(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 180114, 11759]*) (*NotebookOutlinePosition[ 181168, 11795]*) (* CellTagsIndexPosition[ 181124, 11791]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData["A GEOMETRIC DISCOVERY"], "Title", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["The Sum of the Squares of the First n Integers"], "Subtitle", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["\n"], "Input", AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Introduction"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "In beginning calculus it is common to show how to evaluate \ #\.ba\[DotlessI]x\[Currency]dx directly from the definition of the definite \ integral by using the formula [n(n+1)(2n+1)]/6 for the sum of the squares of \ the first n integers, \ 1\[Currency]+2\[Currency]+3\[Currency]+...+n\[Currency]. In most texts this \ formula is presented without consideration of its genesis. The formula is \ usually proved using mathematical induction, but induction really does little \ to explain how the formula was discovered in the first place; in fact, \ students often object that there is a certain \"circularity\" to the process \ since they are being asked to prove something that they have no real reason \ to predict might be true. This project shows that there is a simple geometric \ technique for demonstrating the origin of the formula. This technique can \ even be used by pre-algebra students to find the sum of the first n squares \ for any particular n even if they have not yet learned to use a variable. \ For students who do understand variables, working with the this technique and \ ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" can provide a nice exercise in algebraic identities.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["The Basic Geometric Model of the Sum "], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[{ StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " can display a three dimensional model of the sum \ 1\[Currency]+2\[Currency]+3\[Currency]+4\[Currency] using unit cubes.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Clear[cL]"], "Input", AspectRatioFixed->True], Cell[TextData[ "\n\ncL = Graphics3D[\n\t\t{Text[FontForm[\n\t\t\"SumSquares\",\n\t\t\ {\"Chicago\",10}],{2.5,4,2}],\n\t\tGrayLevel[0.8],Map[Cuboid,Flatten[\n\t\t\ Table[\n\t\tTable[{i,j,k},{i,0,3-k},{j,0,3-k}],{k,0,3}],2]]}];"], "Input", AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Show[cL,Boxed->False,\n\t\tLighting->False,\n\t\tViewPoint->{1.300, -2.400, \ 2.000}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "The top cube in the figure \"SumSquares\" represents 1\[Currency], the next \ layer of four cubes represents 2\[Currency], the third layer of nine cubes \ represents 3\[Currency], and the bottom layer of sixteen cubes represents 4\ \[Currency]."], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData["The Sum Embedded in a Cube"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "A cube 5 units on an edge can be thought of as the sum of three of these \ \"pyramid\" models of the basic sum together with several other unit cubes \ which serve to separate the three \"pyramids\" and complete the larger cube. \ The figure \"BigCube\" shows the three basic sum \"pyramids\" in light gray \ in the lower rear left, the lower rear right, and the top front corners. The \ darker unit cubes separate the three basic sum \"pyramids.\""], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Clear[c1,c2,c3,c4,c5,c6,c7]"], "Input", PageBreakAbove->True, AspectRatioFixed->True], Cell[TextData[ "\n\nc1 = Graphics3D[{GrayLevel[0.8],Map[Cuboid,Flatten[\n\t\tTable[\n\t\t\ Table[\n\t\tTable[{i,j,k},{k,0,3-i}],{j,0,3-i}],{i,0,3}],2]]}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "\n\n\nc2 = Graphics3D[{\n\t\t\ Text[FontForm[\"BigCube\",{\"Chicago\",10}],{7,2,0}],\n\t\t\ GrayLevel[0.8],Map[Cuboid,Flatten[\n\t\tTable[\n\t\tTable[\n\t\t\ Table[{i,j,k},{k,0,j-1}],{i,5-j,4}],{j,1,4}],2]]}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "c3 = Graphics3D[{GrayLevel[0.8],Map[Cuboid,Flatten[\n\t\tTable[\n\t\t\ Table[{i,j,k},{i,5-k,4},{j,0,k-1}],{k,1,4}],2]]}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "\n\nc4 = Graphics3D[{GrayLevel[0.5],Map[Cuboid,Flatten[\n\t\tTable[\n\t\t\ Table[{i,j,k},{i,3-k+1,3-k+1},{j,0,k-1}],{k,1,4}],2]]}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "\n\nc5 = Graphics3D[{GrayLevel[0.5],Map[Cuboid,Flatten[\n\t\tTable[\n\t\t\ Table[{i,j,k},{i,5-k,4},{j,k,k}],{k,1,4}],2]]}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "\n\nc6 = Graphics3D[{GrayLevel[0.5],Map[Cuboid,Flatten[\n\t\tTable[\n\t\t\ Table[{i,j,k},{k,0,3-i},{j,4-i,4-i}],{i,0,3}],2]]}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "\n\nc7 = Graphics3D[{GrayLevel[0.25],Map[Cuboid,Flatten[\n\t\tTable[\n\t\t\ Table[{i,j,k},{k,4-i,4-i},{j,4-i,4-i}],{i,0,4}],2]]}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "Show[c3,c1,c2,c4,c5,c6,c7,\n\t\tLighting->False,\n\t\tBoxed->False,\n\t\t\ ViewPoint->{1.500, -1.500, 1.300}];"], "Input", PageBreakBelow->True, AspectRatioFixed->True]}, Open]]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData["View of the Three Basic Sum \"Pyramids.\""], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "The figure \"3Pyramids\" shows the three basic sum pyramids moved apart \ slightly and without the separating unit cubes."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["\n\nClear[c1L,c2L,c3L,c4L];"], "Input", AspectRatioFixed->True], Cell[TextData[ "\n\nc1L = Graphics3D[{\n\t\tGrayLevel[0.8],Map[Cuboid,Flatten[\n\t\tTable[\n\ \t\tTable[\n\t\tTable[{i,j,k},{k,0,3-i}],{j,0,3-i}],{i,0,3}],2]]}];"], "Input",\ AspectRatioFixed->True], Cell[TextData[ "\n\n\nc2L = Graphics3D[{\n\t\tGrayLevel[0.8],Map[Cuboid,Flatten[\n\t\tTable[\ \n\t\tTable[\n\t\tTable[{i,j,k},{k,0,j-1}],{i,5-j,4}],{j,1,4}],2]]}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "\nc3L = Graphics3D[{\n\t\tGrayLevel[0.8],Map[Cuboid,Flatten[\n\t\tTable[\n\t\ \tTable[{i,j,k},{i,5-k,4},{j,0,k-1}],{k,1,4}],2]]}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "\nc4L = Graphics3D[\n\t\tText[FontForm[\n\t\t\"3Pyramids\",\n\t\t\ {\"Chicago\",10}],{5,5,0}]];"], "Input", AspectRatioFixed->True], Cell[TextData["Clear[pc1L,pc2L,pc3L,pc4L]"], "Input", AspectRatioFixed->True], Cell[TextData[ "pc1L = Show[c1L,Boxed->False,\n\t\tLighting->False,\n\t\tViewPoint->{2.500, \ -1.500, 2.500},\n\t\tDisplayFunction->Identity];"], "Input", AspectRatioFixed->True], Cell[TextData[ "\npc2L = Show[c2L,Boxed->False,\n\t\tLighting->False,\n\t\t\ ViewPoint->{1.500, -2.500, 2.500},\n\t\tDisplayFunction->Identity];"], "Input",\ AspectRatioFixed->True], Cell[TextData[ "\n\npc3L = Show[c3L,Boxed->False,\n\t\tLighting->False,\n\t\t\ ViewPoint->{1.500, -1.500, 1.300},\n\t\tDisplayFunction->Identity];"], "Input",\ AspectRatioFixed->True], Cell[TextData[ "\n\npc4L = Show[c4L,Boxed->False,\n\t\tLighting->False,\n\t\t\ ViewPoint->{1.500, -1.500, 1.300},\n\t\tDisplayFunction->Identity];"], "Input",\ AspectRatioFixed->True], Cell[TextData[ "Show[Graphics[{\n\t\t\tRectangle[Scaled[{0,0}],Scaled[{.66,.66}],pc1L],\n\t\t\ \tRectangle[Scaled[{.34,0}],Scaled[{1,.66}],pc2L],\n\t\t\t\ Rectangle[Scaled[{.17,.34}],Scaled[{.83,1}],pc3L],\n\t\t\t\ Rectangle[Scaled[{.25,0}],Scaled[{.75,.5}],pc4L]}],\n\t\t\t\ DisplayFunction->$DisplayFunction];\n"], "Input", AspectRatioFixed->True]}, Open]]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData[ "View of the Three Basic Sum \"Pyramids\" with the Separating Cubes"], "Section", Evaluatable->False, PageBreakAbove->True, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "The figure \"3SeparatedPyramids\" shows the three basic sum \"pyramids\" in \ light gray and the separating unit cubes in medium and dark gray. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Clear[c1,c2,c3,c4,c5,c6,c7,c8]"], "Input", AspectRatioFixed->True], Cell[TextData[ "\nc1 = Graphics3D[{\n\t\tGrayLevel[0.8],Map[Cuboid,\n\t\tFlatten[\n\t\t\ Table[\n\t\tTable[\n\t\tTable[{i,j,k},{k,0,3-i}],{j,0,3-i}],{i,0,3}],\n\t\t\ 2]]}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "\nc2 = Graphics3D[{\n\t\tGrayLevel[0.8],Map[Cuboid,Flatten[\n\t\tTable[\n\t\t\ Table[\n\t\tTable[{i,j,k},{k,0,j-1}],{i,5-j,4}],{j,1,4}],2]]}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "\n\nc3 = Graphics3D[{Text[FontForm[\n\t\t\"3SeparatedPyramids\",\n\t\t\ {\"Chicago\",10}],{5,0,6}],\n\t\tGrayLevel[0.8],Map[Cuboid,Flatten[\n\t\t\ Table[\n\t\tTable[{i,j,k},{i,5-k,4},{j,0,k-1}],{k,1,4}],2]]}];\n"], "Input", AspectRatioFixed->True], Cell[TextData[ "\nc4 = Graphics3D[{\n\t\tGrayLevel[0.5],Map[Cuboid,\n\t\tFlatten[\n\t\t\ Table[\n\t\tTable[{i,j,k},{i,3-k+1,3-k+1},{j,0,k-1}],{k,1,4}],\n\t\t2]]}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "\n\nc5 = Graphics3D[{GrayLevel[0.5],Map[Cuboid,Flatten[\n\t\tTable[\n\t\t\ Table[{i,j,k},{i,5-k,4},{j,k,k}],{k,1,4}],\n\t\t2]]}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "\n\nc6 = Graphics3D[{GrayLevel[0.5],Map[Cuboid,Flatten[\n\t\tTable[\n\t\t\ Table[{i,j,k},{k,0,3-i},{j,4-i,4-i}],{i,0,3}],\n\t\t2]]}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "\n\nc7 = Graphics3D[{GrayLevel[.25],Map[Cuboid,Flatten[\n\t\tTable[\n\t\t\ Table[{i,j,k},{k,4-i,4-i},{j,4-i,4-i}],{i,0,4}],\n\t\t2]]}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "\n\np41 = Show[c1,\n\t\tLighting->False,\n\t\tBoxed->False,\n\t\t\ ViewPoint->{2.500, -1.500, 1.300},\n\t\tDisplayFunction->Identity];"], "Input",\ AspectRatioFixed->True], Cell[TextData[ "\np42 = Show[c2,\n\t\tLighting->False,\n\t\tBoxed->False,\n\t\t\ ViewPoint->{1.500, -2.500, 1.300},\n\t\tDisplayFunction->Identity];"], "Input",\ AspectRatioFixed->True], Cell[TextData[ "\np43 = Show[c3,\n\t\tLighting->False,\n\t\tBoxed->False,\n\t\t\ ViewPoint->{1.500, -1.500, -1.400},\n\t\tDisplayFunction->Identity];"], "Input", AspectRatioFixed->True], Cell[TextData[ "\np44 = Show[c4,c5,c6,c7,\n\t\tLighting->False,\n\t\tBoxed->False,\n\t\t\ ViewPoint->{1.500, -1.500, 1.300},\n\t\tDisplayFunction->Identity];"], "Input",\ AspectRatioFixed->True], Cell[TextData[ "Show[Graphics[{\n\t\t\tRectangle[Scaled[{0,0}],Scaled[{.5,.5}],p41],\n\t\t\t\ Rectangle[Scaled[{.5,0}],Scaled[{1,.5}],p42],\n\t\t\t\ Rectangle[Scaled[{.25,.55}],Scaled[{.75,1}],p43],\n\t\t\t\ Rectangle[Scaled[{.25,.12}],Scaled[{.75,.62}],p44]\n\t\t\t}],\n\t\t\t\ DisplayFunction->$DisplayFunction];\n\n\n"], "Input", AspectRatioFixed->True]}, Open]]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData["Completely Separated"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "The figure \"AllSeparated\" shows the 3 basic sum \"pyramids\" in light \ gray, the unit cubes which are the interfaces separating two \"pyramids\" in \ medium gray, and the main diagonal of 5 unit cubes in dark gray. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Clear[c1,c2,c3,c4,c5,c6,c7]"], "Input", AspectRatioFixed->True], Cell[TextData[ "\nc1 = Graphics3D[{\n\t\tGrayLevel[0.8],Map[Cuboid,\n\t\tFlatten[\n\t\t\ Table[\n\t\tTable[\n\t\tTable[{i,j,k},{k,0,3-i}],{j,0,3-i}],{i,0,3}],\n\t\t\ 2]]}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "\nc2 = Graphics3D[{\n\t\tGrayLevel[0.8],Map[Cuboid,Flatten[\n\t\tTable[\n\t\t\ Table[\n\t\tTable[{i,j,k},{k,0,j-1}],{i,5-j,4}],{j,1,4}],2]]}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "\n\nc3 = Graphics3D[{Text[FontForm[\n\t\t\"AllSeparated\",\n\t\t\ {\"Chicago\",10}],{5,0,6}],\n\t\tGrayLevel[0.8],Map[Cuboid,Flatten[\n\t\t\ Table[\n\t\tTable[{i,j,k},{i,5-k,4},{j,0,k-1}],{k,1,4}],2]]}];\n"], "Input", AspectRatioFixed->True], Cell[TextData[ "\nc4 = Graphics3D[{\n\t\tGrayLevel[0.5],Map[Cuboid,\n\t\tFlatten[\n\t\t\ Table[\n\t\tTable[{i,j,k},{i,3-k+1,3-k+1},{j,0,k-1}],{k,1,4}],\n\t\t2]]}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "\n\nc5 = Graphics3D[{GrayLevel[0.5],Map[Cuboid,Flatten[\n\t\tTable[\n\t\t\ Table[{i,j,k},{i,5-k,4},{j,k,k}],{k,1,4}],\n\t\t2]]}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "\n\nc6 = Graphics3D[{GrayLevel[0.5],Map[Cuboid,Flatten[\n\t\tTable[\n\t\t\ Table[{i,j,k},{k,0,3-i},{j,4-i,4-i}],{i,0,4}],\n\t\t2]]}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "\n\nc7 = Graphics3D[{GrayLevel[.25],Map[Cuboid,Flatten[\n\t\tTable[\n\t\t\ Table[{i,j,k},{k,4-i,4-i},{j,4-i,4-i}],{i,0,4}],\n\t\t2]]}];"], "Input", AspectRatioFixed->True], Cell[TextData["Clear[p71,p72,p73,p74,p75,p76,p77]"], "Input", AspectRatioFixed->True], Cell[TextData[ "p71 = Show[c1,\n\t\tLighting->False,\n\t\tBoxed->False,\n\t\t\ ViewPoint->{2.500, -1.500, 1.300},\n\t\tDisplayFunction->Identity];"], "Input",\ AspectRatioFixed->True], Cell[TextData[ "\np72 = Show[c2,\n\t\tLighting->False,\n\t\tBoxed->False,\n\t\t\ ViewPoint->{1.500, -2.500, 1.300},\n\t\tDisplayFunction->Identity];"], "Input",\ AspectRatioFixed->True], Cell[TextData[ "\np73 = Show[c3,\n\t\tLighting->False,\n\t\tBoxed->False,\n\t\t\ ViewPoint->{1.500, -1.500, -1.400},\n\t\tDisplayFunction->Identity];"], "Input", AspectRatioFixed->True], Cell[TextData[ "\np74 = Show[c4,\n\t\tLighting->False,\n\t\tBoxed->False,\n\t\t\ ViewPoint->{1.500, -1.500, 1.300},\n\t\tDisplayFunction->Identity];"], "Input",\ AspectRatioFixed->True], Cell[TextData[ "\np75 = Show[c5,\n\t\tLighting->False,\n\t\tBoxed->False,\n\t\t\ ViewPoint->{1.500, -1.500, 1.300},\n\t\tDisplayFunction->Identity];"], "Input",\ AspectRatioFixed->True], Cell[TextData[ "\np76 = Show[c6,\n\t\tLighting->False,\n\t\tBoxed->False,\n\t\t\ ViewPoint->{1.500, -2.500, 1.300},\n\t\tDisplayFunction->Identity];"], "Input",\ AspectRatioFixed->True], Cell[TextData[ "\np77 = Show[c7,\n\t\tLighting->False,\n\t\tBoxed->False,\n\t\t\ ViewPoint->{2.500, -1.500, 1.300},\n\t\tDisplayFunction->Identity];"], "Input",\ AspectRatioFixed->True], Cell[TextData[ "Show[Graphics[{\n\t\t\tRectangle[Scaled[{0,0}],Scaled[{.5,.5}],p71],\n\t\t\t\ Rectangle[Scaled[{.5,0}],Scaled[{1,.5}],p72],\n\t\t\t\ Rectangle[Scaled[{.25,.55}],Scaled[{.75,1}],p73],\n\t\t\t\ Rectangle[Scaled[{0,.5}],Scaled[{.5,1}],p74],\n\t\t\t\ Rectangle[Scaled[{.5,.5}],Scaled[{1,1}],p75],\n\t\t\t\ Rectangle[Scaled[{.35,0}],Scaled[{.85,.4}],p76],\n\t\t\t\ Rectangle[Scaled[{.15,0}],Scaled[{.65,.5}],p77]\n\t\t\t}],\n\t\t\t\ DisplayFunction->$DisplayFunction];"], "Input", AspectRatioFixed->True]}, Open]]}, Open]]}, Open]], Cell[TextData["\n"], "Input", AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Begin Reassembling the Cube"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "The figure \"ReJoin(1)\" shows some parts of the cube reassembled. 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.56007 L p .5 g F P s P p .002 w .48226 .6619 m .42161 .63362 L .42308 .56007 L p .8 g F P s P p .002 w .54291 .63362 m .48226 .6619 L .42161 .63362 L p .8 g F P s P p .002 w .54446 .71093 m .54291 .63362 L .48226 .6619 L p .8 g F P s P p .002 w .48226 .6619 m .48226 .7377 L .54446 .71093 L p .8 g F P s P p .002 w .42161 .63362 m .48226 .6619 L .48226 .7377 L p .8 g F P s P p .002 w .54291 .63362 m .48226 .6619 L .42161 .63362 L p .8 g F P s P p .002 w .36435 .6619 m .3003 .63362 L .30473 .56007 L p .8 g F P s P p .002 w .42161 .63362 m .36435 .6619 L .3003 .63362 L p .8 g F P s P p .002 w .42005 .71093 m .42161 .63362 L .36435 .6619 L p .8 g F P s P p .002 w .36435 .6619 m .36142 .7377 L .42005 .71093 L p .8 g F P s P p .002 w .3003 .63362 m .36435 .6619 L .36142 .7377 L p .8 g F P s P p .002 w .42161 .63362 m .36435 .6619 L .3003 .63362 L p .8 g F P s P p .002 w .42308 .56007 m .36049 .52874 L .36346 .45748 L p .5 g F P s P p .002 w .42308 .56007 m .36049 .52874 L .36346 .45748 L p .8 g F P s P p .002 w .48226 .52874 m .42308 .56007 L .36049 .52874 L p .5 g F P s P p .002 w .42308 .56007 m .48226 .52874 L .54143 .56007 L p .8 g F P s P p .002 w .42308 .56007 m .48226 .52874 L .54143 .56007 L p .8 g F P s P p .002 w .48226 .60366 m .48226 .52874 L .42308 .56007 L p .5 g F P s P p .002 w .42308 .56007 m .42161 .63362 L .48226 .60366 L p .5 g F P s P p .002 w .30473 .56007 m .36049 .52874 L .42308 .56007 L p .8 g F P s P p .002 w .30473 .56007 m .36049 .52874 L .42308 .56007 L p .8 g F P s P p .002 w .36049 .52874 m .42308 .56007 L .42161 .63362 L p .5 g F P s P p .002 w .48226 .52874 m .42308 .56007 L .36049 .52874 L p .5 g F P s P p .002 w .54143 .56007 m .48226 .52874 L .48226 .45748 L p .8 g F P s P p .002 w .54143 .56007 m .48226 .52874 L .48226 .45748 L p .8 g F P s P p .002 w .60402 .52874 m .54143 .56007 L .48226 .52874 L p .8 g F P s P p .002 w .60715 .60366 m .60402 .52874 L .54143 .56007 L p .8 g F P s P p .002 w .54143 .56007 m .54291 .63362 L .60715 .60366 L p .8 g F P s P p .002 w .48226 .52874 m .54143 .56007 L .54291 .63362 L p .8 g F P s P p .002 w .60402 .52874 m .54143 .56007 L .48226 .52874 L p .8 g F P s P p .002 w .48226 .60366 m .48226 .52874 L .42308 .56007 L p .8 g F P s P p .002 w .42308 .56007 m .42161 .63362 L .48226 .60366 L p .8 g F P s P p .002 w .48226 .52874 m .54143 .56007 L .54291 .63362 L p .8 g F P s P p .002 w .35736 .60366 m .36049 .52874 L .30473 .56007 L p .8 g F P s P p .002 w .30473 .56007 m .3003 .63362 L .35736 .60366 L p .8 g F P s P p .002 w .36049 .52874 m .42308 .56007 L .42161 .63362 L p .8 g F P s P p .002 w .30473 .56007 m .23872 .52874 L .24467 .45748 L p .8 g F P s P p .002 w .36049 .52874 m .30473 .56007 L .23872 .52874 L p .8 g F P s P p .002 w .35736 .60366 m .36049 .52874 L .30473 .56007 L p .8 g F P s P p .002 w .30473 .56007 m .3003 .63362 L .35736 .60366 L p .8 g F P s P p .002 w .23872 .52874 m .30473 .56007 L .3003 .63362 L p .8 g F P s P p .002 w .36049 .52874 m .30473 .56007 L .23872 .52874 L p .8 g F P s P p .002 w .36346 .45748 m .42114 .42305 L .48226 .45748 L p .5 g F P s P p .002 w .41956 .49554 m .42114 .42305 L .36346 .45748 L p .5 g F P s P p .002 w .36346 .45748 m .36049 .52874 L .41956 .49554 L p .5 g F P s P p .002 w .42114 .42305 m .48226 .45748 L .48226 .52874 L p .5 g F P s P p .002 w .66561 .42305 m .60105 .45748 L .54338 .42305 L p .8 g F P s P p .002 w .67034 .49554 m .66561 .42305 L .60105 .45748 L p .8 g F P s P p .002 w .60105 .45748 m .60402 .52874 L .67034 .49554 L p .8 g F P s P p .002 w .48226 .45748 m .54338 .42305 L .60105 .45748 L p .8 g F P s P p .002 w .54338 .42305 m .60105 .45748 L .60402 .52874 L p .8 g F P s P p .002 w .54495 .49554 m .54338 .42305 L .48226 .45748 L p .8 g F P s P p .002 w .48226 .45748 m .48226 .52874 L .54495 .49554 L p .8 g F P s P p .002 w .54338 .42305 m .60105 .45748 L .60402 .52874 L p .8 g F P s P p .002 w .54338 .42305 m .48226 .45748 L .42114 .42305 L p .8 g F P s P p .002 w .54495 .49554 m .54338 .42305 L .48226 .45748 L p .8 g F P s P p .002 w .48226 .45748 m .48226 .52874 L .54495 .49554 L p .8 g F P s P p .002 w .42114 .42305 m .48226 .45748 L .48226 .52874 L p .8 g F P s P p .002 w .42114 .42305 m .36346 .45748 L .2989 .42305 L p .8 g F P s P p .002 w .41956 .49554 m .42114 .42305 L .36346 .45748 L p .8 g F P s P p .002 w .36346 .45748 m .36049 .52874 L .41956 .49554 L p .8 g F P s P p .002 w .24467 .45748 m .2989 .42305 L .36346 .45748 L p .8 g F P s P p .002 w .2989 .42305 m .36346 .45748 L .36049 .52874 L p .8 g F P s P p .002 w .29417 .49554 m .2989 .42305 L .24467 .45748 L p .8 g F P s P p .002 w .24467 .45748 m .23872 .52874 L .29417 .49554 L p .8 g F P s P p .002 w .2989 .42305 m .36346 .45748 L .36049 .52874 L p .8 g F P s P p .002 w .2989 .42305 m .24467 .45748 L .17666 .42305 L p .8 g F P s P p .002 w .29417 .49554 m .2989 .42305 L .24467 .45748 L p .8 g F P s P p .002 w .24467 .45748 m .23872 .52874 L .29417 .49554 L p .8 g F P s P p .002 w .17666 .42305 m .24467 .45748 L .23872 .52874 L p .8 g F P s P p .002 w .42207 .84098 m .35834 .81735 L .36142 .7377 L p .5 g F P s P p .002 w .48226 .81735 m .42207 .84098 L .35834 .81735 L p .25 g F P s P p .002 w .48226 .90117 m .48226 .81735 L .42207 .84098 L p .25 g F P s P p .002 w .42207 .84098 m .42054 .92295 L .48226 .90117 L p .25 g F P s P p .002 w .35834 .81735 m .42207 .84098 L .42054 .92295 L p .25 g F P s P p .002 w .48226 .81735 m .42207 .84098 L .35834 .81735 L p .5 g F P s P p .002 w .36142 .7377 m .42005 .71093 L .48226 .7377 L p .5 g F P s P p .002 w .36142 .7377 m .42005 .71093 L .48226 .7377 L p .5 g F P s P p .002 w .41842 .79229 m .42005 .71093 L .36142 .7377 L p .5 g F P s P p .002 w .36142 .7377 m .35834 .81735 L .41842 .79229 L p .5 g F P s P p .002 w .42005 .71093 m .48226 .7377 L .48226 .81735 L p .5 g F P s P p .002 w .48226 .7377 m .42005 .71093 L .42161 .63362 L p .5 g F P s P p .002 w .48226 .7377 m .42005 .71093 L .42161 .63362 L p .8 g F P s P p .002 w .54446 .71093 m .48226 .7377 L .42005 .71093 L p .8 g F P s P p .002 w .54609 .79229 m .54446 .71093 L .48226 .7377 L p .8 g F P s P p .002 w .48226 .7377 m .48226 .81735 L .54609 .79229 L p .8 g F P s P p .002 w .42005 .71093 m .48226 .7377 L .48226 .81735 L p .8 g F P s P p .002 w .54446 .71093 m .48226 .7377 L .42005 .71093 L p .8 g F P s P p .002 w .36142 .7377 m .29565 .71093 L .3003 .63362 L p .8 g F P s P p .002 w .42005 .71093 m .36142 .7377 L .29565 .71093 L p .8 g F P s P p .002 w .41842 .79229 m .42005 .71093 L .36142 .7377 L p .8 g F P s P p .002 w .36142 .7377 m .35834 .81735 L .41842 .79229 L p .8 g F P s P p .002 w .29565 .71093 m .36142 .7377 L .35834 .81735 L p .8 g F P s P p .002 w .42005 .71093 m .36142 .7377 L .29565 .71093 L p .8 g F P s P p .002 w .42161 .63362 m .35736 .60366 L .36049 .52874 L p .5 g F P s P p .002 w .42161 .63362 m .35736 .60366 L .36049 .52874 L p .8 g F P s P p .002 w .48226 .60366 m .42161 .63362 L .35736 .60366 L p .5 g F P s P p .002 w .42161 .63362 m .48226 .60366 L .54291 .63362 L p .8 g F P s P p .002 w .42161 .63362 m .48226 .60366 L .54291 .63362 L p .8 g F P s P p .002 w .48226 .68253 m .48226 .60366 L .42161 .63362 L p .5 g F P s P p .002 w .42161 .63362 m .42005 .71093 L .48226 .68253 L p .5 g F P s P p .002 w .3003 .63362 m .35736 .60366 L .42161 .63362 L p .8 g F P s P p .002 w .3003 .63362 m .35736 .60366 L .42161 .63362 L p .8 g F P s P p .002 w .35736 .60366 m .42161 .63362 L .42005 .71093 L p .5 g F P s P p .002 w .48226 .60366 m .42161 .63362 L .35736 .60366 L p .5 g F P s P p .002 w .48226 .68253 m .48226 .60366 L .42161 .63362 L p .8 g F P s P p .002 w .42161 .63362 m .42005 .71093 L .48226 .68253 L p .8 g F P s P p .002 w .35407 .68253 m .35736 .60366 L .3003 .63362 L p .8 g F P s P p .002 w .3003 .63362 m .29565 .71093 L .35407 .68253 L p .8 g F P s P p .002 w .35736 .60366 m .42161 .63362 L .42005 .71093 L p .8 g F P s P p .002 w .3003 .63362 m .23247 .60366 L .23872 .52874 L p .8 g F P s P p .002 w .35736 .60366 m .3003 .63362 L .23247 .60366 L p .8 g F P s P p .002 w .35407 .68253 m .35736 .60366 L .3003 .63362 L p .8 g F P s P p .002 w .3003 .63362 m .29565 .71093 L .35407 .68253 L p .8 g F P s P p .002 w .23247 .60366 m .3003 .63362 L .29565 .71093 L p .8 g F P s P p .002 w .35736 .60366 m .3003 .63362 L .23247 .60366 L p .8 g F P s P p .002 w .54291 .63362 m .48226 .60366 L .48226 .52874 L p .8 g F P s P p .002 w .54291 .63362 m .48226 .60366 L .48226 .52874 L p .8 g F P s P p .002 w .60715 .60366 m .54291 .63362 L .48226 .60366 L p .8 g F P s P p .002 w .61044 .68253 m .60715 .60366 L .54291 .63362 L p .8 g F P s P p .002 w .54291 .63362 m .54446 .71093 L .61044 .68253 L p .8 g F P s P p .002 w .48226 .60366 m .54291 .63362 L .54446 .71093 L p .8 g F P s P p .002 w .60715 .60366 m .54291 .63362 L .48226 .60366 L p .8 g F P s P p .002 w .48226 .60366 m .54291 .63362 L .54446 .71093 L p .8 g F P s P p .002 w .36049 .52874 m .41956 .49554 L .48226 .52874 L p .5 g F P s P p .002 w .36049 .52874 m .41956 .49554 L .48226 .52874 L p .5 g F P s P p .002 w .4179 .57187 m .41956 .49554 L .36049 .52874 L p .5 g F P s P p .002 w .36049 .52874 m .35736 .60366 L .4179 .57187 L p .5 g F P s P p .002 w .41956 .49554 m .48226 .52874 L .48226 .60366 L p .5 g F P s P p .002 w .48226 .52874 m .41956 .49554 L .42114 .42305 L p .5 g F P s P p .002 w .60402 .52874 m .54495 .49554 L .54338 .42305 L p .8 g F P s P p .002 w .60402 .52874 m .54495 .49554 L .54338 .42305 L p .8 g F P s P p .002 w .67034 .49554 m .60402 .52874 L .54495 .49554 L p .8 g F P s P p .002 w .67531 .57187 m .67034 .49554 L .60402 .52874 L p .8 g F P s P p .002 w .60402 .52874 m .60715 .60366 L .67531 .57187 L p .8 g F P s P p .002 w .48226 .52874 m .54495 .49554 L .60402 .52874 L p .8 g F P s P p .002 w .48226 .52874 m .54495 .49554 L .60402 .52874 L p .8 g F P s P p .002 w .54495 .49554 m .60402 .52874 L .60715 .60366 L p .8 g F P s P p .002 w .67034 .49554 m .60402 .52874 L .54495 .49554 L p .8 g F P s P p .002 w .54661 .57187 m .54495 .49554 L .48226 .52874 L p .8 g F P s P p .002 w .48226 .52874 m .48226 .60366 L .54661 .57187 L p .8 g F P s P p .002 w .54495 .49554 m .60402 .52874 L .60715 .60366 L p .8 g F P s P p .002 w .48226 .52874 m .41956 .49554 L .42114 .42305 L p .8 g F P s P p .002 w .54495 .49554 m .48226 .52874 L .41956 .49554 L p .8 g F P s P p .002 w .54661 .57187 m .54495 .49554 L .48226 .52874 L p .8 g F P s P p .002 w .48226 .52874 m .48226 .60366 L .54661 .57187 L p .8 g F P s P p .002 w .41956 .49554 m .48226 .52874 L .48226 .60366 L p .8 g F P s P p .002 w .54495 .49554 m .48226 .52874 L .41956 .49554 L p .8 g F P s P p .002 w .36049 .52874 m .29417 .49554 L .2989 .42305 L p .8 g F P s P p .002 w .36049 .52874 m .29417 .49554 L .2989 .42305 L p .8 g F P s P p .002 w .41956 .49554 m .36049 .52874 L .29417 .49554 L p .8 g F P s P p .002 w .4179 .57187 m .41956 .49554 L .36049 .52874 L p .8 g F P s P p .002 w .36049 .52874 m .35736 .60366 L .4179 .57187 L p .8 g F P s P p .002 w .23872 .52874 m .29417 .49554 L .36049 .52874 L p .8 g F P s P p .002 w .23872 .52874 m .29417 .49554 L .36049 .52874 L p .8 g F P s P p .002 w .29417 .49554 m .36049 .52874 L .35736 .60366 L p .8 g F P s P p .002 w .41956 .49554 m .36049 .52874 L .29417 .49554 L p .8 g F P s P p .002 w .2892 .57187 m .29417 .49554 L .23872 .52874 L p .8 g F P s P p .002 w .23872 .52874 m .23247 .60366 L .2892 .57187 L p .8 g F P s P p .002 w .29417 .49554 m .36049 .52874 L .35736 .60366 L p .8 g F P s P p .002 w .23872 .52874 m .16878 .49554 L .17666 .42305 L p .8 g F P s P p .002 w .29417 .49554 m .23872 .52874 L .16878 .49554 L p .8 g F P s P p .002 w .2892 .57187 m .29417 .49554 L .23872 .52874 L p .8 g F P s P p .002 w .23872 .52874 m .23247 .60366 L .2892 .57187 L p .8 g F P s P p .002 w .16878 .49554 m .23872 .52874 L .23247 .60366 L p .8 g F P s P p .002 w .29417 .49554 m .23872 .52874 L .16878 .49554 L p .8 g F P s P p .002 w .48226 .38657 m .42114 .42305 L .35637 .38657 L p .5 g F P s P p .002 w .42114 .42305 m .48226 .38657 L .54338 .42305 L p .8 g F P s P p .002 w .48226 .46031 m .48226 .38657 L .42114 .42305 L p .5 g F P s P p .002 w .42114 .42305 m .41956 .49554 L .48226 .46031 L p .5 g F P s P p .002 w .2989 .42305 m .35637 .38657 L .42114 .42305 L p .8 g F P s P p .002 w .35637 .38657 m .42114 .42305 L .41956 .49554 L p .5 g F P s P p .002 w .73403 .38657 m .66561 .42305 L .60814 .38657 L p .8 g F P s P p .002 w .74072 .46031 m .73403 .38657 L .66561 .42305 L p .8 g F P s P p .002 w .66561 .42305 m .67034 .49554 L .74072 .46031 L p .8 g F P s P p .002 w .54338 .42305 m .60814 .38657 L .66561 .42305 L p .8 g F P s P p .002 w .60814 .38657 m .66561 .42305 L .67034 .49554 L p .8 g F P s P p .002 w .61149 .46031 m .60814 .38657 L .54338 .42305 L p .8 g F P s P p .002 w .54338 .42305 m .54495 .49554 L .61149 .46031 L p .8 g F P s P p .002 w .60814 .38657 m .66561 .42305 L .67034 .49554 L p .8 g F P s P p .002 w .60814 .38657 m .54338 .42305 L .48226 .38657 L p .8 g F P s P p .002 w .61149 .46031 m .60814 .38657 L .54338 .42305 L p .8 g F P s P p .002 w .54338 .42305 m .54495 .49554 L .61149 .46031 L p .8 g F P s P p .002 w .48226 .38657 m .54338 .42305 L .54495 .49554 L p .8 g F P s P p .002 w .48226 .46031 m .48226 .38657 L .42114 .42305 L p .8 g F P s P p .002 w .42114 .42305 m .41956 .49554 L .48226 .46031 L p .8 g F P s P p .002 w .48226 .38657 m .54338 .42305 L .54495 .49554 L p .8 g F P s P p .002 w .35302 .46031 m .35637 .38657 L .2989 .42305 L p .8 g F P s P p .002 w .2989 .42305 m .29417 .49554 L .35302 .46031 L p .8 g F P s P p .002 w .35637 .38657 m .42114 .42305 L .41956 .49554 L p .8 g F P s P p .002 w .35637 .38657 m .2989 .42305 L .23048 .38657 L p .8 g F P s P p .002 w .35302 .46031 m .35637 .38657 L .2989 .42305 L p .8 g F P s P p .002 w .2989 .42305 m .29417 .49554 L .35302 .46031 L p .8 g F P s P p .002 w .17666 .42305 m .23048 .38657 L .2989 .42305 L p .8 g F P s P p .002 w .23048 .38657 m .2989 .42305 L .29417 .49554 L p .8 g F P s P p .002 w .22379 .46031 m .23048 .38657 L .17666 .42305 L p .8 g F P s P p .002 w .17666 .42305 m .16878 .49554 L .22379 .46031 L p .8 g F P s P p .002 w .23048 .38657 m .2989 .42305 L .29417 .49554 L p .8 g F P s P p .002 w .23048 .38657 m .17666 .42305 L .10459 .38657 L p .8 g F P s P p .002 w .22379 .46031 m .23048 .38657 L .17666 .42305 L p .8 g F P s P p .002 w .17666 .42305 m .16878 .49554 L .22379 .46031 L p .8 g F P s P p .002 w .10459 .38657 m .17666 .42305 L .16878 .49554 L p .8 g F P s P p .002 w .42054 .92295 m .3551 .90117 L .35834 .81735 L p .25 g F P s P p .002 w .48226 .90117 m .42054 .92295 L .3551 .90117 L p .25 g F P s P p .002 w .35834 .81735 m .41842 .79229 L .48226 .81735 L p .25 g F P s P p .002 w .35834 .81735 m .41842 .79229 L .48226 .81735 L p .5 g F P s P p .002 w .4167 .87804 m .41842 .79229 L .35834 .81735 L p .25 g F P s P p .002 w .35834 .81735 m .3551 .90117 L .4167 .87804 L p .25 g F P s P p .002 w .41842 .79229 m .48226 .81735 L .48226 .90117 L p .25 g F P s P p .002 w .48226 .81735 m .41842 .79229 L .42005 .71093 L p .5 g F P s P p .002 w .48226 .81735 m .41842 .79229 L .42005 .71093 L p .8 g F P s P p .002 w .54609 .79229 m .48226 .81735 L .41842 .79229 L p .5 g F P s P p .002 w .54781 .87804 m .54609 .79229 L .48226 .81735 L p .5 g F P s P p .002 w .48226 .81735 m .48226 .90117 L .54781 .87804 L p .5 g F P s P p .002 w .41842 .79229 m .48226 .81735 L .48226 .90117 L p .5 g F P s P p .002 w .35834 .81735 m .29076 .79229 L .29565 .71093 L p .8 g F P s P p .002 w .41842 .79229 m .35834 .81735 L .29076 .79229 L p .5 g F P s P p .002 w .4167 .87804 m .41842 .79229 L .35834 .81735 L p .5 g F P s P p .002 w .35834 .81735 m .3551 .90117 L .4167 .87804 L p .5 g F P s P p .002 w .29076 .79229 m .35834 .81735 L .3551 .90117 L p .5 g F P s P p .002 w .54609 .79229 m .48226 .81735 L .41842 .79229 L p .8 g F P s P p .002 w .41842 .79229 m .35834 .81735 L .29076 .79229 L p .8 g F P s P p .002 w .42005 .71093 m .35407 .68253 L .35736 .60366 L p .5 g F P s P p .002 w .42005 .71093 m .35407 .68253 L .35736 .60366 L p .8 g F P s P p .002 w .48226 .68253 m .42005 .71093 L .35407 .68253 L p .25 g F P s P p .002 w .42005 .71093 m .48226 .68253 L .54446 .71093 L p .8 g F P s P p .002 w .42005 .71093 m .48226 .68253 L .54446 .71093 L p .8 g F P s P p .002 w .48226 .76567 m .48226 .68253 L .42005 .71093 L p .25 g F P s P p .002 w .42005 .71093 m .41842 .79229 L .48226 .76567 L p .25 g F P s P p .002 w .29565 .71093 m .35407 .68253 L .42005 .71093 L p .8 g F P s P p .002 w .29565 .71093 m .35407 .68253 L .42005 .71093 L p .8 g F P s P p .002 w .35407 .68253 m .42005 .71093 L .41842 .79229 L p .25 g F P s P p .002 w .48226 .68253 m .42005 .71093 L .35407 .68253 L p .5 g F P s P p .002 w .54446 .71093 m .48226 .68253 L .48226 .60366 L p .8 g F P s P p .002 w .54446 .71093 m .48226 .68253 L .48226 .60366 L p .8 g F P s P p .002 w .61044 .68253 m .54446 .71093 L .48226 .68253 L p .8 g F P s P p .002 w .61391 .76567 m .61044 .68253 L .54446 .71093 L p .8 g F P s P p .002 w .54446 .71093 m .54609 .79229 L .61391 .76567 L p .8 g F P s P p .002 w .48226 .68253 m .54446 .71093 L .54609 .79229 L p .8 g F P s P p .002 w .61044 .68253 m .54446 .71093 L .48226 .68253 L p .8 g F P s P p .002 w .48226 .76567 m .48226 .68253 L .42005 .71093 L p .8 g F P s P p .002 w .42005 .71093 m .41842 .79229 L .48226 .76567 L p .8 g F P s P p .002 w .48226 .68253 m .54446 .71093 L .54609 .79229 L p .8 g F P s P p .002 w .3506 .76567 m .35407 .68253 L .29565 .71093 L p .8 g F P s P p .002 w .29565 .71093 m .29076 .79229 L .3506 .76567 L p .8 g F P s P p .002 w .35407 .68253 m .42005 .71093 L .41842 .79229 L p .8 g F P s P p .002 w .29565 .71093 m .22589 .68253 L .23247 .60366 L p .8 g F P s P p .002 w .35407 .68253 m .29565 .71093 L .22589 .68253 L p .8 g F P s P p .002 w .3506 .76567 m .35407 .68253 L .29565 .71093 L p .8 g F P s P p .002 w .29565 .71093 m .29076 .79229 L .3506 .76567 L p .8 g F P s P p .002 w .22589 .68253 m .29565 .71093 L .29076 .79229 L p .8 g F P s P p .002 w .35407 .68253 m .29565 .71093 L .22589 .68253 L p .8 g F P s P p .002 w .35736 .60366 m .4179 .57187 L .48226 .60366 L p .5 g F P s P p .002 w .35736 .60366 m .4179 .57187 L .48226 .60366 L p .5 g F P s P p .002 w .41615 .65235 m .4179 .57187 L .35736 .60366 L p .5 g F P s P p .002 w .35736 .60366 m .35407 .68253 L .41615 .65235 L p .5 g F P s P p .002 w .4179 .57187 m .48226 .60366 L .48226 .68253 L p .5 g F P s P p .002 w .48226 .60366 m .4179 .57187 L .41956 .49554 L p .5 g F P s P p .002 w .60715 .60366 m .54661 .57187 L .54495 .49554 L p .8 g F P s P p .002 w .60715 .60366 m .54661 .57187 L .54495 .49554 L p .8 g F P s P p .002 w .67531 .57187 m .60715 .60366 L .54661 .57187 L p .8 g F P s P p .002 w .68056 .65235 m .67531 .57187 L .60715 .60366 L p .8 g F P s P p .002 w .60715 .60366 m .61044 .68253 L .68056 .65235 L p .8 g F P s P p .002 w .48226 .60366 m .54661 .57187 L .60715 .60366 L p .8 g F P s P p .002 w .48226 .60366 m .54661 .57187 L .60715 .60366 L p .8 g F P s P p .002 w .54661 .57187 m .60715 .60366 L .61044 .68253 L p .8 g F P s P p .002 w .67531 .57187 m .60715 .60366 L .54661 .57187 L p .8 g F P s P p .002 w .54836 .65235 m .54661 .57187 L .48226 .60366 L p .8 g F P s P p .002 w .48226 .60366 m .48226 .68253 L .54836 .65235 L p .8 g F P s P p .002 w .54661 .57187 m .60715 .60366 L .61044 .68253 L p .8 g F P s P p .002 w .48226 .60366 m .4179 .57187 L .41956 .49554 L p .8 g F P s P p .002 w .54661 .57187 m .48226 .60366 L .4179 .57187 L p .8 g F P s P p .002 w .54836 .65235 m .54661 .57187 L .48226 .60366 L p .8 g F P s P p .002 w .48226 .60366 m .48226 .68253 L .54836 .65235 L p .8 g F P s P p .002 w .4179 .57187 m .48226 .60366 L .48226 .68253 L p .8 g F P s P p .002 w .54661 .57187 m .48226 .60366 L .4179 .57187 L p .8 g F P s P p .002 w .35736 .60366 m .2892 .57187 L .29417 .49554 L p .8 g F P s P p .002 w .35736 .60366 m .2892 .57187 L .29417 .49554 L p .8 g F P s P p .002 w .4179 .57187 m .35736 .60366 L .2892 .57187 L p .8 g F P s P p .002 w .41615 .65235 m .4179 .57187 L .35736 .60366 L p .8 g F P s P p .002 w .35736 .60366 m .35407 .68253 L .41615 .65235 L p .8 g F P s P p .002 w .23247 .60366 m .2892 .57187 L .35736 .60366 L p .8 g F P s P p .002 w .23247 .60366 m .2892 .57187 L .35736 .60366 L p .8 g F P s P p .002 w .2892 .57187 m .35736 .60366 L .35407 .68253 L p .8 g F P s P p .002 w .4179 .57187 m .35736 .60366 L .2892 .57187 L p .8 g F P s P p .002 w .28395 .65235 m .2892 .57187 L .23247 .60366 L p .8 g F P s P p .002 w .23247 .60366 m .22589 .68253 L .28395 .65235 L p .8 g F P s P p .002 w .2892 .57187 m .35736 .60366 L .35407 .68253 L p .8 g F P s P p .002 w .23247 .60366 m .16049 .57187 L .16878 .49554 L p .8 g F P s P p .002 w .2892 .57187 m .23247 .60366 L .16049 .57187 L p .8 g F P s P p .002 w .28395 .65235 m .2892 .57187 L .23247 .60366 L p .8 g F P s P p .002 w .23247 .60366 m .22589 .68253 L .28395 .65235 L p .8 g F P s P p .002 w .16049 .57187 m .23247 .60366 L .22589 .68253 L p .8 g F P s P p .002 w .2892 .57187 m .23247 .60366 L .16049 .57187 L p .8 g F P s P p .002 w .41956 .49554 m .35302 .46031 L .35637 .38657 L p .5 g F P s P p .002 w .41956 .49554 m .35302 .46031 L .35637 .38657 L p .8 g F P s P p .002 w .48226 .46031 m .41956 .49554 L .35302 .46031 L p .5 g F P s P p .002 w .41956 .49554 m .48226 .46031 L .54495 .49554 L p .8 g F P s P p .002 w .41956 .49554 m .48226 .46031 L .54495 .49554 L p .8 g F P s P p .002 w .48226 .53809 m .48226 .46031 L .41956 .49554 L p .5 g F P s P p .002 w .41956 .49554 m .4179 .57187 L .48226 .53809 L p .5 g F P s P p .002 w .29417 .49554 m .35302 .46031 L .41956 .49554 L p .8 g F P s P p .002 w .29417 .49554 m .35302 .46031 L .41956 .49554 L p .8 g F P s P p .002 w .35302 .46031 m .41956 .49554 L .4179 .57187 L p .5 g F P s P p .002 w .48226 .46031 m .41956 .49554 L .35302 .46031 L p .5 g F P s P p .002 w .67034 .49554 m .61149 .46031 L .60814 .38657 L p .8 g F P s P p .002 w .67034 .49554 m .61149 .46031 L .60814 .38657 L p .8 g F P s P p .002 w .74072 .46031 m .67034 .49554 L .61149 .46031 L p .8 g F P s P p .002 w .74777 .53809 m .74072 .46031 L .67034 .49554 L p .8 g F P s P p .002 w .67034 .49554 m .67531 .57187 L .74777 .53809 L p .8 g F P s P p .002 w .54495 .49554 m .61149 .46031 L .67034 .49554 L p .8 g F P s P p .002 w .54495 .49554 m .61149 .46031 L .67034 .49554 L p .8 g F P s P p .002 w .61149 .46031 m .67034 .49554 L .67531 .57187 L p .8 g F P s P p .002 w .74072 .46031 m .67034 .49554 L .61149 .46031 L p .8 g F P s P p .002 w .61501 .53809 m .61149 .46031 L .54495 .49554 L p .8 g F P s P p .002 w .54495 .49554 m .54661 .57187 L .61501 .53809 L p .8 g F P s P p .002 w .61149 .46031 m .67034 .49554 L .67531 .57187 L p .8 g F P s P p .002 w .54495 .49554 m .48226 .46031 L .48226 .38657 L p .8 g F P s P p .002 w .54495 .49554 m .48226 .46031 L .48226 .38657 L p .8 g F P s P p .002 w .61149 .46031 m .54495 .49554 L .48226 .46031 L p .8 g F P s P p .002 w .61501 .53809 m .61149 .46031 L .54495 .49554 L p .8 g F P s P p .002 w .54495 .49554 m .54661 .57187 L .61501 .53809 L p .8 g F P s P p .002 w .48226 .46031 m .54495 .49554 L .54661 .57187 L p .8 g F P s P p .002 w .61149 .46031 m .54495 .49554 L .48226 .46031 L p .8 g F P s P p .002 w .48226 .53809 m .48226 .46031 L .41956 .49554 L p .8 g F P s P p .002 w .41956 .49554 m .4179 .57187 L .48226 .53809 L p .8 g F P s P p .002 w .48226 .46031 m .54495 .49554 L .54661 .57187 L p .8 g F P s P p .002 w .3495 .53809 m .35302 .46031 L .29417 .49554 L p .8 g F P s P p .002 w .29417 .49554 m .2892 .57187 L .3495 .53809 L p .8 g F P s P p .002 w .35302 .46031 m .41956 .49554 L .4179 .57187 L p .8 g F P s P p .002 w .29417 .49554 m .22379 .46031 L .23048 .38657 L p .8 g F P s P p .002 w .29417 .49554 m .22379 .46031 L .23048 .38657 L p .8 g F P s P p .002 w .35302 .46031 m .29417 .49554 L .22379 .46031 L p .8 g F P s P p .002 w .3495 .53809 m .35302 .46031 L .29417 .49554 L p .8 g F P s P p .002 w .29417 .49554 m .2892 .57187 L .3495 .53809 L p .8 g F P s P p .002 w .16878 .49554 m .22379 .46031 L .29417 .49554 L p .8 g F P s P p .002 w .16878 .49554 m .22379 .46031 L .29417 .49554 L p .8 g F P s P p .002 w .22379 .46031 m .29417 .49554 L .2892 .57187 L p .8 g F P s P p .002 w .35302 .46031 m .29417 .49554 L .22379 .46031 L p .8 g F P s P p .002 w .21674 .53809 m .22379 .46031 L .16878 .49554 L p .8 g F P s P p .002 w .16878 .49554 m .16049 .57187 L .21674 .53809 L p .8 g F P s P p .002 w .22379 .46031 m .29417 .49554 L .2892 .57187 L p .8 g F P s P p .002 w .16878 .49554 m .09456 .46031 L .10459 .38657 L p .8 g F P s P p .002 w .22379 .46031 m .16878 .49554 L .09456 .46031 L p .8 g F P s P p .002 w .21674 .53809 m .22379 .46031 L .16878 .49554 L p .8 g F P s P p .002 w .16878 .49554 m .16049 .57187 L .21674 .53809 L p .8 g F P s P p .002 w .09456 .46031 m .16878 .49554 L .16049 .57187 L p .8 g F P s P p .002 w .22379 .46031 m .16878 .49554 L .09456 .46031 L p .8 g F P s P p .002 w .35637 .38657 m .41738 .34784 L .48226 .38657 L p .5 g F P s P p .002 w .4156 .42286 m .41738 .34784 L .35637 .38657 L p .5 g F P s P p .002 w .35637 .38657 m .35302 .46031 L .4156 .42286 L p .5 g F P s P p .002 w .41738 .34784 m .48226 .38657 L .48226 .46031 L p .5 g F P s P p .002 w .60814 .38657 m .6769 .34784 L .73403 .38657 L p .8 g F P s P p .002 w .68223 .42286 m .6769 .34784 L .60814 .38657 L p .8 g F P s P p .002 w .60814 .38657 m .61149 .46031 L .68223 .42286 L p .8 g F P s P p .002 w .6769 .34784 m .73403 .38657 L .74072 .46031 L p .8 g F P s P p .002 w .6769 .34784 m .60814 .38657 L .54714 .34784 L p .8 g F P s P p .002 w .68223 .42286 m .6769 .34784 L .60814 .38657 L p .8 g F P s P p .002 w .60814 .38657 m .61149 .46031 L .68223 .42286 L p .8 g F P s P p .002 w .48226 .38657 m .54714 .34784 L .60814 .38657 L p .8 g F P s P p .002 w .54714 .34784 m .60814 .38657 L .61149 .46031 L p .8 g F P s P p .002 w .54891 .42286 m .54714 .34784 L .48226 .38657 L p .8 g F P s P p .002 w .48226 .38657 m .48226 .46031 L .54891 .42286 L p .8 g F P s P p .002 w .54714 .34784 m .60814 .38657 L .61149 .46031 L p .8 g F P s P p .002 w .54714 .34784 m .48226 .38657 L .41738 .34784 L p .8 g F P s P p .002 w .54891 .42286 m .54714 .34784 L .48226 .38657 L p .8 g F P s P p .002 w .48226 .38657 m .48226 .46031 L .54891 .42286 L p .8 g F P s P p .002 w .41738 .34784 m .48226 .38657 L .48226 .46031 L p .8 g F P s P p .002 w .41738 .34784 m .35637 .38657 L .28761 .34784 L p .8 g F P s P p .002 w .4156 .42286 m .41738 .34784 L .35637 .38657 L p .8 g F P s P p .002 w .35637 .38657 m .35302 .46031 L .4156 .42286 L p .8 g F P s P p .002 w .23048 .38657 m .28761 .34784 L .35637 .38657 L p .8 g F P s P p .002 w .28761 .34784 m .35637 .38657 L .35302 .46031 L p .8 g F P s P p .002 w .28228 .42286 m .28761 .34784 L .23048 .38657 L p .8 g F P s P p .002 w .23048 .38657 m .22379 .46031 L .28228 .42286 L p .8 g F P s P p .002 w .28761 .34784 m .35637 .38657 L .35302 .46031 L p .8 g F P s P p .002 w .28761 .34784 m .23048 .38657 L .15785 .34784 L p .8 g F P s P p .002 w .28228 .42286 m .28761 .34784 L .23048 .38657 L p .8 g F P s P p .002 w .23048 .38657 m .22379 .46031 L .28228 .42286 L p .8 g F P s P p .002 w .10459 .38657 m .15785 .34784 L .23048 .38657 L p .8 g F P s P p .002 w .15785 .34784 m .23048 .38657 L .22379 .46031 L p .8 g F P s P p .002 w .14896 .42286 m .15785 .34784 L .10459 .38657 L p .8 g F P s P p .002 w .10459 .38657 m .09456 .46031 L .14896 .42286 L p .8 g F P s P p .002 w .15785 .34784 m .23048 .38657 L .22379 .46031 L p .8 g F P s P p .002 w .3551 .90117 m .4167 .87804 L .48226 .90117 L p .25 g F P s P p .002 w .48226 .90117 m .4167 .87804 L .41842 .79229 L p .25 g F P s P p .002 w .48226 .90117 m .4167 .87804 L .41842 .79229 L p .5 g F P s P p .002 w .54781 .87804 m .48226 .90117 L .4167 .87804 L p .5 g F P s P p .002 w .3551 .90117 m .2856 .87804 L .29076 .79229 L p .5 g F P s P p .002 w .4167 .87804 m .3551 .90117 L .2856 .87804 L p .5 g F P s P p .002 w .41842 .79229 m .3506 .76567 L .35407 .68253 L p .25 g F P s P p .002 w .41842 .79229 m .3506 .76567 L .35407 .68253 L p .8 g F P s P p .002 w .48226 .76567 m .41842 .79229 L .3506 .76567 L p .25 g F P s P p .002 w .54609 .79229 m .48226 .76567 L .48226 .68253 L p .8 g F P s P p .002 w .54609 .79229 m .48226 .76567 L .48226 .68253 L p .8 g F P s P p .002 w .61391 .76567 m .54609 .79229 L .48226 .76567 L p .5 g F P s P p .002 w .61757 .85343 m .61391 .76567 L .54609 .79229 L p .5 g F P s P p .002 w .54609 .79229 m .54781 .87804 L .61757 .85343 L p .5 g F P s P p .002 w .41842 .79229 m .48226 .76567 L .54609 .79229 L p .5 g F P s P p .002 w .41842 .79229 m .48226 .76567 L .54609 .79229 L p .8 g F P s P p .002 w .48226 .76567 m .54609 .79229 L .54781 .87804 L p .5 g F P s P p .002 w .48226 .85343 m .48226 .76567 L .41842 .79229 L p .5 g F P s P p .002 w .41842 .79229 m .4167 .87804 L .48226 .85343 L p .5 g F P s P p .002 w .48226 .76567 m .54609 .79229 L .54781 .87804 L p .5 g F P s P p .002 w .29076 .79229 m .3506 .76567 L .41842 .79229 L p .5 g F P s P p .002 w .29076 .79229 m .3506 .76567 L .41842 .79229 L p .8 g F P s P p .002 w .34694 .85343 m .3506 .76567 L .29076 .79229 L p .5 g F P s P p .002 w .29076 .79229 m .2856 .87804 L .34694 .85343 L p .5 g F P s P p .002 w .3506 .76567 m .41842 .79229 L .4167 .87804 L p .5 g F P s P p .002 w .29076 .79229 m .21895 .76567 L .22589 .68253 L p .8 g F P s P p .002 w .3506 .76567 m .29076 .79229 L .21895 .76567 L p .5 g F P s P p .002 w .34694 .85343 m .3506 .76567 L .29076 .79229 L p .5 g F P s P p .002 w .29076 .79229 m .2856 .87804 L .34694 .85343 L p .5 g F P s P p .002 w .21895 .76567 m .29076 .79229 L .2856 .87804 L p .5 g F P s P p .002 w .61391 .76567 m .54609 .79229 L .48226 .76567 L p .8 g F P s P p .002 w .3506 .76567 m .29076 .79229 L .21895 .76567 L p .8 g F P s P p .002 w .48226 .76567 m .41842 .79229 L .3506 .76567 L p .8 g F P s P p .002 w .48226 .85343 m .48226 .76567 L .41842 .79229 L p .8 g F P s P p .002 w .41842 .79229 m .4167 .87804 L .48226 .85343 L p .8 g F P s P p .002 w .3506 .76567 m .41842 .79229 L .4167 .87804 L p .8 g F P s P p .002 w .35407 .68253 m .41615 .65235 L .48226 .68253 L p .25 g F P s P p .002 w .35407 .68253 m .41615 .65235 L .48226 .68253 L p .5 g F P s P p .002 w .41431 .73732 m .41615 .65235 L .35407 .68253 L p .25 g F P s P p .002 w .35407 .68253 m .3506 .76567 L .41431 .73732 L p .25 g F P s P p .002 w .41615 .65235 m .48226 .68253 L .48226 .76567 L p .25 g F P s P p .002 w .48226 .68253 m .41615 .65235 L .4179 .57187 L p .5 g F P s P p .002 w .48226 .68253 m .41615 .65235 L .4179 .57187 L p .8 g F P s P p .002 w .54836 .65235 m .48226 .68253 L .41615 .65235 L p .5 g F P s P p .002 w .48226 .68253 m .54836 .65235 L .61044 .68253 L p .8 g F P s P p .002 w .48226 .68253 m .54836 .65235 L .61044 .68253 L p .8 g F P s P p .002 w .5502 .73732 m .54836 .65235 L .48226 .68253 L p .5 g F P s P p .002 w .48226 .68253 m .48226 .76567 L .5502 .73732 L p .5 g F P s P p .002 w .41615 .65235 m .48226 .68253 L .48226 .76567 L p .5 g F P s P p .002 w .35407 .68253 m .28395 .65235 L .2892 .57187 L p .8 g F P s P p .002 w .35407 .68253 m .28395 .65235 L .2892 .57187 L p .8 g F P s P p .002 w .41615 .65235 m .35407 .68253 L .28395 .65235 L p .5 g F P s P p .002 w .41431 .73732 m .41615 .65235 L .35407 .68253 L p .5 g F P s P p .002 w .35407 .68253 m .3506 .76567 L .41431 .73732 L p .5 g F P s P p .002 w .22589 .68253 m .28395 .65235 L .35407 .68253 L p .8 g F P s P p .002 w .22589 .68253 m .28395 .65235 L .35407 .68253 L p .8 g F P s P p .002 w .28395 .65235 m .35407 .68253 L .3506 .76567 L p .5 g F P s P p .002 w .61044 .68253 m .54836 .65235 L .54661 .57187 L p .8 g F P s P p .002 w .61044 .68253 m .54836 .65235 L .54661 .57187 L p .8 g F P s P p .002 w .68056 .65235 m .61044 .68253 L .54836 .65235 L p .8 g F P s P p .002 w .6861 .73732 m .68056 .65235 L .61044 .68253 L p .8 g F P s P p .002 w .61044 .68253 m .61391 .76567 L .6861 .73732 L p .8 g F P s P p .002 w .54836 .65235 m .61044 .68253 L .61391 .76567 L p .8 g F P s P p .002 w .68056 .65235 m .61044 .68253 L .54836 .65235 L p .8 g F P s P p .002 w .5502 .73732 m .54836 .65235 L .48226 .68253 L p .8 g F P s P p .002 w .48226 .68253 m .48226 .76567 L .5502 .73732 L p .8 g F P s P p .002 w .54836 .65235 m .61044 .68253 L .61391 .76567 L p .8 g F P s P p .002 w .54836 .65235 m .48226 .68253 L .41615 .65235 L p .8 g F P s P p .002 w .41615 .65235 m .35407 .68253 L .28395 .65235 L p .8 g F P s P p .002 w .27841 .73732 m .28395 .65235 L .22589 .68253 L p .8 g F P s P p .002 w .22589 .68253 m .21895 .76567 L .27841 .73732 L p .8 g F P s P p .002 w .28395 .65235 m .35407 .68253 L .3506 .76567 L p .8 g F P s P p .002 w .22589 .68253 m .15175 .65235 L .16049 .57187 L p .8 g F P s P p .002 w .28395 .65235 m .22589 .68253 L .15175 .65235 L p .8 g F P s P p .002 w .27841 .73732 m .28395 .65235 L .22589 .68253 L p .8 g F P s P p .002 w .22589 .68253 m .21895 .76567 L .27841 .73732 L p .8 g F P s P p .002 w .15175 .65235 m .22589 .68253 L .21895 .76567 L p .8 g F P s P p .002 w .28395 .65235 m .22589 .68253 L .15175 .65235 L p .8 g F P s P p .002 w .4179 .57187 m .3495 .53809 L .35302 .46031 L p .5 g F P s P p .002 w .4179 .57187 m .3495 .53809 L .35302 .46031 L p .8 g F P s P p .002 w .48226 .53809 m .4179 .57187 L .3495 .53809 L p .25 g F P s P p .002 w .4179 .57187 m .48226 .53809 L .54661 .57187 L p .8 g F P s P p .002 w .4179 .57187 m .48226 .53809 L .54661 .57187 L p .8 g F P s P p .002 w .48226 .62022 m .48226 .53809 L .4179 .57187 L p .25 g F P s P p .002 w .4179 .57187 m .41615 .65235 L .48226 .62022 L p .25 g F P s P p .002 w .2892 .57187 m .3495 .53809 L .4179 .57187 L p .8 g F P s P p .002 w .2892 .57187 m .3495 .53809 L .4179 .57187 L p .8 g F P s P p .002 w .3495 .53809 m .4179 .57187 L .41615 .65235 L p .25 g F P s P p .002 w .48226 .53809 m .4179 .57187 L .3495 .53809 L p .5 g F P s P p .002 w .67531 .57187 m .61501 .53809 L .61149 .46031 L p .8 g F P s P p .002 w .67531 .57187 m .61501 .53809 L .61149 .46031 L p .8 g F P s P p .002 w .74777 .53809 m .67531 .57187 L .61501 .53809 L p .8 g F P s P p .002 w .75522 .62022 m .74777 .53809 L .67531 .57187 L p .8 g F P s P p .002 w .67531 .57187 m .68056 .65235 L .75522 .62022 L p .8 g F P s P p .002 w .54661 .57187 m .61501 .53809 L .67531 .57187 L p .8 g F P s P p .002 w .54661 .57187 m .61501 .53809 L .67531 .57187 L p .8 g F P s P p .002 w .61501 .53809 m .67531 .57187 L .68056 .65235 L p .8 g F P s P p .002 w .74777 .53809 m .67531 .57187 L .61501 .53809 L p .8 g F P s P p .002 w .61874 .62022 m .61501 .53809 L .54661 .57187 L p .8 g F P s P p .002 w .54661 .57187 m .54836 .65235 L .61874 .62022 L p .8 g F P s P p .002 w .61501 .53809 m .67531 .57187 L .68056 .65235 L p .8 g F P s P p .002 w .54661 .57187 m .48226 .53809 L .48226 .46031 L p .8 g F P s P p .002 w .54661 .57187 m .48226 .53809 L .48226 .46031 L p .8 g F P s P p .002 w .61501 .53809 m .54661 .57187 L .48226 .53809 L p .8 g F P s P p .002 w .61874 .62022 m .61501 .53809 L .54661 .57187 L p .8 g F P s P p .002 w .54661 .57187 m .54836 .65235 L .61874 .62022 L p .8 g F P s P p .002 w .48226 .53809 m .54661 .57187 L .54836 .65235 L p .8 g F P s P p .002 w .61501 .53809 m .54661 .57187 L .48226 .53809 L p .8 g F P s P p .002 w .48226 .62022 m .48226 .53809 L .4179 .57187 L p .8 g F P s P p .002 w .4179 .57187 m .41615 .65235 L .48226 .62022 L p .8 g F P s P p .002 w .48226 .53809 m .54661 .57187 L .54836 .65235 L p .8 g F P s P p .002 w .34578 .62022 m .3495 .53809 L .2892 .57187 L p .8 g F P s P p .002 w .2892 .57187 m .28395 .65235 L .34578 .62022 L p .8 g F P s P p .002 w .3495 .53809 m .4179 .57187 L .41615 .65235 L p .8 g F P s P p .002 w .2892 .57187 m .21674 .53809 L .22379 .46031 L p .8 g F P s P p .002 w .2892 .57187 m .21674 .53809 L .22379 .46031 L p .8 g F P s P p .002 w .3495 .53809 m .2892 .57187 L .21674 .53809 L p .8 g F P s P p .002 w .34578 .62022 m .3495 .53809 L .2892 .57187 L p .8 g F P s P p .002 w .2892 .57187 m .28395 .65235 L .34578 .62022 L p .8 g F P s P p .002 w .16049 .57187 m .21674 .53809 L .2892 .57187 L p .8 g F P s P p .002 w .16049 .57187 m .21674 .53809 L .2892 .57187 L p .8 g F P s P p .002 w .21674 .53809 m .2892 .57187 L .28395 .65235 L p .8 g F P s P p .002 w .3495 .53809 m .2892 .57187 L .21674 .53809 L p .8 g F P s P p .002 w .20929 .62022 m .21674 .53809 L .16049 .57187 L p .8 g F P s P p .002 w .16049 .57187 m .15175 .65235 L .20929 .62022 L p .8 g F P s P p .002 w .21674 .53809 m .2892 .57187 L .28395 .65235 L p .8 g F P s P p .002 w .16049 .57187 m .08398 .53809 L .09456 .46031 L p .8 g F P s P p .002 w .21674 .53809 m .16049 .57187 L .08398 .53809 L p .8 g F P s P p .002 w .20929 .62022 m .21674 .53809 L .16049 .57187 L p .8 g F P s P p .002 w .16049 .57187 m .15175 .65235 L .20929 .62022 L p .8 g F P s P p .002 w .08398 .53809 m .16049 .57187 L .15175 .65235 L p .8 g F P s P p .002 w .21674 .53809 m .16049 .57187 L .08398 .53809 L p .8 g F P s P p .002 w .35302 .46031 m .4156 .42286 L .48226 .46031 L p .5 g F P s P p .002 w .35302 .46031 m .4156 .42286 L .48226 .46031 L p .5 g F P s P p .002 w .41372 .5021 m .4156 .42286 L .35302 .46031 L p .5 g F P s P p .002 w .35302 .46031 m .3495 .53809 L .41372 .5021 L p .5 g F P s P p .002 w .4156 .42286 m .48226 .46031 L .48226 .53809 L p .5 g F P s P p .002 w .48226 .46031 m .4156 .42286 L .41738 .34784 L p .5 g F P s P p .002 w .61149 .46031 m .68223 .42286 L .74072 .46031 L p .8 g F P s P p .002 w .61149 .46031 m .68223 .42286 L .74072 .46031 L p .8 g F P s P p .002 w .68786 .5021 m .68223 .42286 L .61149 .46031 L p .8 g F P s P p .002 w .61149 .46031 m .61501 .53809 L .68786 .5021 L p .8 g F P s P p .002 w .68223 .42286 m .74072 .46031 L .74777 .53809 L p .8 g F P s P p .002 w .74072 .46031 m .68223 .42286 L .6769 .34784 L p .8 g F P s P p .002 w .61149 .46031 m .54891 .42286 L .54714 .34784 L p .8 g F P s P p .002 w .61149 .46031 m .54891 .42286 L .54714 .34784 L p .8 g F P s P p .002 w .68223 .42286 m .61149 .46031 L .54891 .42286 L p .8 g F P s P p .002 w .68786 .5021 m .68223 .42286 L .61149 .46031 L p .8 g F P s P p .002 w .61149 .46031 m .61501 .53809 L .68786 .5021 L p .8 g F P s P p .002 w .48226 .46031 m .54891 .42286 L .61149 .46031 L p .8 g F P s P p .002 w .48226 .46031 m .54891 .42286 L .61149 .46031 L p .8 g F P s P p .002 w .54891 .42286 m .61149 .46031 L .61501 .53809 L p .8 g F P s P p .002 w .68223 .42286 m .61149 .46031 L .54891 .42286 L p .8 g F P s P p .002 w .55079 .5021 m .54891 .42286 L .48226 .46031 L p .8 g F P s P p .002 w .48226 .46031 m .48226 .53809 L .55079 .5021 L p .8 g F P s P p .002 w .54891 .42286 m .61149 .46031 L .61501 .53809 L p .8 g F P s P p .002 w .48226 .46031 m .4156 .42286 L .41738 .34784 L p .8 g F P s P p .002 w .54891 .42286 m .48226 .46031 L .4156 .42286 L p .8 g F P s P p .002 w .55079 .5021 m .54891 .42286 L .48226 .46031 L p .8 g F P s P p .002 w .48226 .46031 m .48226 .53809 L .55079 .5021 L p .8 g F P s P p .002 w .4156 .42286 m .48226 .46031 L .48226 .53809 L p .8 g F P s P p .002 w .54891 .42286 m .48226 .46031 L .4156 .42286 L p .8 g F P s P p .002 w .35302 .46031 m .28228 .42286 L .28761 .34784 L p .8 g F P s P p .002 w .35302 .46031 m .28228 .42286 L .28761 .34784 L p .8 g F P s P p .002 w .4156 .42286 m .35302 .46031 L .28228 .42286 L p .8 g F P s P p .002 w .41372 .5021 m .4156 .42286 L .35302 .46031 L p .8 g F P s P p .002 w .35302 .46031 m .3495 .53809 L .41372 .5021 L p .8 g F P s P p .002 w .22379 .46031 m .28228 .42286 L .35302 .46031 L p .8 g F P s P p .002 w .22379 .46031 m .28228 .42286 L .35302 .46031 L p .8 g F P s P p .002 w .28228 .42286 m .35302 .46031 L .3495 .53809 L p .8 g F P s P p .002 w .4156 .42286 m .35302 .46031 L .28228 .42286 L p .8 g F P s P p .002 w .27665 .5021 m .28228 .42286 L .22379 .46031 L p .8 g F P s P p .002 w .22379 .46031 m .21674 .53809 L .27665 .5021 L p .8 g F P s P p .002 w .28228 .42286 m .35302 .46031 L .3495 .53809 L p .8 g F P s P p .002 w .22379 .46031 m .14896 .42286 L .15785 .34784 L p .8 g F P s P p .002 w .22379 .46031 m .14896 .42286 L .15785 .34784 L p .8 g F P s P p .002 w .28228 .42286 m .22379 .46031 L .14896 .42286 L p .8 g F P s P p .002 w .27665 .5021 m .28228 .42286 L .22379 .46031 L p .8 g F P s P p .002 w .22379 .46031 m .21674 .53809 L .27665 .5021 L p .8 g F P s P p .002 w .09456 .46031 m .14896 .42286 L .22379 .46031 L p .8 g F P s P p .002 w .09456 .46031 m .14896 .42286 L .22379 .46031 L p .8 g F P s P p .002 w .14896 .42286 m .22379 .46031 L .21674 .53809 L p .8 g F P s P p .002 w .28228 .42286 m .22379 .46031 L .14896 .42286 L p .8 g F P s P p .002 w .13958 .5021 m .14896 .42286 L .09456 .46031 L p .8 g F P s P p .002 w .09456 .46031 m .08398 .53809 L .13958 .5021 L p .8 g F P s P p .002 w .14896 .42286 m .22379 .46031 L .21674 .53809 L p .8 g F P s P p .002 w .48226 .30665 m .41738 .34784 L .34837 .30665 L p .5 g F P s P p .002 w .41738 .34784 m .48226 .30665 L .54714 .34784 L p .8 g F P s P p .002 w .48226 .38295 m .48226 .30665 L .41738 .34784 L p .5 g F P s P p .002 w .41738 .34784 m .4156 .42286 L .48226 .38295 L p .5 g F P s P p .002 w .28761 .34784 m .34837 .30665 L .41738 .34784 L p .8 g F P s P p .002 w .34837 .30665 m .41738 .34784 L .4156 .42286 L p .5 g F P s P p .002 w .54714 .34784 m .61614 .30665 L .6769 .34784 L p .8 g F P s P p .002 w .61992 .38295 m .61614 .30665 L .54714 .34784 L p .8 g F P s P p .002 w .54714 .34784 m .54891 .42286 L .61992 .38295 L p .8 g F P s P p .002 w .61614 .30665 m .6769 .34784 L .68223 .42286 L p .8 g F P s P p .002 w .61614 .30665 m .54714 .34784 L .48226 .30665 L p .8 g F P s P p .002 w .61992 .38295 m .61614 .30665 L .54714 .34784 L p .8 g F P s P p .002 w .54714 .34784 m .54891 .42286 L .61992 .38295 L p .8 g F P s P p .002 w .48226 .30665 m .54714 .34784 L .54891 .42286 L p .8 g F P s P p .002 w .48226 .38295 m .48226 .30665 L .41738 .34784 L p .8 g F P s P p .002 w .41738 .34784 m .4156 .42286 L .48226 .38295 L p .8 g F P s P p .002 w .48226 .30665 m .54714 .34784 L .54891 .42286 L p .8 g F P s P p .002 w .34459 .38295 m .34837 .30665 L .28761 .34784 L p .8 g F P s P p .002 w .28761 .34784 m .28228 .42286 L .34459 .38295 L p .8 g F P s P p .002 w .34837 .30665 m .41738 .34784 L .4156 .42286 L p .8 g F P s P p .002 w .34837 .30665 m .28761 .34784 L .21449 .30665 L p .8 g F P s P p .002 w .34459 .38295 m .34837 .30665 L .28761 .34784 L p .8 g F P s P p .002 w .28761 .34784 m .28228 .42286 L .34459 .38295 L p .8 g F P s P p .002 w .15785 .34784 m .21449 .30665 L .28761 .34784 L p .8 g F P s P p .002 w .21449 .30665 m .28761 .34784 L .28228 .42286 L p .8 g F P s P p .002 w .20692 .38295 m .21449 .30665 L .15785 .34784 L p .8 g F P s P p .002 w .15785 .34784 m .14896 .42286 L .20692 .38295 L p .8 g F P s P p .002 w .21449 .30665 m .28761 .34784 L .28228 .42286 L p .8 g F P s P p .002 w .54781 .87804 m .48226 .85343 L .48226 .76567 L p .5 g F P s P p .002 w .54781 .87804 m .48226 .85343 L .48226 .76567 L p .5 g F P s P p .002 w .61757 .85343 m .54781 .87804 L .48226 .85343 L p .5 g F P s P p .002 w .4167 .87804 m .48226 .85343 L .54781 .87804 L p .5 g F P s P p .002 w .2856 .87804 m .34694 .85343 L .4167 .87804 L p .5 g F P s P p .002 w .4167 .87804 m .34694 .85343 L .3506 .76567 L p .5 g F P s P p .002 w .2856 .87804 m .21163 .85343 L .21895 .76567 L p .5 g F P s P p .002 w .34694 .85343 m .2856 .87804 L .21163 .85343 L p .5 g F P s P p .002 w .4167 .87804 m .34694 .85343 L .3506 .76567 L p .8 g F P s P p .002 w .48226 .85343 m .4167 .87804 L .34694 .85343 L p .8 g F P s P p .002 w .3506 .76567 m .41431 .73732 L .48226 .76567 L p .25 g F P s P p .002 w .21895 .76567 m .27841 .73732 L .3506 .76567 L p .5 g F P s P p .002 w .21895 .76567 m .27841 .73732 L .3506 .76567 L p .8 g F P s P p .002 w .27841 .73732 m .3506 .76567 L .34694 .85343 L p .5 g F P s P p .002 w .3506 .76567 m .27841 .73732 L .28395 .65235 L p .5 g F P s P p .002 w .3506 .76567 m .27841 .73732 L .28395 .65235 L p .8 g F P s P p .002 w .41431 .73732 m .3506 .76567 L .27841 .73732 L p .5 g F P s P p .002 w .3506 .76567 m .41431 .73732 L .48226 .76567 L p .8 g F P s P p .002 w .41236 .82718 m .41431 .73732 L .3506 .76567 L p .8 g F P s P p .002 w .3506 .76567 m .34694 .85343 L .41236 .82718 L p .8 g F P s P p .002 w .41431 .73732 m .3506 .76567 L .27841 .73732 L p .8 g F P s P p .002 w .41236 .82718 m .41431 .73732 L .3506 .76567 L p .8 g F P s P p .002 w .3506 .76567 m .34694 .85343 L .41236 .82718 L p .8 g F P s P p .002 w .27841 .73732 m .3506 .76567 L .34694 .85343 L p .8 g F P s P p .002 w .48226 .76567 m .41431 .73732 L .41615 .65235 L p .25 g F P s P p .002 w .61391 .76567 m .5502 .73732 L .54836 .65235 L p .8 g F P s P p .002 w .61391 .76567 m .5502 .73732 L .54836 .65235 L p .8 g F P s P p .002 w .6861 .73732 m .61391 .76567 L .5502 .73732 L p .5 g F P s P p .002 w .69195 .82718 m .6861 .73732 L .61391 .76567 L p .5 g F P s P p .002 w .61391 .76567 m .61757 .85343 L .69195 .82718 L p .5 g F P s P p .002 w .48226 .76567 m .5502 .73732 L .61391 .76567 L p .5 g F P s P p .002 w .48226 .76567 m .5502 .73732 L .61391 .76567 L p .8 g F P s P p .002 w .5502 .73732 m .61391 .76567 L .61757 .85343 L p .5 g F P s P p .002 w .55216 .82718 m .5502 .73732 L .48226 .76567 L p .5 g F P s P p .002 w .48226 .76567 m .48226 .85343 L .55216 .82718 L p .5 g F P s P p .002 w .5502 .73732 m .61391 .76567 L .61757 .85343 L p .5 g F P s P p .002 w .48226 .76567 m .41431 .73732 L .41615 .65235 L p .5 g F P s P p .002 w .5502 .73732 m .48226 .76567 L .41431 .73732 L p .5 g F P s P p .002 w .27256 .82718 m .27841 .73732 L .21895 .76567 L p .5 g F P s P p .002 w .21895 .76567 m .21163 .85343 L .27256 .82718 L p .5 g F P s P p .002 w .21895 .76567 m .14252 .73732 L .15175 .65235 L p .8 g F P s P p .002 w .27841 .73732 m .21895 .76567 L .14252 .73732 L p .5 g F P s P p .002 w .27256 .82718 m .27841 .73732 L .21895 .76567 L p .5 g F P s P p .002 w .21895 .76567 m .21163 .85343 L .27256 .82718 L p .5 g F P s P p .002 w .14252 .73732 m .21895 .76567 L .21163 .85343 L p .5 g F P s P p .002 w .6861 .73732 m .61391 .76567 L .5502 .73732 L p .8 g F P s P p .002 w .27841 .73732 m .21895 .76567 L .14252 .73732 L p .8 g F P s P p .002 w .5502 .73732 m .48226 .76567 L .41431 .73732 L p .8 g F P s P p .002 w .55216 .82718 m .5502 .73732 L .48226 .76567 L p .8 g F P s P p .002 w .48226 .76567 m .48226 .85343 L .55216 .82718 L p .8 g F P s P p .002 w .41431 .73732 m .48226 .76567 L .48226 .85343 L p .8 g F P s P p .002 w .41431 .73732 m .48226 .76567 L .48226 .85343 L p .8 g F P s P p .002 w .41615 .65235 m .34578 .62022 L .3495 .53809 L p .25 g F P s P p .002 w .41615 .65235 m .34578 .62022 L .3495 .53809 L p .8 g F P s P p .002 w .48226 .62022 m .41615 .65235 L .34578 .62022 L p .25 g F P s P p .002 w .54836 .65235 m .48226 .62022 L .48226 .53809 L p .8 g F P s P p .002 w .54836 .65235 m .48226 .62022 L .48226 .53809 L p .8 g F P s P p .002 w .61874 .62022 m .54836 .65235 L .48226 .62022 L p .5 g F P s P p .002 w .54836 .65235 m .61874 .62022 L .68056 .65235 L p .8 g F P s P p .002 w .54836 .65235 m .61874 .62022 L .68056 .65235 L p .8 g F P s P p .002 w .62268 .7071 m .61874 .62022 L .54836 .65235 L p .5 g F P s P p .002 w .54836 .65235 m .5502 .73732 L .62268 .7071 L p .5 g F P s P p .002 w .41615 .65235 m .48226 .62022 L .54836 .65235 L p .5 g F P s P p .002 w .41615 .65235 m .48226 .62022 L .54836 .65235 L p .8 g F P s P p .002 w .48226 .62022 m .54836 .65235 L .5502 .73732 L p .5 g F P s P p .002 w .48226 .7071 m .48226 .62022 L .41615 .65235 L p .5 g F P s P p .002 w .41615 .65235 m .41431 .73732 L .48226 .7071 L p .5 g F P s P p .002 w .48226 .62022 m .54836 .65235 L .5502 .73732 L p .5 g F P s P p .002 w .28395 .65235 m .34578 .62022 L .41615 .65235 L p .5 g F P s P p .002 w .28395 .65235 m .34578 .62022 L .41615 .65235 L p .8 g F P s P p .002 w .34184 .7071 m .34578 .62022 L .28395 .65235 L p .5 g F P s P p .002 w .28395 .65235 m .27841 .73732 L .34184 .7071 L p .5 g F P s P p .002 w .34578 .62022 m .41615 .65235 L .41431 .73732 L p .5 g F P s P p .002 w .28395 .65235 m .20929 .62022 L .21674 .53809 L p .8 g F P s P p .002 w .28395 .65235 m .20929 .62022 L .21674 .53809 L p .8 g F P s P p .002 w .34578 .62022 m .28395 .65235 L .20929 .62022 L p .5 g F P s P p .002 w .34184 .7071 m .34578 .62022 L .28395 .65235 L p .5 g F P s P p .002 w .28395 .65235 m .27841 .73732 L .34184 .7071 L p .5 g F P s P p .002 w .15175 .65235 m .20929 .62022 L .28395 .65235 L p .8 g F P s P p .002 w .15175 .65235 m .20929 .62022 L .28395 .65235 L p .8 g F P s P p .002 w .20929 .62022 m .28395 .65235 L .27841 .73732 L p .5 g F P s P p .002 w .68056 .65235 m .61874 .62022 L .61501 .53809 L p .8 g F P s P p .002 w .68056 .65235 m .61874 .62022 L .61501 .53809 L p .8 g F P s P p .002 w .75522 .62022 m .68056 .65235 L .61874 .62022 L p .8 g F P s P p .002 w .76309 .7071 m .75522 .62022 L .68056 .65235 L p .8 g F P s P p .002 w .68056 .65235 m .6861 .73732 L .76309 .7071 L p .8 g F P s P p .002 w .61874 .62022 m .68056 .65235 L .6861 .73732 L p .8 g F P s P p .002 w .75522 .62022 m .68056 .65235 L .61874 .62022 L p .8 g F P s P p .002 w .62268 .7071 m .61874 .62022 L .54836 .65235 L p .8 g F P s P p .002 w .54836 .65235 m .5502 .73732 L .62268 .7071 L p .8 g F P s P p .002 w .61874 .62022 m .68056 .65235 L .6861 .73732 L p .8 g F P s P p .002 w .61874 .62022 m .54836 .65235 L .48226 .62022 L p .8 g F P s P p .002 w .34578 .62022 m .28395 .65235 L .20929 .62022 L p .8 g F P s P p .002 w .20142 .7071 m .20929 .62022 L .15175 .65235 L p .8 g F P s P p .002 w .15175 .65235 m .14252 .73732 L .20142 .7071 L p .8 g F P s P p .002 w .20929 .62022 m .28395 .65235 L .27841 .73732 L p .8 g F P s P p .002 w .15175 .65235 m .07281 .62022 L .08398 .53809 L p .8 g F P s P p .002 w .20929 .62022 m .15175 .65235 L .07281 .62022 L p .8 g F P s P p .002 w .20142 .7071 m .20929 .62022 L .15175 .65235 L p .8 g F P s P p .002 w .15175 .65235 m .14252 .73732 L .20142 .7071 L p .8 g F P s P p .002 w .07281 .62022 m .15175 .65235 L .14252 .73732 L p .8 g F P s P p .002 w .20929 .62022 m .15175 .65235 L .07281 .62022 L p .8 g F P s P p .002 w .48226 .62022 m .41615 .65235 L .34578 .62022 L p .8 g F P s P p .002 w .48226 .7071 m .48226 .62022 L .41615 .65235 L p .8 g F P s P p .002 w .41615 .65235 m .41431 .73732 L .48226 .7071 L p .8 g F P s P p .002 w .34578 .62022 m .41615 .65235 L .41431 .73732 L p .8 g F P s P p .002 w .3495 .53809 m .41372 .5021 L .48226 .53809 L p .25 g F P s P p .002 w .3495 .53809 m .41372 .5021 L .48226 .53809 L p .5 g F P s P p .002 w .41173 .58594 m .41372 .5021 L .3495 .53809 L p .25 g F P s P p .002 w .3495 .53809 m .34578 .62022 L .41173 .58594 L p .25 g F P s P p .002 w .41372 .5021 m .48226 .53809 L .48226 .62022 L p .25 g F P s P p .002 w .48226 .53809 m .41372 .5021 L .4156 .42286 L p .5 g F P s P p .002 w .48226 .53809 m .41372 .5021 L .4156 .42286 L p .8 g F P s P p .002 w .55079 .5021 m .48226 .53809 L .41372 .5021 L p .5 g F P s P p .002 w .48226 .53809 m .55079 .5021 L .61501 .53809 L p .8 g F P s P p .002 w .48226 .53809 m .55079 .5021 L .61501 .53809 L p .8 g F P s P p .002 w .55278 .58594 m .55079 .5021 L .48226 .53809 L p .5 g F P s P p .002 w .48226 .53809 m .48226 .62022 L .55278 .58594 L p .5 g F P s P p .002 w .41372 .5021 m .48226 .53809 L .48226 .62022 L p .5 g F P s P p .002 w .3495 .53809 m .27665 .5021 L .28228 .42286 L p .8 g F P s P p .002 w .3495 .53809 m .27665 .5021 L .28228 .42286 L p .8 g F P s P p .002 w .41372 .5021 m .3495 .53809 L .27665 .5021 L p .5 g F P s P p .002 w .41173 .58594 m .41372 .5021 L .3495 .53809 L p .5 g F P s P p .002 w .3495 .53809 m .34578 .62022 L .41173 .58594 L p .5 g F P s P p .002 w .21674 .53809 m .27665 .5021 L .3495 .53809 L p .8 g F P s P p .002 w .21674 .53809 m .27665 .5021 L .3495 .53809 L p .8 g F P s P p .002 w .27665 .5021 m .3495 .53809 L .34578 .62022 L p .5 g F P s P p .002 w .61501 .53809 m .68786 .5021 L .74777 .53809 L p .8 g F P s P p .002 w .61501 .53809 m .68786 .5021 L .74777 .53809 L p .8 g F P s P p .002 w .69382 .58594 m .68786 .5021 L .61501 .53809 L p .8 g F P s P p .002 w .61501 .53809 m .61874 .62022 L .69382 .58594 L p .8 g F P s P p .002 w .68786 .5021 m .74777 .53809 L .75522 .62022 L p .8 g F P s P p .002 w .74777 .53809 m .68786 .5021 L .68223 .42286 L p .8 g F P s P p .002 w .61501 .53809 m .55079 .5021 L .54891 .42286 L p .8 g F P s P p .002 w .61501 .53809 m .55079 .5021 L .54891 .42286 L p .8 g F P s P p .002 w .68786 .5021 m .61501 .53809 L .55079 .5021 L p .8 g F P s P p .002 w .69382 .58594 m .68786 .5021 L .61501 .53809 L p .8 g F P s P p .002 w .61501 .53809 m .61874 .62022 L .69382 .58594 L p .8 g F P s P p .002 w .55079 .5021 m .61501 .53809 L .61874 .62022 L p .8 g F P s P p .002 w .68786 .5021 m .61501 .53809 L .55079 .5021 L p .8 g F P s P p .002 w .55278 .58594 m .55079 .5021 L .48226 .53809 L p .8 g F P s P p .002 w .48226 .53809 m .48226 .62022 L .55278 .58594 L p .8 g F P s P p .002 w .55079 .5021 m .61501 .53809 L .61874 .62022 L p .8 g F P s P p .002 w .55079 .5021 m .48226 .53809 L .41372 .5021 L p .8 g F P s P p .002 w .41372 .5021 m .3495 .53809 L .27665 .5021 L p .8 g F P s P p .002 w .27069 .58594 m .27665 .5021 L .21674 .53809 L p .8 g F P s P p .002 w .21674 .53809 m .20929 .62022 L .27069 .58594 L p .8 g F P s P p .002 w .27665 .5021 m .3495 .53809 L .34578 .62022 L p .8 g F P s P p .002 w .21674 .53809 m .13958 .5021 L .14896 .42286 L p .8 g F P s P p .002 w .21674 .53809 m .13958 .5021 L .14896 .42286 L p .8 g F P s P p .002 w .27665 .5021 m .21674 .53809 L .13958 .5021 L p .8 g F P s P p .002 w .27069 .58594 m .27665 .5021 L .21674 .53809 L p .8 g F P s P p .002 w .21674 .53809 m .20929 .62022 L .27069 .58594 L p .8 g F P s P p .002 w .08398 .53809 m .13958 .5021 L .21674 .53809 L p .8 g F P s P p .002 w .08398 .53809 m .13958 .5021 L .21674 .53809 L p .8 g F P s P p .002 w .13958 .5021 m .21674 .53809 L .20929 .62022 L p .8 g F P s P p .002 w .27665 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m .4156 .42286 L .34459 .38295 L p .5 g F P s P p .002 w .54891 .42286 m .61992 .38295 L .68223 .42286 L p .8 g F P s P p .002 w .54891 .42286 m .61992 .38295 L .68223 .42286 L p .8 g F P s P p .002 w .62393 .4637 m .61992 .38295 L .54891 .42286 L p .8 g F P s P p .002 w .54891 .42286 m .55079 .5021 L .62393 .4637 L p .8 g F P s P p .002 w .61992 .38295 m .68223 .42286 L .68786 .5021 L p .8 g F P s P p .002 w .68223 .42286 m .61992 .38295 L .61614 .30665 L p .8 g F P s P p .002 w .54891 .42286 m .48226 .38295 L .48226 .30665 L p .8 g F P s P p .002 w .54891 .42286 m .48226 .38295 L .48226 .30665 L p .8 g F P s P p .002 w .61992 .38295 m .54891 .42286 L .48226 .38295 L p .8 g F P s P p .002 w .62393 .4637 m .61992 .38295 L .54891 .42286 L p .8 g F P s P p .002 w .54891 .42286 m .55079 .5021 L .62393 .4637 L p .8 g F P s P p .002 w .48226 .38295 m .54891 .42286 L .55079 .5021 L p .8 g F P s P p .002 w .61992 .38295 m .54891 .42286 L .48226 .38295 L p .8 g F P s P p .002 w .48226 .4637 m .48226 .38295 L .4156 .42286 L p .8 g F P s P p .002 w .4156 .42286 m .41372 .5021 L .48226 .4637 L p .8 g F P s P p .002 w .48226 .38295 m .54891 .42286 L .55079 .5021 L p .8 g F P s P p .002 w .34058 .4637 m .34459 .38295 L .28228 .42286 L p .8 g F P s P p .002 w .28228 .42286 m .27665 .5021 L .34058 .4637 L p .8 g F P s P p .002 w .34459 .38295 m .4156 .42286 L .41372 .5021 L p .8 g F P s P p .002 w .28228 .42286 m .20692 .38295 L .21449 .30665 L p .8 g F P s P p .002 w .28228 .42286 m .20692 .38295 L .21449 .30665 L p .8 g F P s P p .002 w .34459 .38295 m .28228 .42286 L .20692 .38295 L p .8 g F P s P p .002 w .34058 .4637 m .34459 .38295 L .28228 .42286 L p .8 g F P s P p .002 w .28228 .42286 m .27665 .5021 L .34058 .4637 L p .8 g F P s P p .002 w .14896 .42286 m .20692 .38295 L .28228 .42286 L p .8 g F P s P p .002 w .14896 .42286 m .20692 .38295 L .28228 .42286 L p .8 g F P s P p .002 w .20692 .38295 m .28228 .42286 L .27665 .5021 L p .8 g F P s P p .002 w .34459 .38295 m .28228 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.38295 L .55341 .34036 L p .8 g F P s P p .002 w .41312 .26276 m .48226 .30665 L .48226 .38295 L p .8 g F P s P p .002 w .41312 .26276 m .34837 .30665 L .27485 .26276 L p .8 g F P s P p .002 w .4111 .34036 m .41312 .26276 L .34837 .30665 L p .8 g F P s P p .002 w .34837 .30665 m .34459 .38295 L .4111 .34036 L p .8 g F P s P p .002 w .21449 .30665 m .27485 .26276 L .34837 .30665 L p .8 g F P s P p .002 w .27485 .26276 m .34837 .30665 L .34459 .38295 L p .8 g F P s P p .002 w .26878 .34036 m .27485 .26276 L .21449 .30665 L p .8 g F P s P p .002 w .21449 .30665 m .20692 .38295 L .26878 .34036 L p .8 g F P s P p .002 w .27485 .26276 m .34837 .30665 L .34459 .38295 L p .8 g F P s P p .002 w .61757 .85343 m .55216 .82718 L .5502 .73732 L p .5 g F P s P p .002 w .61757 .85343 m .55216 .82718 L .5502 .73732 L p .5 g F P s P p .002 w .69195 .82718 m .61757 .85343 L .55216 .82718 L p .5 g F P s P p .002 w .48226 .85343 m .55216 .82718 L .61757 .85343 L p .5 g F P s P p .002 w .21163 .85343 m .27256 .82718 L .34694 .85343 L p .5 g F P s P p .002 w .34694 .85343 m .27256 .82718 L .27841 .73732 L p .5 g F P s P p .002 w .21163 .85343 m .13276 .82718 L .14252 .73732 L p .5 g F P s P p .002 w .27256 .82718 m .21163 .85343 L .13276 .82718 L p .5 g F P s P p .002 w .48226 .85343 m .41236 .82718 L .41431 .73732 L p .8 g F P s P p .002 w .48226 .85343 m .41236 .82718 L .41431 .73732 L p .8 g F P s P p .002 w .55216 .82718 m .48226 .85343 L .41236 .82718 L p .8 g F P s P p .002 w .34694 .85343 m .41236 .82718 L .48226 .85343 L p .8 g F P s P p .002 w .34694 .85343 m .27256 .82718 L .27841 .73732 L p .8 g F P s P p .002 w .41236 .82718 m .34694 .85343 L .27256 .82718 L p .8 g F P s P p .002 w .6861 .73732 m .62268 .7071 L .61874 .62022 L p .8 g F P s P p .002 w .6861 .73732 m .62268 .7071 L .61874 .62022 L p .8 g F P s P p .002 w .76309 .7071 m .6861 .73732 L .62268 .7071 L p .5 g F P s P p .002 w .77144 .79913 m .76309 .7071 L .6861 .73732 L p .5 g F P s P p .002 w .6861 .73732 m .69195 .82718 L .77144 .79913 L p .5 g F P s P p .002 w .5502 .73732 m .62268 .7071 L .6861 .73732 L p .5 g F P s P p .002 w .5502 .73732 m .62268 .7071 L .6861 .73732 L p .8 g F P s P p .002 w .62268 .7071 m .6861 .73732 L .69195 .82718 L p .5 g F P s P p .002 w .62685 .79913 m .62268 .7071 L .5502 .73732 L p .5 g F P s P p .002 w .5502 .73732 m .55216 .82718 L .62685 .79913 L p .5 g F P s P p .002 w .62268 .7071 m .6861 .73732 L .69195 .82718 L p .5 g F P s P p .002 w .5502 .73732 m .48226 .7071 L .48226 .62022 L p .5 g F P s P p .002 w .5502 .73732 m .48226 .7071 L .48226 .62022 L p .5 g F P s P p .002 w .62268 .7071 m .5502 .73732 L .48226 .7071 L p .5 g F P s P p .002 w .41431 .73732 m .48226 .7071 L .5502 .73732 L p .5 g F P s P p .002 w .14252 .73732 m .20142 .7071 L .27841 .73732 L p .5 g F P s P p .002 w .14252 .73732 m .20142 .7071 L .27841 .73732 L p .8 g F P s P p .002 w .19307 .79913 m .20142 .7071 L .14252 .73732 L p .5 g F P s P p .002 w .14252 .73732 m .13276 .82718 L 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g F P s P p .002 w .62685 .79913 m .62268 .7071 L .5502 .73732 L p .8 g F P s P p .002 w .5502 .73732 m .55216 .82718 L .62685 .79913 L p .8 g F P s P p .002 w .41431 .73732 m .48226 .7071 L .5502 .73732 L p .8 g F P s P p .002 w .48226 .7071 m .5502 .73732 L .55216 .82718 L p .8 g F P s P p .002 w .48226 .79913 m .48226 .7071 L .41431 .73732 L p .8 g F P s P p .002 w .41431 .73732 m .41236 .82718 L .48226 .79913 L p .8 g F P s P p .002 w .48226 .7071 m .5502 .73732 L .55216 .82718 L p .8 g F P s P p .002 w .41431 .73732 m .34184 .7071 L .34578 .62022 L p .8 g F P s P p .002 w .48226 .7071 m .41431 .73732 L .34184 .7071 L p .8 g F P s P p .002 w .48226 .79913 m .48226 .7071 L .41431 .73732 L p .8 g F P s P p .002 w .41431 .73732 m .41236 .82718 L .48226 .79913 L p .8 g F P s P p .002 w .27841 .73732 m .34184 .7071 L .41431 .73732 L p .8 g F P s P p .002 w .34184 .7071 m .41431 .73732 L .41236 .82718 L p .8 g F P s P p .002 w .33766 .79913 m .34184 .7071 L .27841 .73732 L p .8 g F P s P p .002 w .27841 .73732 m .27256 .82718 L .33766 .79913 L p .8 g F P s P p .002 w .34184 .7071 m .41431 .73732 L .41236 .82718 L p .8 g F P s P p .002 w .34184 .7071 m .27841 .73732 L .20142 .7071 L p .8 g F P s P p .002 w .33766 .79913 m .34184 .7071 L .27841 .73732 L p .8 g F P s P p .002 w .27841 .73732 m .27256 .82718 L .33766 .79913 L p .8 g F P s P p .002 w .20142 .7071 m .27841 .73732 L .27256 .82718 L p .8 g F P s P p .002 w .48226 .7071 m .41431 .73732 L .34184 .7071 L p .8 g F P s P p .002 w .34578 .62022 m .41173 .58594 L .48226 .62022 L p .25 g F P s P p .002 w .48226 .62022 m .41173 .58594 L .41372 .5021 L p .25 g F P s P p .002 w .61874 .62022 m .55278 .58594 L .55079 .5021 L p .8 g F P s P p .002 w .61874 .62022 m .55278 .58594 L .55079 .5021 L p .8 g F P s P p .002 w .69382 .58594 m .61874 .62022 L .55278 .58594 L p .5 g F P s P p .002 w .61874 .62022 m .69382 .58594 L .75522 .62022 L p .8 g F P s P p .002 w .61874 .62022 m .69382 .58594 L .75522 .62022 L p .8 g F P s P p .002 w .70014 .67478 m .69382 .58594 L .61874 .62022 L p .5 g F P s P p .002 w .61874 .62022 m .62268 .7071 L .70014 .67478 L p .5 g F P s P p .002 w .48226 .62022 m .55278 .58594 L .61874 .62022 L p .5 g F P s P p .002 w .48226 .62022 m .55278 .58594 L .61874 .62022 L p .8 g F P s P p .002 w .55278 .58594 m .61874 .62022 L .62268 .7071 L p .5 g F P s P p .002 w .55488 .67478 m .55278 .58594 L .48226 .62022 L p .5 g F P s P p .002 w .48226 .62022 m .48226 .7071 L .55488 .67478 L p .5 g F P s P p .002 w .55278 .58594 m .61874 .62022 L .62268 .7071 L p .5 g F P s P p .002 w .48226 .62022 m .41173 .58594 L .41372 .5021 L p .5 g F P s P p .002 w .55278 .58594 m .48226 .62022 L .41173 .58594 L p .5 g F P s P p .002 w .20929 .62022 m .27069 .58594 L .34578 .62022 L p .5 g F P s P p .002 w .20929 .62022 m .27069 .58594 L .34578 .62022 L p .8 g F P s P p .002 w .26437 .67478 m .27069 .58594 L .20929 .62022 L p .5 g F P s P p .002 w .20929 .62022 m .20142 .7071 L .26437 .67478 L p .5 g F P s P p .002 w .27069 .58594 m .34578 .62022 L .34184 .7071 L p .5 g F P s P p .002 w .20929 .62022 m .12964 .58594 L .13958 .5021 L p .8 g F P s P p .002 w .20929 .62022 m .12964 .58594 L .13958 .5021 L p .8 g F P s P p .002 w .27069 .58594 m .20929 .62022 L .12964 .58594 L p .5 g F P s P p .002 w .26437 .67478 m .27069 .58594 L .20929 .62022 L p .5 g F P s P p .002 w .20929 .62022 m .20142 .7071 L .26437 .67478 L p .5 g F P s P p .002 w .07281 .62022 m .12964 .58594 L .20929 .62022 L p .8 g F P s P p .002 w .07281 .62022 m .12964 .58594 L .20929 .62022 L p .8 g F P s P p .002 w .12964 .58594 m .20929 .62022 L .20142 .7071 L p .5 g F P s P p .002 w .34578 .62022 m .27069 .58594 L .27665 .5021 L p .5 g F P s P p .002 w .34578 .62022 m .27069 .58594 L .27665 .5021 L p .8 g F P s P p .002 w .41173 .58594 m .34578 .62022 L .27069 .58594 L p .5 g F P s P p .002 w .70014 .67478 m .69382 .58594 L .61874 .62022 L p .8 g F P s P p .002 w .61874 .62022 m .62268 .7071 L .70014 .67478 L p .8 g F P s P p .002 w .69382 .58594 m .75522 .62022 L .76309 .7071 L p .8 g F P s P p .002 w .75522 .62022 m .69382 .58594 L .68786 .5021 L p .8 g F P s P p .002 w .69382 .58594 m .61874 .62022 L .55278 .58594 L p .8 g F P s P p .002 w .27069 .58594 m .20929 .62022 L .12964 .58594 L p .8 g F P s P p .002 w .11911 .67478 m .12964 .58594 L .07281 .62022 L p .8 g F P s P p .002 w .07281 .62022 m .061 .7071 L .11911 .67478 L p .8 g F P s P p .002 w .12964 .58594 m .20929 .62022 L .20142 .7071 L p .8 g F P s P p .002 w .55278 .58594 m .48226 .62022 L .41173 .58594 L p .8 g F P s P p .002 w .55488 .67478 m .55278 .58594 L .48226 .62022 L p .8 g F P s P p .002 w .48226 .62022 m .48226 .7071 L .55488 .67478 L p .8 g F P s P p .002 w .34578 .62022 m .41173 .58594 L .48226 .62022 L p .8 g F P s P p .002 w .41173 .58594 m .48226 .62022 L .48226 .7071 L p .8 g F P s P p .002 w .40963 .67478 m .41173 .58594 L .34578 .62022 L p .8 g F P s P p .002 w .34578 .62022 m .34184 .7071 L .40963 .67478 L p .8 g F P s P p .002 w .41173 .58594 m .48226 .62022 L .48226 .7071 L p .8 g F P s P p .002 w .41173 .58594 m .34578 .62022 L .27069 .58594 L p .8 g F P s P p .002 w .40963 .67478 m .41173 .58594 L .34578 .62022 L p .8 g F P s P p .002 w .34578 .62022 m .34184 .7071 L .40963 .67478 L p .8 g F P s P p .002 w .27069 .58594 m .34578 .62022 L .34184 .7071 L p .8 g F P s P p .002 w .41372 .5021 m .34058 .4637 L .34459 .38295 L p .25 g F P s P p .002 w .41372 .5021 m .34058 .4637 L .34459 .38295 L p .8 g F P s P p .002 w .48226 .4637 m .41372 .5021 L .34058 .4637 L p .25 g F P s P p .002 w .55079 .5021 m .48226 .4637 L .48226 .38295 L p .8 g F P s P p .002 w .55079 .5021 m .48226 .4637 L .48226 .38295 L p .8 g F P s P p .002 w .62393 .4637 m .55079 .5021 L .48226 .4637 L p .5 g F P s P p .002 w .55079 .5021 m .62393 .4637 L .68786 .5021 L p .8 g F P s P p .002 w .55079 .5021 m .62393 .4637 L .68786 .5021 L p .8 g F P s P p .002 w .62818 .54929 m .62393 .4637 L .55079 .5021 L p .5 g F P s P p .002 w .55079 .5021 m .55278 .58594 L .62818 .54929 L p .5 g F P s P p .002 w .41372 .5021 m .48226 .4637 L .55079 .5021 L p .5 g F P s P p .002 w .41372 .5021 m .48226 .4637 L .55079 .5021 L p .8 g F P s P p .002 w .48226 .4637 m .55079 .5021 L .55278 .58594 L p .5 g F P s P p .002 w .48226 .54929 m .48226 .4637 L .41372 .5021 L p .5 g F P s P p .002 w .41372 .5021 m .41173 .58594 L .48226 .54929 L p .5 g F P s P p .002 w .48226 .4637 m .55079 .5021 L .55278 .58594 L p .5 g F P s P p .002 w .27665 .5021 m .34058 .4637 L .41372 .5021 L p .5 g F P s P p .002 w .27665 .5021 m .34058 .4637 L .41372 .5021 L p .8 g F P s P p .002 w .33633 .54929 m .34058 .4637 L .27665 .5021 L p .5 g F P s P p .002 w .27665 .5021 m .27069 .58594 L .33633 .54929 L p .5 g F P s P p .002 w .34058 .4637 m .41372 .5021 L .41173 .58594 L p .5 g F P s P p .002 w .27665 .5021 m .1989 .4637 L .20692 .38295 L p .8 g F P s P p .002 w .27665 .5021 m .1989 .4637 L .20692 .38295 L p .8 g F P s P p .002 w .34058 .4637 m .27665 .5021 L .1989 .4637 L p .5 g F P s P p .002 w .33633 .54929 m .34058 .4637 L .27665 .5021 L p .5 g F P s P p .002 w .27665 .5021 m .27069 .58594 L .33633 .54929 L p .5 g F P s P p .002 w .13958 .5021 m .1989 .4637 L .27665 .5021 L p .8 g F P s P p .002 w .13958 .5021 m .1989 .4637 L .27665 .5021 L p .8 g F P s P p .002 w .1989 .4637 m .27665 .5021 L .27069 .58594 L p .5 g F P s P p .002 w .62818 .54929 m .62393 .4637 L .55079 .5021 L p .8 g F P s P p .002 w .55079 .5021 m .55278 .58594 L .62818 .54929 L p .8 g F P s P p .002 w .62393 .4637 m .68786 .5021 L .69382 .58594 L p .8 g F P s P p .002 w .68786 .5021 m .62393 .4637 L .61992 .38295 L p .8 g F P s P p .002 w .62393 .4637 m .55079 .5021 L .48226 .4637 L p .8 g F P s P p .002 w .34058 .4637 m .27665 .5021 L .1989 .4637 L p .8 g F P s P p .002 w .1904 .54929 m .1989 .4637 L .13958 .5021 L p .8 g F P s P p .002 w .13958 .5021 m .12964 .58594 L .1904 .54929 L p .8 g F P s P p .002 w .1989 .4637 m .27665 .5021 L .27069 .58594 L p .8 g F P s P p .002 w .48226 .4637 m .41372 .5021 L .34058 .4637 L p .8 g F P s P p .002 w .48226 .54929 m .48226 .4637 L .41372 .5021 L p .8 g F P s P p .002 w .41372 .5021 m .41173 .58594 L .48226 .54929 L p .8 g F P s P p .002 w .34058 .4637 m .41372 .5021 L .41173 .58594 L p .8 g F P s P p .002 w .34459 .38295 m .4111 .34036 L .48226 .38295 L p .25 g F P s P p .002 w .34459 .38295 m .4111 .34036 L .48226 .38295 L p .5 g F P s P p .002 w .4111 .34036 m .48226 .38295 L .48226 .4637 L p .25 g F P s P p .002 w .48226 .38295 m .4111 .34036 L .41312 .26276 L p .5 g F P s P p .002 w .48226 .38295 m .4111 .34036 L .41312 .26276 L p .8 g F P s P p .002 w .55341 .34036 m .48226 .38295 L .4111 .34036 L p .5 g F P s P p .002 w .48226 .38295 m .55341 .34036 L .61992 .38295 L p .8 g F P s P p .002 w .48226 .38295 m .55341 .34036 L .61992 .38295 L p .8 g F P s P p .002 w .55556 .42263 m .55341 .34036 L .48226 .38295 L p .5 g F P s P p .002 w .48226 .38295 m .48226 .4637 L .55556 .42263 L p .5 g F P s P p .002 w .4111 .34036 m .48226 .38295 L .48226 .4637 L p .5 g F P s P p .002 w .55556 .42263 m .55341 .34036 L .48226 .38295 L p .8 g F P s P p .002 w .48226 .38295 m .48226 .4637 L .55556 .42263 L p .8 g F P s P p .002 w .55341 .34036 m .61992 .38295 L .62393 .4637 L p .8 g F P s P p .002 w .61992 .38295 m .55341 .34036 L .55139 .26276 L p .8 g F P s P p .002 w .55341 .34036 m .48226 .38295 L .4111 .34036 L p .8 g F P s P p .002 w .20692 .38295 m .26878 .34036 L .34459 .38295 L p .8 g F P s P p .002 w .20692 .38295 m .26878 .34036 L .34459 .38295 L p .8 g F P s P p .002 w .26235 .42263 m .26878 .34036 L .20692 .38295 L p .8 g F P s P p .002 w .20692 .38295 m .1989 .4637 L .26235 .42263 L p .8 g F P s P p .002 w .40895 .42263 m .4111 .34036 L .34459 .38295 L p .25 g F P s P p .002 w .34459 .38295 m .34058 .4637 L .40895 .42263 L p .25 g F P s P p .002 w .34459 .38295 m .26878 .34036 L .27485 .26276 L p .8 g F P s P p .002 w .34459 .38295 m .26878 .34036 L .27485 .26276 L p .8 g F P s P p .002 w .4111 .34036 m .34459 .38295 L .26878 .34036 L p .5 g F P s P p .002 w .40895 .42263 m .4111 .34036 L .34459 .38295 L p .5 g F P s P p .002 w .34459 .38295 m .34058 .4637 L .40895 .42263 L p .5 g F P s P p .002 w .26878 .34036 m .34459 .38295 L .34058 .4637 L p .5 g F P s P p .002 w .4111 .34036 m .34459 .38295 L .26878 .34036 L p .8 g F P s P p .002 w .26878 .34036 m .34459 .38295 L .34058 .4637 L p .8 g F P s P p .002 w .48226 .2159 m .41312 .26276 L .3393 .2159 L p .25 g F P s P p .002 w .41312 .26276 m .48226 .2159 L .55139 .26276 L p .8 g F P s P p .002 w .48226 .29479 m .48226 .2159 L .41312 .26276 L p .25 g F P s P p .002 w .41312 .26276 m .4111 .34036 L .48226 .29479 L p .25 g F P s P p .002 w .27485 .26276 m .3393 .2159 L .41312 .26276 L p .8 g F P s P p .002 w .3393 .2159 m .41312 .26276 L .4111 .34036 L p .25 g F P s P p .002 w .48226 .29479 m .48226 .2159 L .41312 .26276 L p .8 g F P s P p .002 w .41312 .26276 m .4111 .34036 L .48226 .29479 L p .8 g F P s P p .002 w 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m .55488 .67478 L .55278 .58594 L p .5 g F P s P p .002 w .70014 .67478 m .62268 .7071 L .55488 .67478 L p .5 g F P s P p .002 w .48226 .7071 m .55488 .67478 L .62268 .7071 L p .5 g F P s P p .002 w .061 .7071 m .11911 .67478 L .20142 .7071 L p .5 g F P s P p .002 w .061 .7071 m .11911 .67478 L .20142 .7071 L p .8 g F P s P p .002 w .10794 .7691 m .11911 .67478 L .061 .7071 L p .5 g F P s P p .002 w .061 .7071 m .04848 .79913 L .10794 .7691 L p .5 g F P s P p .002 w .11911 .67478 m .20142 .7071 L .19307 .79913 L p .5 g F P s P p .002 w .20142 .7071 m .26437 .67478 L .34184 .7071 L p .5 g F P s P p .002 w .34184 .7071 m .26437 .67478 L .27069 .58594 L p .5 g F P s P p .002 w .20142 .7071 m .11911 .67478 L .12964 .58594 L p .5 g F P s P p .002 w .20142 .7071 m .11911 .67478 L .12964 .58594 L p .8 g F P s P p .002 w .26437 .67478 m .20142 .7071 L .11911 .67478 L p .5 g F P s P p .002 w .76309 .7071 m .70014 .67478 L .69382 .58594 L p .8 g F P s P p .002 w .70014 .67478 m .62268 .7071 L 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P p .002 w .25767 .7691 m .26437 .67478 L .20142 .7071 L p .8 g F P s P p .002 w .20142 .7071 m .19307 .79913 L .25767 .7691 L p .8 g F P s P p .002 w .11911 .67478 m .20142 .7071 L .19307 .79913 L p .8 g F P s P p .002 w .55488 .67478 m .48226 .7071 L .40963 .67478 L p .8 g F P s P p .002 w .40963 .67478 m .34184 .7071 L .26437 .67478 L p .8 g F P s P p .002 w .55278 .58594 m .62818 .54929 L .69382 .58594 L p .5 g F P s P p .002 w .55278 .58594 m .62818 .54929 L .69382 .58594 L p .8 g F P s P p .002 w .63269 .64017 m .62818 .54929 L .55278 .58594 L p .5 g F P s P p .002 w .55278 .58594 m .55488 .67478 L .63269 .64017 L p .5 g F P s P p .002 w .62818 .54929 m .69382 .58594 L .70014 .67478 L p .5 g F P s P p .002 w .55278 .58594 m .48226 .54929 L .48226 .4637 L p .5 g F P s P p .002 w .55278 .58594 m .48226 .54929 L .48226 .4637 L p .5 g F P s P p .002 w .62818 .54929 m .55278 .58594 L .48226 .54929 L p .5 g F P s P p .002 w .41173 .58594 m .48226 .54929 L .55278 .58594 L p .5 g F P s P 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represents the sum 1\[Currency]+2\[Currency]+3\[Currency]+4\[Currency]. The \ three medium-gray sheets that separate the three basic sum \"pyramids\" each \ represent the sum 1+2+3+4. "], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Small Digression"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Students studying sums like 1\[Currency]+2\[Currency]+3\[Currency]+4\ \[Currency] are probably already familiar with sums like 1+2+3+4, but, if \ not, the the figure \"IntegerSum,\" provides a simple two-dimensional \ analogue to our three-dimensional technique. Clearly 1+2+3+4 is [(5)(5) - \ 5]/2, which easily generalizes to 1+2+3+...+n = \n[(n+1)(n+1) - (n+1)]/2. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Clear[s1,s2,s3,s4]"], "Input", AspectRatioFixed->True], Cell[TextData[ "s1=Graphics[\n{Line[{{0,0},{5,0}}],Line[{{0,1},{4,1}}],\n\ Line[{{0,2},{3,2}}],Line[{{0,3},{2,3}}],\nLine[{{0,4},{1,4}}],\n\ Line[{{0,0},{0,5}}],Line[{{1,0},{1,4}}],\n\ Line[{{2,0},{2,3}}],Line[{{3,0},{3,2}}],\nLine[{{4,0},{4,1}}]}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "s2=Graphics[{GrayLevel[.25],\nRectangle[{0,4},{1,5}],\n\ Rectangle[{1,3},{2,4}],\nRectangle[{2,2},{3,3}],\nRectangle[{3,1},{4,2}],\n\ Rectangle[{4,0},{5,1}]}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "s3=Graphics[\n{Line[{{0,5},{5,5}}],Line[{{0,4},{5,4}}],\n\ Line[{{2,3},{5,3}}],Line[{{3,2},{5,2}}],\nLine[{{4,1},{5,1}}],\n\ Line[{{1,4},{1,5}}],Line[{{2,3},{2,5}}],\n\ Line[{{3,2},{3,5}}],Line[{{4,0},{4,5}}],\nLine[{{5,0},{5,5}}]}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "s4=Graphics[Text[FontForm[\n\t\t\t\t\"IntegerSum\",{\"Chicago\",10}],\n\t\t\t\ \t{2.5,5.3}]];"], "Input", AspectRatioFixed->True], Cell[TextData["Show[s1,s2,s3,s4,AspectRatio->Automatic];"], "Input", AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData[{ StyleBox["Notice that ", Evaluatable->False, PageBreakAbove->True, GroupPageBreakWithin->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, PageBreakAbove->True, GroupPageBreakWithin->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " can easily convert \n[(n+1)(n+1) - (n+1)]/2 to more familiar forms:", Evaluatable->False, PageBreakAbove->True, GroupPageBreakWithin->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, PageBreakAbove->True, GroupPageBreakWithin->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Simplify[((n+1)(n+1)-(n+1))/2]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ (n + n^2)/2\ \>", "\<\ 2 n + n ------ 2\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["or"], "Text", Evaluatable->False, GroupPageBreakWithin->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Factor[%]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ (n*(1 + n))/2\ \>", "\<\ n (1 + n) --------- 2\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]]}, Open]]}, Open]]}, Open]], Cell[TextData["End Small Digression"], "Subsection", Evaluatable->False, PageBreakBelow->Automatic, GroupPageBreakWithin->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Finding the Sum 1\[Currency]+2\[Currency]+3\[Currency]+...+n\[Currency], the \ Legend Continues"], "Section", Evaluatable->False, PageBreakAbove->False, GroupPageBreakWithin->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[{ StyleBox[ "Anyway, the entire \"BigCube\" is made up of three light-gray basic sum \ \"pyramids,\" three \"1+2+3+4\" medium-gray separating sheets, and one \ \"1+1+1+1+1 = 5\" dark-gray main diagonal. Therefore 1\[Currency]+2\ \[Currency]+3\[Currency]+4\[Currency] is \[IAcute]\[Sterling]\:2215 of [5\ \:2039 - 3(\[IAcute]\:203a\[IAcute]\[CloseCurlyDoubleQuote]\[IAcute]\ \[Trademark]\[FiLigature]\[IAcute]\[CloseCurlyQuote]) - 5]. Again, this \ easily generalizes to 1\[Currency]+2\[Currency]+3\[Currency]+...+n\[Currency] \ = \n\[IAcute]\[Sterling]\:2215[(n+1)\:2039 - 3(\[IAcute]\:02c6\[IAcute]\ \[CloseCurlyDoubleQuote]\[IAcute]\:02c6\[IAcute]\[Trademark]\[PlusMinus]\ \[IAcute]\:2215\[IAcute]\[CloseCurlyQuote]) - (n+1)]. Let's see if ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" can make this formula look more familiar:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Simplify[((n+1)^3-3(n(n+1)/2)-(n+1))/3]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ n/6 + n^2/2 + n^3/3\ \>", "\<\ 2 3 n n n - + -- + -- 6 2 3\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Together[%]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ (n + 3*n^2 + 2*n^3)/6\ \>", "\<\ 2 3 n + 3 n + 2 n --------------- 6\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Factor[%]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ (n*(1 + n)*(1 + 2*n))/6\ \>", "\<\ n (1 + n) (1 + 2 n) ------------------- 6\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "There you go; the old familiar 1\[Currency]+2\[Currency]+3\[Currency]+...+n\ \[Currency] = \[IAcute]\:02c6\[IAcute]\[CloseCurlyDoubleQuote]\[IAcute]\:02c6\ \[IAcute]\[PlusMinus]\[IAcute]\:2215\[IAcute]\[Section]\[CloseCurlyQuote]\ \[IAcute]\[CloseCurlyDoubleQuote]\[IAcute]\[Currency]\[IAcute]\:02c6\[IAcute]\ \[PlusMinus]\[IAcute]\:2215\[IAcute]\[CloseCurlyQuote]."], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]]}, Open]]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData["Q.E.D."], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "[Which is an abbreviation for the semi-irrelevant (but pleasingly \ pretentious) Latin phrase \"Quod erat demonstrandum,\" or \"That which was to \ be demonstrated.\"]\n"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["[I did not make that up.]"], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 640}, {0, 460}}, WindowToolbars->{}, CellGrouping->Manual, WindowSize->{520, 365}, WindowMargins->{{4, Automatic}, {15, Automatic}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, -1}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, MacintoshSystemPageSetup->"\<\ AVU/IFiQKFD000000V:^/09R]g0000000OVaH097bCP0AP1Y06`0I@1^0642HZj` 0V:gT0000001nK500TO9>000000000000000009R[[0000000000000000000000 00000000000000000000000000000000\>" ] (*********************************************************************** Cached data follows. 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