgAJyenLuvPUerc\n3Ljtuc6ZONi5W7duYezYsVi7dq1Z1yUmJqK6ulr/ysrK6qYSdg9L4s7KysKePXuwb98+5Ofno6Sk\nBIsWLerGUnaPgoICLFiwAO+99x7OnDmDwMBATJ8+Hc3NzZ1e1xvrPC8vD2vWrMGWLVtw8uRJ3Lx5\nE7NmzTJ5/ltvvYVTp07h6NGj+Oabb7Bv3z6sXr26B0tsHebGDQBpaWkG9btgwYIeKq113b59G1Om\nTMGyZctkne8odW5u3IAd17mgXuHKlSsCgLh06VKX586bN0/Mnj27B0rV/eTG3draKpRKpcjJydHv\ny8/PFy4uLqKurq67i2lVM2bMEK+99pp+W6vVCk9PT/Htt9+avKa31nl0dLTIyMjQb//2228CgCgt\nLe1w7o0bN4SLi4s4cuSIfl9ubq7w9/cXOp2uJ4prNebELYQQMTExYvny5T1Uup5RWFgoAIiWlhaT\n5zhSnbeTE7cQ9l3nbHFwUIcPH4ZKpcLIkSOxfPlyh18eXKPRoK6uDlOmTNHvi4mJAQCUlJTYqlgW\nOX36tEEc3t7emDBhAoqLizu9rrfV+b1791BeXm4Qa2hoKEJCQozGevbsWQghEBsbq98XFxeH+vp6\nWY+y7IW5cbfLycmBUqnE6NGj8dFHH6G1tbUnimtTjlLnlrLXOrf/KarIbPHx8Zg1axaCgoJQUVGB\n9PR0VFVVYdeuXbYuWrepra0FAKjVav0+FxcX+Pn56Y/1FrW1tQZxAIBKpeo0jt5Y5/X19Whra5Md\na21tLRQKBdzc3AzObT8WERHRvQW2EnPjBoDk5GSEhoZCpVLh1KlTWLp0KRoaGnrF46hH4Sh1bgl7\nrnMmDjaycOFCZGdnmzweExODoqIii977wWelkZGRUKvViIuLw8cff6z/pbOV7opb9IIJUPtqnZti\nbp0ZO99elrw3hyX/Vt988039/0dFRcHFxQWpqalYvXp1r/wZyOUodW4Je65zJg42sm7dOqxYscLk\ncXd3d6vda8yYMQCAyspKm3+IdFfcAQEBAKRvIb6+vgCA1tZW3Lhxo8M3O1uRG7tare7wzfP69esI\nCwuTfS97qnNTlEolnJ2djcZqrM4CAgLQ0NCAlpYW/TdQYy1N9s7cuI0ZM2YMtFot6urq7LZ+rcFR\n6twa7KnOmTjYiEKhgEKh6JF7lZeXAwBCQkJ65H6d6a64Q0NDoVQqUVhYqP+APX78OABg7NixVr+f\nJeTGPn78eBQWFuKNN94AIPXGLi4uRmpqqux72VOdm+Lu7o5Ro0ahsLAQcXFxAIArV66gsrISEyZM\n6HD+U089BScnJ/z000+YOnUqAGkEir+/P8LDw3u07I/C3LiNKS8vh7e3N5RKZXcW1eYcpc6twa7q\n3IYdM0mGpqYmUVpaKg4dOiQAiO+//16UlpaK+vp6/TkRERHiwIED+vPT09NFcXGxuHLlijh48KAI\nCwsTSUlJtgrBIubGLYQQmZmZ4rHHHhP5+fmiuLhYREZGijlz5tii+I/k2LFjwtXVVezYsUNUVFSI\nWbNmibCwMHHv3j39OY5S57m5ucLHx0ccOHBAlJWVicmTJ4tJkyYJIYS4evWqiIiIEMXFxfrz58yZ\nI0aMGCGKi4tFQUGBGDRokMjMzLRV8S1mTtyXL18Wa9asEWfPnhUajUbs3btXqFQqkZ6ebssQLFZf\nXy9KS0vF9u3bBQBRUlIiSktLRVNTk0PXuTlx23udM3Gwc+1Ddx5+5eXl6c95cPv27dvi2WefFUql\nUvTr10+EhYWJ9PR00dTUZJsALGRu3EII0dLSIpYsWSL8/PyEj4+PmDNnTq+Lu11ubq4IDg4W7u7u\n4plnnhG//vqrwXFHqvO1a9eKwMBA4eHhIV588UVRXV0thLg/FLewsFB/blNTk5g3b57w9fUVAwcO\nFKmpqV0Oa7NXcuOuqqoSkyZNEgqFQnh4eIjhw4eLdevWiebmZhuW3nJ5eXlGf7cLCwsdus7Nidve\n65zLahMREZFsnMeBiIiIZGPiQERERLIxcSAiIiLZmDgQERGRbEwciIiISDYmDkRERCQbEwciIiKS\njYkDERERycbEgYiMio2NhZOTk8Fr4sSJVnnv5ORkpKSkWOW9LLFmzRqMHz8e7u7uePrpp21WDqLe\niItcEZFJaWlpWLp0qX67X79+NixNR83NzRaVSafTITk5GcOHD4dGo+mGkhE5LrY4EJFJ3t7eCAwM\n1L/8/PwASEtAz549GwqFAkqlErNnz0Z9fb3+utzcXIwePRre3t4IDg5GZmYmdDodAGDVqlXYvXs3\ndu7cqW/JaN//8Lf/lJQUJCcn67dDQkKwfv16JCYmwsvLC59++ikA4Ouvv8aIESPg6emJyMhI7N+/\nv9O4Vq1ahbfffhuhoaGP/kMi6mOYOBCR2V5++WUAwIkTJ1BUVISGhgaDD/i2tjZs3LgRFRUV2LZt\nG3bs2IGcnBwAwLvvvovExEQkJSWhuroa1dXVZt17w4YNmDZtGioqKvDqq6+ioKAAixcvxgcffIDz\n588jIyMDc+fOxalTp6wXMBHp8VEFEZm0YcMGbN68Wb+dnZ2NoKAgXLhwAfn5+XB1lf6EbN++HYMH\nD8bVq1cxZMgQzJ8/X3/NsGHDkJqaiv3792PRokXw8fGBh4cHXF1dERgYaHaZEhISDN5/7ty5WLly\nJV555RUAQGhoKIqKirBjxw6r9ckgovuYOBCRSfPnz8eSJUv02wEBAfjqq69w/fp1KBSKDudrNBoM\nGTIEJ0+exKpVq3D+/Hk0NjZCp9MhKCjIKmWKjo422D537hx+/vlnLFu2TL+vubmZnR6JugkTByIy\naeDAgQgPDzfYp9VqER4ejkOHDnU4f/DgwWhqasL06dORlJSE1atXw8/PD3v27MGXX37Z6b2cnZ0h\nhDDY19LSou8D0c7Ly6tDeTZu3Ijnn3/eYL+np2dX4RGRBZg4EJFZRo0ahaqqKvTv3x9qtbrD8ZKS\nEjQ0NGD9+vX6Vonff//d4Bw3Nzd9Z8l2KpUKNTU1BvvOnTuHJ598ssvyaDSaDgkOEXUPdo4kIrM8\n99xziIqKwsyZM3HixAloNBocPXoUCxYsAAAMHToUbm5u2Lp1KzQaDbZt24bvvvvO4D2Cg4NRWlqK\nyspK1NXVAQAmTZoEjUaDzz//HJcuXUJGRgYqKyu7LE9GRgY+++wzbNq0CRcvXkR5eTm2bNmCffv2\nmbymqqoKZWVlqKmpwa1bt1BWVoaysjJLfyREfQoTByIyi7OzM3788UdERERg5syZGDlyJBYvXqxv\nXVCr1cjJycHWrVsRFRWFI0eOGPQ/AKS+E35+fhgxYgRUKhUAIDIyEps2bUJWVhbGjRuHtrY2zJgx\no8vyJCQkYO/evdi1axeioqIwdepU/PDDDwgODjZ5zfvvv4/o6GhkZ2ejrKwM0dHRHfpOEJFxTuLh\nh4pEREREJrDFgYiIiGRj4kBERESyMXEgIiIi2Zg4EBERkWxMHIiIiEg2Jg5EREQkGxMHIiIiko2J\nAxEREcnGxIGIiIhkY+JAREREsjFxICIiItn+HzQwiKdQZBzDAAAAAElFTkSuQmCC\n"
          }
        }
      ],
      "source": [
        "def generate_noisy_circle(radius, num_points, noise_std):\n",
        "    theta = np.linspace(0, 2*np.pi, num_points)\n",
        "    x = radius * np.cos(theta) + np.random.normal(0, noise_std, num_points)\n",
        "    y = radius * np.sin(theta) + np.random.normal(0, noise_std, num_points)\n",
        "    return np.column_stack((x, y))\n",
        "\n",
        "# Generate data for inner and outer circles\n",
        "X_1 = generate_noisy_circle(0.5, 100, 0.05)\n",
        "X_2 = generate_noisy_circle(1.5, 100, 0.05)\n",
        "\n",
        "# Create dataframes\n",
        "df_1 = pd.DataFrame({\"f1\": X_1[:, 0], \"f2\": X_1[:, 1], \"k\": 0})\n",
        "df_2 = pd.DataFrame({\"f1\": X_2[:, 0], \"f2\": X_2[:, 1], \"k\": 1})\n",
        "df = pd.concat([df_1, df_2], ignore_index=True)\n",
        "fig, ax = plt.subplots(figsize=(6, 4))\n",
        "sns.scatterplot(data=df, x=\"f1\", y=\"f2\", hue=\"k\", ax=ax)\n",
        "ax.set(xlabel=\"Feature 1\", ylabel=\"Feature 2\", title=\"Latent Groups\")\n",
        "plt.show()"
      ],
      "id": "250015e6"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Note that the `k=0` and `k=1` state are fully separated and the\n",
        "difference is visually obvious. Now lets perform k-means clustering"
      ],
      "id": "b48e10b0-2579-4fc4-9497-5270c7e8867b"
    },
    {
      "cell_type": "code",
      "execution_count": 13,
      "metadata": {},
      "outputs": [
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "C:\\Users\\jesse\\anaconda3\\lib\\site-packages\\sklearn\\cluster\\_kmeans.py:1412: FutureWarning:\n",
            "\n",
            "The default value of `n_init` will change from 10 to 'auto' in 1.4. Set the value of `n_init` explicitly to suppress the warning\n",
            "\n",
            "C:\\Users\\jesse\\anaconda3\\lib\\site-packages\\sklearn\\cluster\\_kmeans.py:1436: UserWarning:\n",
            "\n",
            "KMeans is known to have a memory leak on Windows with MKL, when there are less chunks than available threads. You can avoid it by setting the environment variable OMP_NUM_THREADS=1.\n"
          ]
        },
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "image/png": 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YUR07dtT06dOtz34JDQ3V6dOnJUn79u3TkiVL1KZNG23ZskWSdObMGSUlJVn7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\n0yKQAQAApkUgAwAATItABgAAmNb/A7OAp2RZ7DMQAAAAAElFTkSuQmCC\n"
          }
        }
      ],
      "source": [
        "kmeans = KMeans(n_clusters=2, random_state=0)\n",
        "df[\"k_hat\"] = kmeans.fit_predict(df[[\"f1\", \"f2\"]])\n",
        "\n",
        "# Create confusion matrix\n",
        "cm = confusion_matrix(df[\"k\"], df[\"k_hat\"])\n",
        "\n",
        "# Plot confusion matrix\n",
        "sns.heatmap(cm, annot=True, cmap='Blues')\n",
        "plt.xlabel('Predicted k')\n",
        "plt.ylabel('True k')\n",
        "plt.title('Confusion Matrix for K-Means Clustering')\n",
        "plt.show()"
      ],
      "id": "d86ebaac"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Can you explain why this was a failure using your knowledge of k-means\n",
        "clustering? Be precise if possible."
      ],
      "id": "4ce62c25-cc81-49e8-b51d-7394e43125a7"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "**(double click to add an explanation here)**"
      ],
      "id": "3fc48aaf-3e10-4f0f-922e-6e4a63e67985"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "But all is not lost! Look at the results of the following code which\n",
        "uses a different algorithm"
      ],
      "id": "db9a711f-9871-4034-b779-becb26dd8837"
    },
    {
      "cell_type": "code",
      "execution_count": 14,
      "metadata": {},
      "outputs": [
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "C:\\Users\\jesse\\anaconda3\\lib\\site-packages\\sklearn\\manifold\\_spectral_embedding.py:273: UserWarning:\n",
            "\n",
            "Graph is not fully connected, spectral embedding may not work as expected.\n",
            "\n",
            "C:\\Users\\jesse\\anaconda3\\lib\\site-packages\\sklearn\\cluster\\_kmeans.py:1436: UserWarning:\n",
            "\n",
            "KMeans is known to have a memory leak on Windows with MKL, when there are less chunks than available threads. You can avoid it by setting the environment variable OMP_NUM_THREADS=1.\n"
          ]
        },
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "image/png": 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rVi0NHjxYY8eOVVpamubPn6/ly5e7rA8ELQAAoFgWLFig559/XjExMTpz5oya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\ngaAFAABYwv8HzOF+8vR1bVIAAAAASUVORK5CYII=\n"
          }
        }
      ],
      "source": [
        "from sklearn.cluster import SpectralClustering\n",
        "spectral_clustering = SpectralClustering(n_clusters=2, affinity='nearest_neighbors', random_state=0)\n",
        "df[\"k_hat\"] = spectral_clustering.fit_predict(df[[\"f1\", \"f2\"]])\n",
        "\n",
        "# Create confusion matrix\n",
        "cm = confusion_matrix(df[\"k\"], df[\"k_hat\"])\n",
        "sns.heatmap(cm, annot=True, cmap='Blues')\n",
        "plt.xlabel('Predicted k')\n",
        "plt.ylabel('True k')\n",
        "plt.title('Confusion Matrix for K-Means Clustering')\n",
        "plt.show()"
      ],
      "id": "6b6a2d08"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Lookup [spectral\n",
        "clustering](https://scikit-learn.org/stable/modules/generated/sklearn.cluster.SpectralClustering.html)\n",
        "in skikit-learn docs, etc. and provide a 2 sentence summary of why this\n",
        "worked."
      ],
      "id": "9386fa57-cd9d-444a-92c6-0ace682a44dc"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "**(double click to add your explanation of why spectral clustering\n",
        "worked)**"
      ],
      "id": "ce5fa10f-b638-4375-b0cd-916e4ef7c735"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Q4\n",
        "\n",
        "Consider the following map\n",
        "\n",
        "$$\n",
        "x_{t+1} = f(x_t) = a x_t^2 + (1-a)\n",
        "$$\n",
        "\n",
        "Where we will assume that $a=\\frac{3}{4}$. The code below comes from the\n",
        "lecture notes, and can be used to simulate this map"
      ],
      "id": "1c874d4b-4775-4b50-a314-2f19831024dc"
    },
    {
      "cell_type": "code",
      "execution_count": 15,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "image/png": 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gt2/frrVr12rLli3avXu3WlpatHjx4j63d7vdeuihh666jSSdOnVKZ86c0Zkz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\n9P8B7250fL3H9+wAAAAASUVORK5CYII=\n"
          }
        }
      ],
      "source": [
        "def simulate(f, X_0, T):\n",
        "    X = np.zeros((1, T+1))\n",
        "    X[:,0] = X_0\n",
        "    for t in range(T):\n",
        "        X[:,t+1] = f(X[:,t])\n",
        "    return X\n",
        "def f(x, a):\n",
        "    return a * x**2 + (1-a)\n",
        "T = 10\n",
        "x_0 = 0.1\n",
        "X = simulate(lambda x : f(x, 3/4), x_0, T) # use a=3/4\n",
        "fig, ax = plt.subplots(figsize=(6, 4))\n",
        "ax.plot(range(T+1), X.T)"
      ],
      "id": "ef2487b3"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Q4.1\n",
        "\n",
        "If I solve the equation $x^* = f(x^*)$ for $x^*$ using the quadratic\n",
        "formula you get the following:"
      ],
      "id": "d8338a8f-7bc2-4944-95bd-1880364bb147"
    },
    {
      "cell_type": "code",
      "execution_count": 16,
      "metadata": {},
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "x^* = [1.0, 0.3333333333333333]"
          ]
        }
      ],
      "source": [
        "def x_star(a):\n",
        "  return [(1 + np.sqrt(1 - 4 * a*(1-a)))/(2*a),(1 - np.sqrt(1 - 4 * a*(1-a)))/(2*a)]  \n",
        "print(f\"x^* = {x_star(3/4)}\")"
      ],
      "id": "ee691770"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Adjust the above code to simulate up to $T=5$ from $x_0 = 0.9$ and\n",
        "$x_0 = 1.1$ and plot the results."
      ],
      "id": "b00d5348-7aa3-4869-b183-69ab09755b94"
    },
    {
      "cell_type": "code",
      "execution_count": 17,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "image/png": 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avny51q1bp4KCAs2fP19ZWVnKzMxUaWmphg4dqq1bt0qS6urqdOONN8pms2nF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\nYII=\n"
          }
        }
      ],
      "source": [
        "# Modify this code\n",
        "T = 5\n",
        "X = simulate(lambda x : f(x, 3/4), x_0, T) # use a=3/4\n",
        "fig, ax = plt.subplots(figsize=(6, 4))\n",
        "ax.plot(range(T+1), X.T)"
      ],
      "id": "de847e8e"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "What do you notice about this and how do you intepret the different\n",
        "behavior for various $x_0$?"
      ],
      "id": "3c35bf4c-52c5-4dff-a3f1-e8894e1adb51"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "**(double click to add your explanation)**"
      ],
      "id": "2298ddce-b6bf-47ba-8b20-d6248ede86b0"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Q4.3\n",
        "\n",
        "See the graph below with $x_0 = 1.0$ and a very large $T$."
      ],
      "id": "28a33b2c-809d-4e74-bf2f-0e0737c87b8e"
    },
    {
      "cell_type": "code",
      "execution_count": 19,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "image/png": 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BDwaDamlpUXp6esT2tLQ0BQKBVuNzc3Plcrn06KOP6uTJkwoEAlqyZIkkOZ/b33nnnZo7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\n"
          }
        }
      ],
      "source": [
        "T = 500\n",
        "X = simulate(lambda x : f(x, 3/4), 1.0, T)\n",
        "fig, ax = plt.subplots(figsize=(6, 4))\n",
        "ax.plot(range(T+1), X.T)"
      ],
      "id": "9d384013"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "And $x_0 = 1 + 10^{-15}$ and even a smaller $T$"
      ],
      "id": "69d1197c-ed65-4f03-b2df-a551e08ac61b"
    },
    {
      "cell_type": "code",
      "execution_count": 20,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "image/png": 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nxWIxRSKRcS8AAOyWSovcSLMwya6/v1+JREK5ubnjjufk5Ki3t1eSdOzYMT300ENatGiR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\nEfAAADgQAQ8AgAP9fyXz/3pWGhQ6AAAAAElFTkSuQmCC\n"
          }
        }
      ],
      "source": [
        "T = 89\n",
        "X = simulate(lambda x : f(x, 3/4), 1 + 1e-15, T)\n",
        "fig, ax = plt.subplots(figsize=(6, 4))\n",
        "ax.set_yscale('log')\n",
        "ax.plot(range(T+1), X.T)"
      ],
      "id": "bea204d6"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Notice the `log` scale on the axis. Briefly interpret these results"
      ],
      "id": "de13cab0-72fe-4ac4-911e-b63587fbf5a3"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "**(double click to add your interpretation)**"
      ],
      "id": "265f1d86-6943-4e47-a732-a651f5e9c107"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Q5\n",
        "\n",
        "Find some economic or financial data on the web of your choice\n",
        "\n",
        "1.  Download it and load it into a pandas dataframe\n",
        "2.  Either run PCA or k-means clustering on the data\n",
        "3.  Briefly intrepret the results in whatever way you wish\n",
        "\n",
        "We are not giving you a list of possible datasources because we want you\n",
        "to explore to look for interesting sources of data.\n",
        "\n",
        "The economic rational for looking for the dimension reduction or the\n",
        "clustering doesn’t need to be especially tight, but do your best to\n",
        "explain why you ran the particular specification.\n",
        "\n",
        "For the data, post a link to where you found it but: no need to have it\n",
        "automatically downloaded in the notebook, and no need to post it up with\n",
        "your problem set."
      ],
      "id": "9cceae32-ef81-468d-b69f-b3e488a92854"
    },
    {
      "cell_type": "code",
      "execution_count": 21,
      "metadata": {},
      "outputs": [],
      "source": [
        "# Your code here"
      ],
      "id": "a9d30003"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "**(double click to add a brief interpretation or summary)**"
      ],
      "id": "1854d7a2-5731-41be-a12c-28d38a237af7"
    }
  ],
  "nbformat": 4,
  "nbformat_minor": 5,
  "metadata": {
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3 (ipykernel)",
      "language": "python"
    },
    "language_info": {
      "name": "python",
      "codemirror_mode": {
        "name": "ipython",
        "version": "3"
      },
      "file_extension": ".py",
      "mimetype": "text/x-python",
      "nbconvert_exporter": "python",
      "pygments_lexer": "ipython3",
      "version": "3.9.12"
    }
  }
}