{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "Computing a discrete STFT introduces a frequency grid which resolution depends on the signal's sampling rate and the STFT window size, see Section 2.1.4 of [Müller, FMP, Springer 2015]. In this notebook, we discuss how to make the frequency grid denser by suitably padding the windowed sections in the STFT computation. Often, one loosely says that this procedure increases the frequency resolution. This, however, is not true in a qualitative sense as we explain below. \n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## DFT Frequency Grid\n", "\n", "Let $x\\in \\mathbb{R}^N$ be discrete signal of length $N\\in\\mathbb{N}$ with samples $x(0), x(1), \\ldots, x(N-1)$. Given the sampling rate $F_\\mathrm{s}$, we assume that $x$ is obtained by sampling a continues-time signal $f:\\mathbb{R}\\to\\mathbb{R}$. Then, the **discrete Fourier transform** (DFT) $X := \\mathrm{DFT}_N \\cdot x$ can be interpreted as an approximation of the continuous Fourier transform $\\hat{f}$ for certain frequencies (see Equation (2.132) of [Müller, FMP, Springer 2015]):\n", "\n", "$$ \n", "X(k) := \\sum_{n=0}^{N-1} x(n) \\exp(-2 \\pi i k n / N) \n", "\\approx {F_\\mathrm{s}} \\cdot \\hat{f} \\left(k \\cdot \\frac{F_\\mathrm{s}}{N}\\right)\n", "$$\n", "\n", "for $k\\in[0:N-1]$. Thus, the index $k$ of $X(k)$ corresponds to the physical frequency\n", "\n", "\\begin{equation}\n", " F_\\mathrm{coef}^N(k) := \\frac{k\\cdot F_\\mathrm{s}}{N} \n", "\\end{equation}\n", "\n", "given in Hertz. In other words, the discrete Fourier transform introduces a linear **frequency grid** of resolution $F_\\mathrm{s}/N$, which depends on the size $N$ of the $\\mathrm{DFT}_N$. To increase the density of the frequency grid, one idea is to increase the size of the DFT by artificially appending zeros to the signal. To this end, let $L\\in\\mathbb{N}$ with $L\\geq N$. Then, we apply **zero padding** to the right of the signal $x$ obtaining the signal $\\tilde{x}\\in \\mathbb{R}^L$:\n", "\n", "\\begin{equation}\n", "\\tilde{x}(n) :=\\left\\{\\begin{array}{ll}\n", " x(n) ,& \\,\\,\\mbox{for}\\,\\, n \\in[0:N-1],\\\\\n", " 0, & \\,\\,\\mbox{for}\\,\\, n \\in[N:L-1].\n", "\\end{array}\\right. \n", "\\end{equation}\n", "\n", "Applying a $\\mathrm{DFT}_L$, we obtain:\n", "\n", "$$ \n", "\\tilde{X}(k) = \\mathrm{DFT}_L \\cdot \\tilde{x}\n", "= \\sum_{n=0}^{L-1} \\tilde{x}(n) \\exp(-2 \\pi i k n / L)\n", "= \\sum_{n=0}^{N-1} x(n) \\exp(-2 \\pi i k n / L)\n", "\\approx {F_\\mathrm{s}} \\cdot \\hat{f} \\left(k \\cdot \\frac{F_\\mathrm{s}}{L}\\right)\n", "$$\n", "\n", "for $k\\in[0:L-1]$. The coefficient $\\tilde{X}(k)$ now corresponds to the physical frequency\n", "\n", "\\begin{equation}\n", " F_\\mathrm{coef}^L(k) := \\frac{k\\cdot F_\\mathrm{s}}{L},\n", "\\end{equation}\n", "\n", "thus introducing a linear frequency resolution of $F_\\mathrm{s}/L$. For example, if $L=2N$, the frequency grid resolution is increased by a factor of $2$. In other words, the longer DFT results in more frequency bins that are more closely spaced. However, note that this trick does not improve the approximation quality of the DFT (note that the number of summands in the Riemann approximation is still $N$). Only, the linear sampling of the frequency axis is refined when using $L\\geq N$ and zero padding. The following example compares $\\mathrm{DFT}_N \\cdot x$ with $\\mathrm{DFT}_L \\cdot \\tilde{x}$. " ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "execution": { "iopub.execute_input": "2024-02-15T09:00:54.149319Z", "iopub.status.busy": "2024-02-15T09:00:54.149026Z", "iopub.status.idle": "2024-02-15T09:00:56.733751Z", "shell.execute_reply": "2024-02-15T09:00:56.733062Z" } }, "outputs": [ { "data": { "image/png": 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vxYYNG2TiL4nES0pLS1FQUICJEyeioqLCry9lEok/2bNnD/Lz8zFx4kSUlpYOWCoOfzGk4qWiogIAMGbMGHWbEAD+HDpauXIlzjjjDERFRSE8PBwzZ870q3gRtgrbgePXJK7RVz788EM0NTXhtttuA6AIpPT0dL+KFxFDI7w61113HaKiovDcc8/5rQ6J5GSitLQUEydORH5+Pvr7+1FdXT3UJkkkdjAz9u7di/Hjx2PixIlobW1FY2PjUJvllIATL/n5+QD8J15aW1uxbds2zJs3T902b948lJSU+G062GCIlyVLlmD8+PGYP3++uq2oqMiv4mXr1q2IjIzEuHHjAABxcXG47rrr8P7772MocvNIJMOZzs5OHDhwABMnTsT48eMByBlHksCkubkZbW1tyM/PxymnKCv87Nq1a4itcs6QixeDwQDtYmpxcXFIT0/3m3j55ptvwMx24gWA35K8lZWVITk5GYmJieq2UaNGITg42C/ipaKiAhs3bsSNN94Ig+H4V1ZUVIQ9e/bg2LFjPtcBKOJl6tSpCAoKUrfdfPPN6O3txccff+yXOgaStrY2vPjiiw7dnbt27cJnn302yFYNPuvXr8fo0aP9Hswt8Yzdu3cDgJV4kXEvkkBEPJfC8wIE/oyjIRcvWVlZCA0NtdpeUFDgN/GycuVKREdH49RTT1W3TZs2DXFxcX4bOiorK7PyugBAcHAwcnNz/SJeysvLAQDTp0+32l5UVASz2YydO3f6XIfZbMa2bdvUISPBhAkTEBERodoQyCxduhQ333wz9u3bp1v+5JNPYvFityfCBTS1tbUORdrGjRtRVVWFyspK3fLu7m60tLQMpHkuIaIFRLSXiCqI6AEn+80gIhMR/Wgw7fMHovGfOHEiEhISkJqaKj0vkoBEPJf5+flISUlBYmKiFC/OqKystBoyEhQUFGDPnj1+CRhauXIlZs2ahZCQEHVbUFAQ5syZ4xfxwsy64gVQho4cdSCeIMbJc3NzrbYLoeGPoaOKigocPXrUTrwQEXJycobFWL0Yo3U0HNjY2IiWlpZhHzRZX1+P3Nxch14kcf2O7sODDz6Is88+e8DscwURBQF4HsBCABMAXE1EExzs9wSALwbXQv9QWlqK8PBw9Xc7fvx46XmRBCR79+5FeHg4srKyQERq0G4gM+SeFz3xkp+fjyNHjqCurs6n89fW1mLfvn1WQ0aCefPmobq62udOuampSR0rtGXMmDGoqKjwWYRVV1cjLCxMnZElyMzMRGJiol/EiwjWFbOYtOTm5p4Q4kVsH+7xO9XV1TAajQ69YeL6HF1neXk5ysvLB3wpCyecCqCCmauYuQ/A+wAu0dnvNgAfAwj8XOU6iJlGYhg2Pz9fel4kAcmePXswbtw4NSxhOMw4GjLx0traitbWVoeeF8D34DbhWXEkXrT7eIuw0ZHn5ciRIz53ltXV1cjOzraKdwEUr4i/gna3bt2K0NBQTJhg9wIcMOKlpaUFCQkJWLNmjW65K3EitjsSNwsWLMD999/vB0sHloaGBgCuRZqzcpPJNKDLWLggA8BBzedayzYVIsoAcBmAl5ydaKgXfHWGmGkkGD9+PFpaWobyvkskuohp0oKJEyeio6MD9fX1Q2iVc4ZMvIjhFGfixde4l++++w6JiYmYNGmSbh2pqak+J6vTm2kk8NeMo+rqarshI0FRURF27drl80rZW7duxaRJk+zijwBFvHR0dAz5qrh79uxBe3u7Q7HmrNNmZped+qZNm/DDDz/4ydqBQ4gXR1MZXXlexHFDuPAm6WyzfcV7BsD9zOzUPeTOgq9DQXt7O+rq6qzEi+gc5NCRJJDo7e1FdXW1GlQOYFgE7Q6ZeBEd+ujRo+3K0tLSEBcX57N4qaqqQn5+vp3HAlC8FgUFBT4nwysrK0NUVBRGjRplVzZY4qW/v9+nh4yZ1WUB9BB1D7X3RQwjis7bFtEp63XanZ2d6O/vt9pPS29vL1pbW30eqvQXXV1dqr22CNHhi+cFcHwfjx07NtBrWtUC0P5gMgHYvuJNB/A+EdUA+BGAF4jo0oE0yp9oZxoJ5HRpSSBSUVEBs9ls53kBpHjRRXToeXl5dmVCWPgqXmpqapCdne2wPDs7W13x1VvKysqQn58PIvuXyZycHBgMBp/Ei/B4OBMvAHxapPHAgQNobW0NePFSW1sLwLHHwNmwkbYj1+vUhaBxNotnMDnttNPw0EMP6Za5GjZy5nnp6upCV1eX1XlsOf/883HHHXd4bLMHbAIwlohyiSgUwFUArKKPmTmXmXOYOQfARwBuZuZPB9Iof6KdaSTIyclBaGio9LxIAgrtNGlBcnIykpOTpXjRo7KyEhkZGYiMjNQt91W8GI1GHDx40CqHjC05OTmor6/36S3T0UwjAAgNDUVWVpZPM44czTQS5OXlISYmxqe4F2fButq6A0W86HW6fX196rCWXqeu7cj1ysU5e3p60Nra6hd7vaW7uxulpaUoKSnRLXc2bNTb26vmd9G7Tu02R+Jl586d2LJli8d2uwszGwHcCmUWURmApcxcSkQ3EtGNA1bxIFJaWorIyEir9icoKAhjx46VnhdJQCGeR614ARDwM46G1POiF+8iKCgoQGNjo9dxFvX19TAajS7FCzPj4MGDDvdxxpEjR1BbW+tQvADHZxx5iyvxYjAYUFhY6JN42bp1K4KCgjB58mTd8vj4eMTHxw+5eHE2bKQVJ648L3qdvvacQz10tH//fgDHxZotWs+LrZdIm7/F1X3Q82B1dnais7NzwL9rZl7GzOOYeTQzP2bZ9hIz2wXoMvMiZh5Wy5uXlpZiwoQJdkPW+fn50vMiCSj27t2LjIwMREdHW22fOHEidu/eHRCeaD0CWrwA3gftig7AlXgB4PXQkbOZRoKBFi+AMnS0Y8cOr/OXbN26FQUFBYiIiHC4z2DNOHrjjTewceNG3TJnnhfRKScnJzv1vDgq155TTzQwM/70pz/hwIEDblyFb4g4LEfiRYiO/v5+uyy64tpyc3O98rwI4dbS0oLOzk4vrJcA9jONBOPHj0dlZaXDeCbJcebNm4fHH3/cq2O/+OIL5ObmDrkXdThgO9NIMHHiRHR2djpsh4aaIREvZrMZjY2NbokXb12sQpC4innR7usp7ooXMS3cG6qrqxETE4MRI0Y43GfKlCno7u72Ovh4165dmDJlitN9BkO89Pf348Ybb8TTTz+tWy5+RM3NzXaNv+iUTznlFLS0tMBsNuuWT5w4UbdT13oh9H6s1dXV+M1vfoPXX3/dgyvyDvE9Hj58GD09PVZl4rcjnl1bL5IQaRMnTkRXV5fd0hFi/+zsbKfiBRj6YcLhSltbGw4dOqQrXvLz82E0Gn2eKHCi09bWhlWrVuGvf/2rV8uf/PnPf0ZNTY3DFyGJgnZBRlsCPWh3SMSLaJD1ZhoJcnJyEBYW5rXnRQiSrKwsh/tkZmbCYDB4LV7KysoQHBzs9DqEQPM27kXMNNILCBaIoGfhbfIEo9GI2tpa3cBpLbm5uaipqRlQF2J5eTn6+vp0O02TyYRDhw6p60fZChDRKZ9yyikwm812YrG5uRkxMTHIyspyOGwUHx8Pg8GgO2wkOhtHcSj+RNux2dpy+PBhmEwmVWza3getSAPsh45E+ZQpU3TFi1a4yQ7WO/SCdQVyxpF77NixA4AiYj744AOPjt29ezdWr14NANi+fbufLTuxaGxsREdHh0PPCyDFixUiJ4kzz0tQUBDGjRvnk3gZOXIkwsPDHe4TEhKCzMxMrzp9QBEvY8aMsVp6wBZfp0s7myYt8MWDVFtbC5PJ5NRDBSjipaenx2GQpz8QazTpdZpNTU0wGo3q+k62dmg9L4B+p52SkoLU1FTdWJGGhgZkZmYiLS1N1/MibPLHOlKu0Io3W1vEdTsSL1rPi/azoKmpCbGxscjJydGNeZGeF9/RmyYtkAs0uoeYRDBq1Ci8+OKLHh370ksvITQ0FCkpKVK8uEA8h3riJTExEampqVK8aBHixZnHAvBtxlFNTY3TeBdBTk6OT54XZ0NGwHGviDeeF2ZGTU2NS69IRkYGgoKCvLoOcYyreyVsGMgOTXg1mpubcfToUasy0YlPmzYNgL54CQsLU58pvU49OTkZKSkp6O3txZEjR6zKGxoakJaWhoyMDKfipbKyUp1q7Atr167Fv//9b92yqqoqVfTael6E4BDB1bZepKamJoSEhKjH63moUlJSkJaWhs7OTjuXfG1tLRITExEfH68rIpkZzzzzTMCOgwcCtbW1MBgMyMzMtCuLi4tDWlqa9Ly4YPv27Rg5ciTuu+8+bNq0CZs2bXLruKNHj+Kf//wnrrjiCpx55pk+pZA4GXA000iQnZ0dsL/1IRMvKSkpiI2NdbpfQUEBqqur0d3d7XEd+/fvd+lNALzP9dLX14eKigqX4iUiIgIZGRleeV6amppw7Ngxl56X4OBgjBo1akDFy2BMl9YOydjWIzpx4Xmx9RqITjklJQWAY8+LKLft9A8dOoS0tDRkZmbqDhsJe5hZfbP2hbvuugs33XST3XZmRlVVFWbNmgXAsefllFNOARE5FGmpqanqZy3iPqSlpQGwvw+1tbXIzMxEXl6ernipqKjAXXfdhaeeesqTyz2paGxsRFJSkrqmkS1yxpFrtm/fjsLCQlx77bWIiopy2/vyr3/9C52dnbj55ptRWFiI8vJyuxchyXH27t2LiIgIXaENQPVUByJDFvPibMhIUFBQAGbGvn37PDq/yWTCgQMH3Pa81NXVeRz9X1FRAZPJ5FK8AN7POHJnppEgJyfHq+Gv/fv3g4h0MwTbnl9r00Cwc+dO1XNi23GKTtzZsFFqaipEinhXnbq2nJnR0NCAkSNHOvW8CNt8HTpqbGzE5s2b0djYaGdnS0sLjh49ismTJyM2NtaheBGLcurFvIgEU7bXKT6npqaqi3za3se6ujpkZmYiNzdXV7wIgbl8+XJPL/ukobGxUX3O9MjOzvY6PcPJQG9vL3bv3o2pU6ciLi4O11xzDd577z2Xkx6YGc8//zymTp2K008/HVOnTgUzD0qc2nDl4MGDuuvmCVJTUx0uQzLUDJnnxV3xAnge3Hbo0CH09/e7LV7MZrPHrjFhk95YoS2DIV689SDV1NQgPT1dd00jLeHh4Rg5cuSAiZfOzk7s378fl1yiLC6sJ15CQkKQkZGBhIQEXfGSkpKCpKQkANYeB2ZGc3OzledF26l3dHSgt7dX9bx0dHTYva1VVVVh/vz5iIyM9Lkx/OKLL9S/bc8l7m9eXh4yMjLsvEANDQ2Ijo5GdHQ0UlJSdGcbpaSkIDo6GuHh4XaeF+2wEWDvwaqtrUVGRgby8vJQU1NjN2tLCLe9e/f6lHzxRMaVeElLS0NjY2PA5s8YakpLS2E0GjF16lQAwE033YSenh68+eabTo9bv349du7ciZtvvhlEpB4vh44cI4bLHZGamorm5ma7diAQGBLx0t/f7zLeBTgeZ+HprAd3h0K0+3ja8Qub3LmOMWPGoLGx0S7OwhWiI3P3Ourr6z1eoNHd2CAADt/GPaG/v183f8iuXbsAALNnz0ZsbKxdPXV1dcjIyIDBYEBaWpqdeBEdRnBwMEaMGGElTtrb22E0Gq08L9pOX5xLiBdRn6CjowOtra0YPXo0TjnlFIeeF3dXC162bBni4uIA2IsXcd15eXnIzMy0E9VieAsAUlJSHHpeiMgup43JZEJLS4uVeNHex97eXjQ3N6vDRr29vXbipqSkRLVdel/0EQLREWlpaejr60N7e/vgGTWMsM34PWXKFMycORMvvvii0070hRdeQGxsLH7yk58AUIJ9R4wYIYN2ndDY2OhUvKSkpAz1CvQOGbIkde54XqKjo5GcnOzx274YPnE35gXwXLxUV1cjPj4eCQkJLvcV1+ppx19dXY2UlBRERUW53NfbbMGeihdfPC+dnZ2YPn065syZY1cmBMHkyZORl5dnV4+IxQCUxl/bqYoVo0WHkZKSorscgNYzo5dpVgTsivoEWm/IpEmTUFJSYvfWvGzZMiQlJeG1115zeg+MRiO+/PJLXHbZZUhJSbETQuIZycnJ0RUv2jclvfFo4XnRuw+HDx8GM6vDawaDwUq81NcrayMK8aK1R7Bz507MmzcPY8aMkeLFAWJozhF6wlFynO3btyM6OtpqosJNN92EiooKfP3117rHNDc348MPP8R1112ntpfC+yI9L45xx/MCOF7BfigJaPECeNdhupOgTjBq1CgQkVfixZ3hHMD76dKe1CEEiCdxLyLHiyfi5eDBg15lBzUajbjqqquwc+dObNu2zW7IoaSkBDExMcjOztYNFhXDGQAwcuRIq4a/o6MD/f39aqdt63HQZtcNDQ1FQkKCbqbZkSNHqgJJL99JXl4eJk+ejJaWFrsf80cfKdnrb7zxRqxatcrhffjhhx/Q1taG4uJiTJ48WXfYKDU1FVFRUcjIyEBDQ4NV5mQRmwPAbtiop6cHR44cUeNdbO+D2DclJQVBQUFISUnRTc6XkZGhG6Dd1dWFyspKTJ48GcXFxVi1apVXwfQnMmLhSylevGf79u2YMmWKVRzGj370IwQFBeHbb7/VPWbDhg3o6+vDVVddZbW9sLAQJSUlXmcfP5Hp6urCkSNH3BIvgRi06xfxQkQLiGgvEVUQ0QPuHDPQ4iU1NdVpuntBaGgoMjIyPA529URYiKGlgRQv3niQxPpP7og8QPkuzGazV8GGd911F5YvX4777rsPgP2QQ0lJCSZNmgQiUr9z4d1gZjWQFIA6bCTKRacsfmjJycm6nhfRqdsGoWmHjYRA0g4bCfGSm5uLSZMmAbAO2jWbzVi+fDmKi4sxfvx4XH755Q7jtJYtW4agoCCce+65mDRpEkpLS2EymazqEm+cmZmZMJvNVp2credFxOsAx0WaKw+UuE+2w29CvGRmZiI7OxtEZCUixTonkyZNQnFxMXp6erBmzRrd6zxZsX0W9ZDixTFms1mdaaQlLCwMaWlpDtcdE9tt27KpU6eit7dXzu7SQTyrJ63nhYiCADwPYCGACQCuJqIJzo4JCgpymu5eS25uLg4cOGDVwLvCk6EQwPNcLyL/irvCIiYmBikpKR6JFzFjyt06vMkW7ElsEOD9dOlnn30Wzz33HH71q1/hiSeewNixY7Fs2TK1nJmxc+dOVRjk5eVZJcRra2tDd3e3lXg5duyYGlSrHRYS/+t5XhyVNzQ0IDQ0FPHx8YiIiMCIESPsho0SEhIQHx+v2qj1mGzfvh0NDQ248sor8fnnnyM0NBQXXHCB7sKIy5cvx5lnnqmeq7u728oLVVVVpd5nWy9QT08P2tvbrWJetNdvK9KceV7EfdRbkDIzMxOhoaEYNWqUlXjRDu3Nnj0bERERVt+jxD3xIsqkeLGnqqoKR48eVYNttegFsAvq6upgMBjs7rsQQXLoyB7x/LnzrJ6Q4gXAqQAqmLmKmfsAvA/gEmcHhIWFuX3y3Nxc9Pf3e7TSb01NjdveBMDzmToNDQ3o6elxW1gAiqepvLzc7f1ra2thNBrdrkNkCw408fL111/jrrvuwiWXXIInnngCAFBcXIxvvvlGHXKora1FR0eHmnjNNt5CO5wBwG6mjK14SU5OVtPoa8tFvIuteBFBsGIJBttYE603JCkpCSNHjrTyvAgv0oIFC5CTk4PPPvsM9fX1uOyyy6zWJqqvr8e2bdtQXFwM4HiiOSGE+vv7ceDAAbUuWy+Q1kOkvV5xLXoirbu7W02qZ3uf9Dwv0dHRav4l2wDtkpISREZGIi8vD+Hh4Zg7d64ULzbY3mM94uPjERoaKsWLDrbBuloyMjLUuCxb6uvrkZaWhuDgYKvt48ePR3h4uAza1cG2PdEjPj4ewcHBJ6x4yQCgHUeotWyzgoh+QUSbiWizoznlenjaYZrNZrdzvAhycnJUseAOnkxhFowZM8ajqaXe1OFprhexr7P1n7RkZmYiKCjII/Hy7rvvIiEhAe+++66atGvhwoXo6elR1x8RnbfwaohrFh2n1iMA2Lvdbd92U1JSwMxqXojm5mbExcWpollv2Ej7A7ZNVKf1hgg7tZ6XZcuWYfr06WqHddppp+Gtt97CunXrsHjxYnV4a8WKFQCgipcJEybAYDCo5zp48CDMZrPVsBFgv5q2iHmxHY/W87zYlgcHB6tB5iJ2SMzg0AZFA7ALnC4pKcHEiRPVWITi4mJUVlZ6JMpPdNzxvBCR7ow5ieLFDA4OxoQJ9s779PR0p56X9PR0u+3BwcGYNGmSFC86uCNeDAaDbkqGQMAf4kVvxUC7BAbM/A9mns7M0x2lItbDU/HS0NCAvr4+j8WLyWRy27vjrXipra11O8DRmzo89SCJ9Z/c9YQFBwcjKyvLI/FSWVmJCRMmWM2Ysh1yEF4MsS6RbbyFNhYDgF2CNVvPil6nLbYBirhpa2tDX1+feh5xTgBWierMZrPdEg2TJk3C7t27YTQacfjwYWzYsEEVJIIf//jHePzxx/Gvf/0LjzzyCADFQ5ORkaGKtIiICIwZM8ZuTSfxnScmJiIsLMxOvNh6XkTDoud50W4XU3iFhyktLQ1Go1EVedq4IkARL/X19eju7laH9oS3CFBEKADpfdFgOzTnCCle9Nm2bRsKCgp016TLyMhAe3u77irT9fX1qqfSFjHjSObVsaahoQEGg8GqbdQjULPs+kO81ALQpmfNBKDv2/OCrKwsEJHbHaanQyHafd3t+D3JvyIQAcruXkd1dTUMBoPbXhFhT11dndopu8LT2CDA8wDqiooKu+Bs7ZCDyIA5atQo1SMQHh6OjIwMtZ7a2lr1bRWw97w0NTUhMTFRdRnbdtra6cPa8paWFvU8tp6XpqYm9Pb2or6+Hn19fVbiZfLkyejt7UV5eTm+/PJLmM1mtSPX8sADD2DRokV45JFH8Oabb+LLL7/EwoULrVYI18440s5qApQ3dO04v6tho6amJoSGhiImJgaAvojT3gfb+6id0aW1o6amBo2NjWhpaVGFF6A8C/n5+VK8aGhsbERCQoLLpI9SvOijF6wrEJ4VvaEjR54XQBEvra2tAbtGz1DR0NCA5ORkh8tYCAI1y64/xMsmAGOJKJeIQgFcBeAzP5wXgDIbKDMz02Px4mnMi/ZYV1RXVyMtLc2t2UwCT6dLV1dXIzMz0+mK1bZ4mi14oMVLV1cXDh06pDuzrLi4GFVVVSgvL1dnGtnWox02SktLU+/FiBEjEBwcrMa82GY0deV50Qah9ff3o7m52U68AEosjK03BIBV0O7y5cuRmJiIGTNm2F0jEeHll1/G7Nmzcf3116Ozs9POQzNp0iR1scfq6mo1i7DWFvF9Hjp0SE0+Byh5kCIjI61iXrSeFVsRZ5t/RCteTCYTDh06ZOV50Xo9hcDSel4A5XuUM46O4ypBnUCKF3saGxtx6NAh3WBdwD4GTNDd3Y3W1laHnhcZtKuPqxwvghN22IiZjQBuBfAFgDIAS5nZr2toe9JhepKgTiC8G56IF0+GcwDPp0t7U4cnuV5MJhMOHjzolXhpbGzUdd3aImJ89MSL8FR8+umnKCsrsxMv2lwvtrEYYlaB1vOi51lx5XlpampSO37tj1ibqM7WGwIoy1YEBQVhx44dWL58ORYsWODw7SU0NBSffPIJxo4di9DQUMybN8+qfNKkSWBmlJaWoqqqCtnZ2Vbn0oqXhoYGpKSkWAUlahsWW5FmK+JsO1YxVHbo0CE0NjbCZDLZDRsBikdIDG3Zfk8LFy70OKsz4Dq9AhH9PyLaafn3PRFN8biSIcBVgjpBWloampubZf4RDSIuxZHnRfwubT0v4iXGkXgRKRhk3Is17ooX4XkJtGE3v+R5YeZlzDyOmUcz82P+OKcWT8RLTU0NkpOT3cpKKwgLC0N6errbwa7eCIsRI0YgISFhQMWLJx4ksf6TJyIPOP427k4d4lr1llAQQw7PPfccjEaj3Ru9iLfo6emxG84ArBPV2YqXxMREdcVls9mMlpYWu5gXcZxtECxgHSirN3wXHh6OcePG4e2330ZLS4vukJGWESNGYO3atVi3bp3dSuraGUfaWU0CMWwkFo+0bWy0M6dsRVpUVBQiIyPR3Nxsl4UYsPa82M7oEueOjIxEVVUVSkpKkJaWpsYVCc4++2z85S9/cXr9triZXqEawGxmngzgDwD+4VElQ4SrdY0EaWlpYGZ16FJyXLxMmaKvU8WwkK3nRXx2NGwUHR2NcePGSfFig6ulAQSpqano6+vTXdZlKBmyDLuekJub6/a6Pd4MhQDu53oxGo04ePCgx8ICcH+Bxu7ubtTX13tchyfZgr2JDQLsZwI5w5l4AZQhB5HwTm/YiJmxf/9+u0BSwNrtbtthBAUFITExEc3NzWhra4PJZLLqtLXDRnoR91r3dFVVlZr3RMukSZNw8OBBEBHOP/98l/ciJSVFXRHb9jqjoqKwc+dOXfGSmZmJvr4+tLS06IoXbTCdrecFOJ7rpaurC93d3Vb3SQw7NTQ02M3oAqAmDBSeF1uBCSjC/5577nF5/Ta4TK/AzN8zc5vl4wYosXQBjyfiBZC5XrRs27YN2dnZDpdciY2NRVRUlEPx4sjzAkAuE2CDo5chPQI118uwES+iI3OFpzleBO7O1Dl48CBMJpPX4sWd6dL79u0DoOQo8ASRLXggxYuwyZ2VvisrK5GUlIT4+HjdchH/ERISYnetohPfuXMn2tvbdcXLoUOH0Nvbi46ODrs4A9Fp204fBpSkgWFhYWhqarJa10gQFxeHqKgoddjIVlAAxz0mp512mp03whMMBgMmTpyIdevWobW11e650nqBtIsyCrTDRraeF1He3NysOwtGO2XXdkaXIC8vD+Xl5di9e7edwPQBt9IraFgMQHchJW0KBr2kgINJb28v2tvb3Yp5kYnq7Nm1a5euQBaIAHbbYSPx2Zl4mTJlCmpqajxeHPdEpb29HX19fW4JbdtZjYHCsBEvgOuZOmazGfv37/fa8yKEiTO8mcIsGDNmDGpqalzOBiorKwOgxFZ4iru5XjzN8SIYMWIEUlJSVBudoTfTSMtZZ52FqKgo5Ofn23k2hGBYu3YtAPtOVcQMiMbfUadtO30YUBpBMdyil2WSiNRYE0fiRXTkroaM3GHy5MnYsmULAOgOGwGKaHbkeWlublbX1HHkebFdGkAgRGBtbS1CQ0PthFheXh52796N3t5epx2Lh7iVXgEAiOgcKOLlfr1ybQoGV1M+BxrxrEnPi+eYTCZUVFQgPz/f6X56uV7q6uoQERGhrnauh3g58nSJlhMVd3K8CKTnxQfcFS9iequ34sVoNDrM4CjwVbwIgeWMsrIyGAwGjBs3zuM63PUgebL+ky0FBQVueV5ciZewsDD8/ve/x2233WZXlpaWhvDwcHz33XcA7N+qRo4cCbPZjF27dgGw7zCceV7E/mLYKCEhwS6vREZGBsrLy9HQ0KD7Xc+ePRuXX345rrvuOofX5y5aj4besBGgvJX29/dbxeYAiigzGo2qt86RiHOU+VXEDtXV1SEjI8NqGretPX70vLiVXoGIJgN4FcAlzHzYX5UPFO4kqBNIz4s1Bw8eRG9vr8s2T2+JAEfPrhZxXvE7OdmR4mWQSE9PR2hoqEvx4s00aYG7wa7V1dUICgrCqFGjnO6nh7szjvbs2YPc3FzdRE2ucDdbsLexQYAiXsrKypxGn/f09ODgwYMO410E9957L37+85/bbRfxFiLITs/zAhxPcOeJ50V8Fp4XvR9wZmamem49z0tcXBw++ugjr541W2xzp2hJS0tDUFAQNm/erH62vQ4Aqohz5HlxlDxNO2xke4+19gQFBXnlCXSAy/QKRJQF4BMA1zLzsOhxPBEvUVFRiImJkeLFghAV7oiX+vp6q7bHWYI6wejRo0FEUrxY8ES8JCUlqRMgAolhIV4MBgOys7PdFi/eel6053BEdXU1Ro0aZbeGhju4m+ulrKzM645CZAt2levFV/HS1tbm9GEWq0K7u3q4Hnl5eWojZds4iR/djh07ANh3ymJ9IxHTYjscIsSLXhwJoIgXUbeeePEnQrzEx8fbBSsGBQUhLS3NoXgRHaUQL3oirbe3V4210hMvbW1tqKys1O0AxLWPGzfOKzGth6P0CkR0IxHdaNntIQCJAF4gou1EtNkvlQ8gnogXQOZ60eKueElPT0dfXx8OHz7uiHOWoE4QERGBrKwsKV4seCJegoODkZSUJD0v3uLOdGlfPC8i9sNVHd5MYRakpKQgOjraadCuyWTCvn37fBIvgPNcL96s/6RFjEs7i3txluPFXUTHOWLECLvhLVvxYtthiE66rKwMCQkJdsn+xCwdR+JFL9PsQCEWe3RUT2ZmpjoryxvPiygXCwJqEedz5Xnx45ARAP30Csz8EjO/ZPn7Z8ycwMxTLf/sp2oFGO4syqhFipfj7Nu3DzExMS6Fn22uF2Z2y/MCKMJIiheFhoYGhIaGOpxMYUsgJqo7ocRLSUkJMjMz1fTonhAREYHc3Fy1E3CEL+KFiFxOl66urkZvb6/X4sWd4S+x/pO3Qx7CNmfiRVyjL+JF3Ge9TlV0uvv27UNERIRdXh/RaZeWluqu3ZGSkoK+vj7s37/fLo5EW2dkZKTLtT/8we23344bbrhBt0x7/XoxL4Bzz4so1+tU9fLbaImMjMQvf/lLXHvtte5cxklNY2MjoqKi3M4xJcXLcfbt24dx48Y5jVsB7HO9tLW1oaenx6XnBTguXgIt2dpQIIbLXd1vQSAuETCsxMvhw4edTnXbunUrioqKvK6jqKgIW7dudVje3d3tMIDTXVyJFyEIXEXdO8KdbMG+DK8BSicXHR3tUrzExcUhMTHRqzoA2K2urCUyMhKxsbEwm81WKfEFoqMuLy/X7bTFNrPZ7HDYSNjg7g/cFx544AHccsstumXirTIiIsJOmCcmJsJgMODAgQMIDw93KOIOHDig+1arl9/GlpdeegkXXnih+xdzkuJujheBFC/HEeLFFbZLBLiT40Uwbtw4tLe3y8SAcD+7rkCKFx9wNePo6NGj2Lt3r8/ipbKyEh0dHbrlotP3VbxUVVU5nJLtyzRp4Hi24IEUL0SE/Px8l+JlzJgxPnX8Qrw4apjEj0+vwxCdttFo1PWc6K3xo0XUOdBDRu4ghJTem1JQUJAaz+NMxNn+LdBb00niHd6Il46ODvT09AygVYFPb28vampq3BIvwlMoho3cyfEikDOOjuNudl1BIK4sfcKIl507d4KZHa6L4Q5C+DhKI+3LNGnB6NGj0d/fr8Yw2FJWVoa0tDS3xyL1cJXrxZv1n2xxNV26oqLC5UwjV+Tm5iIkJMShgLBdXVmLq05bL02+luTkZERHR3ucKHAg0IoXPcS16Ik0vWURbI8VgkeKF99wd1FGQaBOQR1sKisrwcxuiZfQ0FAkJyfbeV7cGTYaO3YsACleAMXz4onQTklJwdGjR91a026wOGHEixju8cXzIoSPo6Ejf4gXVzOOfJlpJHCV60Ws/xQZGel1HQUFBaitrdUdxuvv70dNTY1P8S6Akr5+/fr1DodTnHleRowYAYNBeby98bwYDAasXbsWv/71r72y3Z+It0q92Bzg+LXodZwRERGIjo622k9LSEgIkpKSYDAYPHoTO5mpr6/H9OnTsXfvXqvt7i7KKJCJ6hTcnWkk0OZ68US8ZGdnIyQk5KQXLyaTCc3NzR57XgBrod3d3Y2zzz4b33zzjd9tdIdhI14SExMRHR3tVLwkJye75T50RGpqKtLT052Kl/DwcJ8aedGh6804Yma/iJfc3FwcOHDAoTu6vLzc6yEjgbBRz/ty4MABmEwmn8ULAEybNs1hALbozPU6bYPBYDWcYot26rQjUTBlyhSH66wMJr54XrTbHXkF0tLSkJaW5tX0/5ORzz//HFu2bMHHH3+sbjOZTGhpaZHixQuEmBCeEVdolwior69HUlISwsLCXB4XHByM0aNHn/Tipbm52WGsnyP0xMv333+PtWvX4p133vG7je4wbMSLSFrmTLwUFRX5HFzpLGi3uroaOTk5PtWRnp6O8PBwXc9LQ0MDOjs7fRYvp59+OkwmE77//nu7st7eXmzYsAFnnHGGT3U4my7tj5lG7uBs2Ag43mnrderBwcFITExEcHAwRowYMXBG+oGMjAxERUU5vJ/i+h3dB1flY8aMCYjhseHC6tWrrf4HgJaWFpjNZr+IlzvuuAMfffSRz3YOF/bt24fU1FSn6f21aJcIcCfHi5aTbbp0aWkpLrvsMnR1danbPMnxItATL3q/g8Fk2IgXwPF06Z6eHpSWlvo0ZCQoKirCnj17rL5sgS/TpAUGgwGjR4/WFS++BusKZs+ejaCgIKxcudKubP369eju7sa8efN8qmP06NEIDg4OCPHiqMNwJl7EcampqerwUqASFhaGXbt24aabbtItF9fvyvPi6D69+uqr+OCDD/xg6YkPM2PNmjUAgHXr1qG/vx8AHGYwdobYVyteamtr8eyzz+Kpp57yl8kBj7szjQQZGRloampCX1+fujSAu4wbNw7l5eUwm83emDrseOWVV/Dpp5/iiy++ULd5I17Es6oN2hW/g6qqKocxnANJYLfaNgjxYjtPf9euXTAajT4F6wqKiopgNpvV1PBa/CFeAAy4eImNjcWMGTN0xcvKlSthMBgwe/Zsn+oICQnB2LFjHYqXyMjIAY+hEEs0OHrzcuVxyMjIGDZBqjk5OQ4z3Po6bDRixIhByWVzIlBRUYH6+nqce+65OHbsmJr52NHCl84Q8UZa8bJixQoAwA8//HDSTOn1VLyI33tDQwPq6+s99rz09vYOSWc7FCxfvtzqf8A38SJEend3N3744Qece+65AI4LmcFk2ImXrq4uux/1tm3bAPgWrCsQ5xDnFLS3t6O9vd0v4mXMmDFqhL2WsrIyxMbGOozB8IR58+Zh06ZNdtO+V65ciRkzZrjtonWGoxlHYqbRQOdHOeecc/Dpp59i1qxZuuWuPC9LlizBa6+9NmD2DRbOAna12z3pWE92lixZovsCI1zkDz74oNVnT5cGENjmelm2bBnCwsLAzPjyyy+9sHx40dHRgcbGRo89L4Aya7KxsdFjzwtwcsw4qqysxL59+xAWFobly5er/Y143jx5VsPDwxEXF6c+5xs2bEBfXx9uu+02JCQk6A4dtba24uGHHx6wVADDSryIKbOlpaVW27du3Yq4uDi/5OTIzMxEUlKSXdyLqNMfdYwZMwbd3d3qujsCEazrj05/3rx5MJvNVoq4s7MTGzdu9HnISFBQUICKigr09fVZba+srBzwISNAGYK75JJLHN6v8ePHIyEhwW5dI235xIkTB9LEQWHmzJm45pprcOaZZ+qWX3755bjzzjsRGxs7yJYNT3bv3o3bb79dd6bZmjVrkJqairPOOgsTJ05Uf1/+EC99fX34+uuvce211yI5OdnqbflEpby8HID7M42A4+Jly5YtYGYpXhwgnp9f/epXqKurQ0lJCQBFvMTExLidCVqgTVS3evVqGAwGzJo1C2effbau5+W5557DI488gvfff9/HK9FnWImXOXPmICYmBq+++qrV9q1bt6KwsNAvnT4RobCw0E68vPrqq4iMjMTcuXN9rsPRdOmysjKvM+vacsYZZyA8PNxq6Ojbb7+FyWTyq3gxmUxW12EymQZNvLjipptuQnl5+Qk/iyYhIQFvv/22w5lRM2bMwNNPPz0omYKHC5988gmWLl2qW/bmm28CUIZwxKwWQIl3Wb16NWbPng0iwpw5c7B27Vr09/ejsbERoaGhHns0teJl3bp1OHLkCC688EKcf/75WLFihV0yS2bG8uXLh2Viu/3796vDYgJPp0kDx4eNNm3aZPXZHdLS0hAdHW0nXrZt24aNGze6fZ5AgZmxYsUK3edh2bJlGDt2LG699Vb1M+B5dl2Bdn2jNWvWoLCwEHFxcZgzZw4qKirUIGpAyVwufkdvvPGG7vk2bdrkNLart7fXqT3DSrzExMRg0aJFWLp0qXoT+/v7sWPHDr8MGQmKioqwa9cu9ea1tLTgvffew09/+lOfkscJ9MRLR0cHDh065HO8iyA8PBxnn322lXhZuXIlwsPDHb6he4reGkd1dXXo6+sLCPEiZhRJTj4++OADnHfeeboNYEdHB66//nosXrwYbW1tVmVGoxFvv/02CgsLYTabraaBVlVVoa6uDnPmzAGgvEx1dXVhy5YtaoI6TwViamoqGhoaVFESEhKCuXPnori4GC0tLWpMjeDzzz9HcXExHn/8cY/qCQQWL16MCy64wKrd27dvH4jIo4SWSUlJCA0NVcWLJ54XIrKbcdTX14eLL74YF1xwQUAlYXOHFStWYOHChfjDH/5gtb27uxvffPMNFi5ciJEjR6KwsFD1xHiaoE4gsuz29PRgw4YNVr8DwDru5bvvvkN1dTUKCwvx7bff2qUGYWbcdNNNuOeee/DDDz/o1nfVVVc5tWdYiRcAuPXWW9Hf349//OMfAJQ8I729vX4J1hUUFRWhv79fHSp69dVX0dvbqypYXxk1ahRCQkKsfsT+CtbVMm/ePJSWlqpvditXrsRZZ53lMPDTU8T0Wq14GayZRhKJI7q7u3H33Xfjq6++0o1pevnll9HZ2YmjR4/ihRdesCr78ssv0dDQgIceeggzZ87Em2++qcYKiHF90ViLWKs1a9Z4nKBOkJaWhp6eHhw5cgTLli3DrFmzEBMTg/POOw8Gg8Fq6IiZ8dhjjwFQYnI6OzvtzsfMVm/AgcLGjRuxcuVKmM1mPPnkk+r2ffv2IScnx608LQIiQnp6ujrk5GluL1vx8s4776C2thYtLS12Xv1AQMysskX7PDz//PNW8Y2rV69GT08PiouLAQALFy7EunXr0N7e7vHSAAIxbLRhwwb09vaqv4PJkycjLi7OKu7lzTffRExMDD744AMYDAa89dZbVudatWoVtmzZAgB44okn7OrasmULPv30U+cGMfOg/5s2bRr7wvnnn8/p6enc19fH//znPxkAl5aW+nROLeXl5QyAX331Ve7v7+dRo0bx3Llz/XZ+ZuZx48bxjBkzuKOjg5mZX3/9dQbA+/bt81sdmzZtYgD87rvvcmNjIwPgP/3pT347PzNzVlYW/+QnP2FmZrPZzLfddhsD4JqaGr/WIwlsAGzmIWhLtP9Eu/L0008zAM7KyuL09HTu7u5W7ezp6eGRI0fy/PnzeeHChZycnMzHjh1Ty3/0ox9xUlIS9/b28quvvsoAeMOGDczMfM0113BKSgqbzWZ1/wkTJvCCBQu4qKiIi4uLPb5vb7/9NgPgr776igHwX//6V7XsjDPO4BkzZqifV65cyQB48eLFDICfeOIJu/P99re/ZSLir7/+Wre+3t5ePnz4sMd2ukNtbS3/7ne/U9s0LZdeeiknJCTwNddcwyEhIVxbW8vMzNOmTePzzz/f47rOPPNMBsAhISFsMpk8Ovahhx5ig8HAPT09bDQaeezYsVxYWMhnnXUWZ2Zmcm9vr9X+ZrOZ//KXv/APP/zgsZ3u0N3dzW1tbbpltbW1HB8fz/Pnz7d67piZV69ebfU8PPbYY2rZbbfdxhEREeqzv3btWgbAS5cu5fj4eL711ls9tvORRx5hAPyb3/yGicjK5osuuojHjh3LzMxHjhzhqKgo/tnPfsbMSn+dnZ1t9T3Nnz+f09LS+N5772Ui4j179ljVVVxczAkJCU7blWEpXj7//HMGwO+//z7fcccdHBERwUaj0adzajGZTBwbG8s333wzf/zxxwyA//3vf/vt/MyKWAkKCuL8/HwuKyvj++67j0NDQ7m/v99vdRiNRo6Pj+cbbriB33vvPQbAGzdu9Nv5mZUHs6ioiDs7O/myyy5jAPzTn/7U7ocmObEJFPFy9OhRTklJ4XPOOYdXrVrFAPiZZ55R7XzllVdUsSAa/xdeeIGZmVtaWjg0NJTvuOMOZmbu6OjgiIgIvvHGG9lsNvOoUaP4xz/+sdV133TTTRwdHc2pqal8/fXXe3zfhGj5yU9+wgB49+7datmjjz7KRMSNjY3MzDx37lxOS0vj7u5uPvfcczk1NdVKeG3ZsoWDgoKYiDg3N5ePHj1qVVdvby/PnDmTR4wYwZWVlXa2GI1Gvuuuu/j555/XtbW+vp4XLFjA//3vf+3K+vv7eebMmQyAr7nmGquyXbt2MQD+/e9/z5WVlRwUFMR33303m81mjomJ4dtuu839G2bhxz/+sSpQPeWdd95R7/UHH3zAAPjDDz/kZcuWMQB+/fXXrfZ/7rnnGACnpqZyQ0OD3fnKysp4/vz5/P333+vW9/XXX/PVV1/Nra2tdmVdXV1cWFjIqampqqATmM1mvuiii5iIGAC/8sorVuXnnXcep6Sk8LFjx3jhwoWclJTEXV1dzMw8evRovuCCC9R9+/v7OT4+nq+++moGwH/84x/du1kaXnrpJQbAY8eO5aKiIquyv/71rwyA6+rq+I033mAAvHbtWmZmte9ZuXIlMzNv3ryZAfCf//xnbmxs5PDwcF68eLF6rvXr1zMAfvzxx0888WI0GjkvL49nzpzJZ599Np9xxhk+nU+P2bNn8+mnn85z5szhrKwsv4oKwapVqzg5OZljYmJ47NixfMopp/i9jssuu4yzs7P5Zz/7GcfHx/tV5DEz33nnnRwREcEFBQUcFBTETz31lBQuJyGBIl6efPJJq4bznHPO4dTUVO7q6lLfsouKithsNrPZbObTTjuNc3Nzub+/X+2ktm/frl7XNddcw3FxcVxaWsoA7Dp20fkB4Pvvv9/j+1ZSUsIAODw8nLOzs61+O6KRf+utt9QG/S9/+QszH3/rfu6555iZua+vj6dOncppaWn86aefMgBVhAluueUWBsCRkZE8depUtaMT3H333eq1vPvuu1ZlXV1dPH36dAbAERERvHnzZqvyhx56iAHw/PnzVZu19zAqKopbWlqsPgtRs2TJEo/v25133skAvGr7N27cqL6QTp06lceNG8dGo5HNZrPVZ2bmHTt2cFhYGJ955pkcHh7OCxYssPIgNDU1cV5eHgPg5ORkO1G4fft2jo6OZgA8Z84cK6+O2Wzma6+9lomIIyIi+PTTT+eenh61/F//+hcD4CeffJJnz57NsbGxqsAR1/DnP/+ZmZm/++47Vajv27fP6tkQXHnllRweHq6OKnjKJ598oj4fd999t1WZeFb/9a9/8ezZs3ns2LHqs3zs2DGOi4vja6+9lpmZr7jiCo6NjeX29nZmZr755putvHHnnnsuJycn85EjRwZOvAD4MYBSAGYA0909zlfxwsz81FNPMQAODg7mW265xefz2XLXXXdxcHCw1QMyEBw4cEBtFGzf6vyBaJDj4uL40ksv9fv5X375ZQbASUlJvGrVKr+fXzI88FS8AFgAYC+ACgAP6JQTgGct5TsBFLk6Z2FhIScmJloNQwh3+ZNPPskfffQRA+APPvhALRcN8nvvvcfTpk3jqVOnWl2XGKo577zzGADv2rXLqvzQoUNqg/63v/3N4/vW3NysHn/TTTdZlZlMJk5JSeGrr76aL774Yk5ISOAjR44ws9LxnXnmmZyVlcV9fX38xz/+kQHwJ598wsxKh0BEvG7dOmZmfvPNNxkA33vvvbxs2TImIr722mvVDka0EzfffDPPnj2bQ0ND+dtvv1Xt+L//+z8mIn7llVc4OzubR44cyQcOHGBmRUgZDAa+7rrr2Gg08qxZszg6Opr37dvHVVVVqqdFIETLOeecwwD4iy++8Pi+CZF6+eWXe3xsW1ubVf1aT8vSpUtVT0xXVxcXFBRwWloaNzU18YsvvsgA+KmnnmJmZbhn5syZHB4ezm+//TYnJCRwQUGBOpxy8OBBzsjI4MzMTLW/0rvnjzzyCH/44YdWz0BTUxMnJSXxjBkzuL+/n8vLyzkiIoIvuugiNpvNfOmll3J8fLzVEN2sWbM4IyNDvTe2Qko8AwD4888/9/i+rVu3Tj3+P//5j1WZ0Wjk2NhYPvfcc+2GsJiZb7zxRo6IiOCtW7eywWDg++67Ty2rrKxkg8HA99xzD3/77bdWw6cDKV4KAIwHsHqwxUtbWxtHRkZ6rSJd8dZbb6lvROKNYaDo7u7mRx991KHb0RfKysrUB86bNxxXtLS08N1338379+/3+7klwwdPxAuAIACVAPIAhALYAWCCzT7FAJZbRMzpAH5wdd709HQGYBebcP7553NiYiIXFhby6NGjrbyPJpOJx48fz5mZmQyA//73v1sdazKZOCsrSxXoel7F/Px8XW+FO5hMJvUlSW845rrrrlPbuYcfftiq7H//+x8DUIectS8/nZ2dnJWVxfn5+fz9999zeHg4n3POOaoH+dFHH1XbhP/+979sMBj44osvZqPRyK2trTx+/HgeMWIE7927l++//34GwE8//TQzK96imJgYnjJlCtfU1HBmZiaPHTuWOzs7mVl5IUtISOBp06bx4sWLOTQ01G5I5JJLLlHbperqao/v27vvvssA+Pbbb/f4WGbmlJQUBsCjRo2y8oYYjUYeN24cFxYW8s9//nOr+CGz2cyXXXYZh4SE8KZNm9ShPiGGV69ezSEhITx//nxubW3lKVOmcExMDO/YsYOZj9/zhx9+mNeuXcvBwcF84YUXqp6ce++9lwHwG2+8wVdffTWHhIRwSUmJapsYmvnd737HAPjBBx+0uqYVK1YwAI6KiuL8/Hy7a25oaFDvua3nzB0qKioYABOR7hDYBRdcoJYfPHjQqmzDhg0MgDMzMzk0NJTr6+utyq+66iqOiYnh0047jdPS0lSv4ICJF/UkQyBemJl/+ctfMgDesmWLX86nRbwd3HDDDX4/92BiNptZNOra8XSJxJ94KF7OAPCF5vOvAfzaZp+XAVyt+bwXwEhn5w0KCuKLLrrIzrYffvhBbbRffPFFu3IRmBsSEsLNzc125Q8++KDTt/wbb7yRATgMknVFRkYGh4aG2sWoMDO///77DICjo6PtAm3FMAcAHjFihF08hujMQkJCeNSoUdzU1KSWmUwmvuiiizg4OJgjIyNZxAsJKisrOTk5mZOSklSPgFa4rVixgoOCgjgqKopDQkLsOkPtEMPPf/5zu+sSnVlYWJhXQ9nffPONT17xs846S1esMjO/9tprqu0PPPCAVdnhw4c5MzOTo6KidD0MIt4jKSmJg4KCeMWKFWqZ2WzmRYsWqZ7w0aNHWwW99vf389y5c1UxaytWjUYjn3rqqapAsX2pNpvNPG3aNAbAd911l+51i3JbMekOlmEcO++k4C9/+YvqpbTFbDarIl8E8mrZunWres+fffZZdXtAiBcAvwCwGcBmb4Ks9Dh06NCAxViYzWZ++umnua6uzu/nHmx+8YtfcF5enoxFkQwYHoqXHwF4VfP5WgDP2ezzOYCzNJ9X6rUx2nYFAG/btk3XvksuuYTT09OtAlwFPT09nJWVxVdccYXusZWVlRwSEmIXMCn4z3/+wwaDgauqqty5VXacc845fPHFF+uWtba2cmRkJP/617/WLf/kk0+YiPidd97RLb/hhhs4LCxMN1C/vb2dx40bx1lZWXZvwsyKwIiIiODzzjtPN+bvhRdesBpGseXmm2/msLAwLi8v1y0///zz+dRTT9Utc0VtbS0HBwfzZ5995tXxt912m9Ubvpbe3l7Oy8vj008/nfv6+uzK16xZw0FBQbxo0SLdNvU3v/kNA+CXX35Z99zz5s3jqKgo3rlzp115U1MTZ2Vl8ZQpU+xmPTErL9Xh4eF2okrwn//8hwHwmjVrdMsfe+wxjo2N1b0uV5jNZh45cqTDZ3HHjh0MgD/66CPd8r/97W8cEhLCe/fu1S0vLi7mrKwsq9mBPokXAF8D2KXz7xLNPkPieZG4R1dX14APfUlObjwULz/WES9LbPb5n454mebsvOPGjXNoX1dXl5XnwZbm5mZdz4egtrbW6ZRcb95kBe3t7Wosix51dXVOvRN63iKByWRyet1HjhzRndosaGxsdDpZQU/0CMxms9O6jx49qjv84C719fVev5AdO3bMaZvY1tamKx4Ehw4dcli32Wx2+tLb39+vziDTo7Oz0+mz2Nzc7PS6RSySo7r1Zky5S1NTk9P74qxuk8lkN5yk5ciRI3bfibN2hZRy3yCi1QDuYebNrvYFgOnTp7Nt5kiJRDJ8IaItzDzdzX3PAPAwM59v+fxrAGDmP2n2eRnAamZ+z/J5L4A5zHxI55QAZLsikZxoOGtXhl2GXYlEMuzZBGAsEeUSUSiAqwB8ZrPPZwB+SgqnA+hwJlwkEsnJhU+eFyK6DMASAMkA2gFsF29TLo47AiUALxBJAtAy1EY4IFBtC1S7AGmbt3hqWzYzJ7u7MxEVA3gGysyj15n5MSK6EQCY+SVSFgl6DsqU6mMArnfl2ZXtitdI27wjUG0LVLsAP7Yrfhk28hQi2uyui3mwkbZ5TqDaBUjbvCWQbXNEINssbfMOaZvnBKpdgH9tk8NGEolEIpFIhhVSvEgkEolEIhlWDJV4+ccQ1esO0jbPCVS7AGmbtwSybY4IZJulbd4hbfOcQLUL8KNtQxLzIpFIJBKJROItcthIIpFIJBLJsEKKF4lEIpFIJMOKQRUvRLSAiPYSUQURPTCYdTuw53UiaiKiXZptI4joKyIqt/yfMAR2jSKib4iojIhKieiOALItnIg2EtEOi22PBIptFjuCiGgbEX0eSHZZbKkhohIi2k5EmwPFPiKKJ6KPiGiP5Zk7IxDs8oRAaltku+KVbQHdrlhsCci2JVDbFYsdA9a2DJp4IaIgAM8DWAhgAoCriWjCYNXvgDehJMHS8gCAlcw8Fsp6KkPREBoB/IqZCwCcDuAWy70KBNt6Acxl5ikApgJYQEoG1ECwDQDuAFCm+RwodgnOYeapmlwHgWDf3wGsYOZ8AFOg3L9AsMstArBteROyXfGUQG9XgMBuWwKxXQEGsm1xtOiRv/8BOAPAF5rPvwbw68Gq34ldOQB2aT7vBTDS8vdIAHsDwMb/ADg30GwDEAlgK4DTAsE2AJmWH8NcAJ8H2vcJoAZAks22IbUPQCyAaliC9wPFLg+vIeDaFtmu+GRXQLUrlroDtm0JxHbFUu+Ati2DOWyUAeCg5nOtZVugkcqWNVQs/6cMpTFElAOgEMAPCBDbLO7T7QCaAHzFzIFi2zMA7gNg1mwLBLsEDOBLItpCRL+wbBtq+/IANAN4w+ISf5WIogLALk8YDm1LQN1P2a54zDMI3LYlENsVYIDblsEUL6SzTc7TdgIRRQP4GMCdzNw51PYImNnEzFOhvI2cSkSnDLFJIKILATQx85ahtsUJM5m5CMrwxi1ENGuoDQIQDKAIwIvMXAigCwE8ROQA2bZ4gGxXPGMYtC2B2K4AA9y2DKZ4qQUwSvM5E0D9INbvLo1ENBIALP83DYURRBQCpYF5l5k/CSTbBMzcDmA1lPH9obZtJoCLiagGwPsA5hLROwFglwoz11v+bwLwbwCnBoB9tQBqLW+5APARlAZnqO3yhOHQtgTE/ZTtilcEdNsSoO0KMMBty2CKl00AxhJRLhGFArgKyrL3gcZnAK6z/H0dlHHhQYWICMBrAMqY+W8BZlsyEcVb/o4AMB/AnqG2jZl/zcyZzJwD5dlaxczXDLVdAiKKIqIY8TeA8wDsGmr7mLkBwEEiGm/ZNA/A7qG2y0OGQ9sy5PdTtiveEchtS6C2K8AgtC2DHMBTDGAfgEoAvx3Muh3Y8x6AQwD6oajExQASoQRmlVv+HzEEdp0Fxe29E8B2y7/iALFtMoBtFtt2AXjIsn3IbdPYOAfHg+oCwi4o4787LP9KxfMfCPZBmd2x2fKdfgogIRDs8vAaAqZtke2KV7YFfLtisSeg2pZAblcsdgxY2yKXB5BIJBKJRDKskBl2JRKJRCKRDCukeJFIJBKJRDKskOJFIpFIJBLJsEKKF4lEIpFIJMMKKV4kEolEIpEMK6R4OQEgokTLiqLbiaiBiOosfx8lohcGqM47ieinA3FuF/UuIqLnvDgumYhWDIRNEsmJBBGZNO3JdstSAicERFRIRK9a/n6YiO7R2SediD7y8vx/JaK5vtopcU3wUBsg8R1mPgxlPj2I6GEAR5n5rwNVHxEFA7gBSrbEYQEzNxPRISKayczrhtoeiSSA6WYlTb8dlkR3xMxmvfJhwG8A/NHZDqxkrP2Rl+dfAuAVAKu8PF7iJtLzcgJDRHOI6HPL3w8T0T+J6EsiqiGi/yOiJ4mohIhWWNKGg4imEdEayyJfX4g0zjbMBbCVmY2WY24not1EtJOI3rdsO5WIvrcsyPW9yLJo8Zx8SkT/JaJqIrqViO627LeBiEZY9ltNRM9Yjt1FRKfqXF8yEX1MRJss/2Zats/WvDVuExkooSRJ+n9+vckSyQkOEeUQUZnFi7sVwCgiutfym9tJRI9o9v0tEe0loq+J6D3h2bD8nqdb/k4iJdW+WIzxL5pz/dKyfY7lmI+IaA8RvWsRTiCiGZZ2YQcRbSSiGCL6joimauxYR0STba4jBsBkZt6h2TyFiFYRUTkR/Vxzvbssfy8iok8sbWQ5ET2psftNS9tUQkR3AQAz7weQSERpfvwKJDpIz8vJxWgA5wCYAGA9gMuZ+T4i+jeAC4jof1DeHC6xeCquBPAYFC+LlpkAtIuUPQAgl5l7yZLiG0pq71nMbCSi+QAeB3C5pewUKCvahgOoAHA/MxcS0dMAfgplBVcAiGLmM0lZaOx1y3Fa/g7gaWZeS0RZAL4AUADgHgC3MPM6Uhah67Hsvxku3rokEgkiSFndGQCqAdwFYDyA65n5ZiI6D8BYKGvoEIDPLL/RLijp8wuh9C1bYd1O6LEYQAczzyCiMADriOhLS1khgIlQ1qlaB2AmEW0E8AGAK5l5ExHFAugG8CqARQDuJKJxAMKYeadNXdOhZO/VMhnA6QCiAGyztIG2TLXY0gtgLxEtgbIScgYznwIAmnYPluueCWUNKckAIcXLycVyZu4nohIAQQBEDEgJgBwoDdQpAL6yvOQEQUlzbstIAGWazzsBvEtEn0LxbgBAHIB/EtFYKCnJQzT7f8PMRwAcIaIOAP/V2KF9W3oPAJj5WyKKtWkgAGX9kwkWWwEg1vJ2tQ7A34joXQCfMHOtpbwJQLrO9UgkkuNYDRuREvOyn5k3WDadZ/m3zfI5GoqYiQHwb2Y+ZjnOnfWlzgMwmYjEME2c5Vx9ADaK365FTOUA6ABwiJk3AQBbVsUmog8BPEhE90J52XpTp66RAJpttv2HmbsBdBPRN1AE2XabfVYyc4elnt0AsqGk4s+zCJn/AfhSs79sZwYBKV5OLnoBgJnNRNTPx9eGMEN5FghAKTOf4eI83VC8JoILAMwCcDGUBmQigD9AESmXWRq/1bZ2aOru1fytfSZt166w/WwAcIal8dHyZ8sbVDGADUQ0n5n3WGy23VcikbimS/M3AfgTM7+s3YGI7oT9b1RgxPEwBW3bQQBuY+YvbM41B9bthAnH2yi7Opj5GBF9BeASAFdA8bLYYttuQedcevbb2cHMbUQ0BcD5AG6x1Ck81LKdGQRkzItEy14AyUR0BgAQUYhFiNhSBmCMZR8DgFHM/A2A+wDEQ3kTiwNQZ9l/kZf2XGmp4yworuUOm/IvAdwqPogxbyIazcwlzPwElKGifMsu42DvNpZIJJ7xBYAbLEOyIKIMIkoB8C2Ay4gowuIBvUhzTA2AaZa/f2RzrpvoeMzdOFJWR3bEHgDpRDTDsn8MKRMIAGXo6FkAm5i5VedYtd3ScAkRhRNRIpRFFzc5qVuFiJIAGJj5YwAPwnrygmxnBgHpeZGoMHOfxX37LBHFQXk+noHiItWyHMDblr+DALxj2Z+gxKC0WwLb/klEd8P7yPs2IvoeQCzs424A4HYAzxPRTout3wK4Ecq49zlQ3pJ2W+wFlHgfvTFtiUTiJsz8JREVAFhvGbI9CuAaZt5KRB9AGXbZD+A7zWF/BbCUiK6FdXvwKpThoK2WgNxmAJc6qbvPEou3hIgioHg45kOZYbmFiDoBvOHg2D1EFEdEMZZhawDYCKVNyALwB2auJ/emhmcAeMPy8gYAvwaUFz4oAmmzG+eQ+IBcVVriFZYg3/uYuXyAzr8awD3M7LdGgIi+hRKM3Oavc0okEn1oENI22NSXDmV4Ot/RVG7LrKAjzPzqANlwGYAiZn5wIM4vOY4cNpJ4ywNQAuCGBUSUDOBvUrhIJCcepCTM/AHAb13koHkR1jEs/iYYwFMDeH6JBel5kUgkEolEMqyQnheJRCKRSCTDCileJBKJRCKRDCukeJFIJBKJRDKskOJFIpFIJBLJsEKKF4lEIpFIJMOK/w8bX6uJrHcZJwAAAABJRU5ErkJggg==\n", 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