CS4248_Tutorial_4

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National University of Singapore *

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4248

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Computer Science

Date

Nov 24, 2024

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8oefG;:opqG;:A54 ZM*mnoqrsnde*VI UH*[QI<qrs*I<@43 aWnde_U9@43fghfghPC[QI< lmnqrsmno_UndePC*VI G;::. oef@43i_`{qrs@43[Q9@43fgh *[Q=1 ?32[Q9_U=1@43nde₺<0@439_U=1@43nde BBBBBBBBBRf@NNNONNrL kjjjh@@@@@@@@Pd>CCCCCCCCSgAcccdcca kjjjhnnokjjjhRRRSRRvPaaabaa_CCCCCCCCSgA±IIIIIJIYmG... st33333uWǷIIIIIJIYmGIIIIIJIYmG UUUVUUtySaaabaa_±,,,kjjjhCCCCCCCCSgA,,,33333uW kjjjh33333uW,,,@@@@@@@@Pd>NNNONNrLCCCCCCCCSgA,,,±IIIIIJIYmG kjjjhRRRSRRvPUUUVUUtySCCCCCCCCSgA,,,cccdcca 8888886z\6aaabaa_RRRSRRvPLLLMLLpJ RRRSRRvPnnoaaabaa_ rs33333uW33333uWGGGGGGHWkE {eefeec ±NNNONNrL0000rT {44444vX2 IIIIIJIYmG33333uW,,,kjjjhnnoaaabaa_33333uWcccdcca--- pcdcebA4j]^]_{qnqA4\OPOQ:A4dedfc ±NNNONNrL0000rT @3\OPOQ:`STSU>1A4obcbda₸ =A4:`STSU>1A4obcbda YYYZYYx}Wx TTTUTTsxRIIIIIJIYmG33333uW±cccdcca33333uW ,,,RRRSRRvPLLLMLLpJ33333uW kjjjhRRRSRRvP xRRRSRRvPnnoaaabaa_ kjjjhnnokjjjhRRRSRRvPaaabaa_CCCCCCCCSgA±IIIIIJIYmG cccdcca33333uWcccdccacccdccaCCCCCCCCSgARRRSRRvPNNNONNrL stCCCCCCCCSgAkjjjh@@@@@@@@Pd> xRRRSRRvPnnoaaabaa_ ±kjjjhkjjjh33333uWLLLMLLpJUUUVUUtySkjjjhcccdcca kjjjhRRRSRRvP kjjjh@@@@@@@@Pd>33333uWcccdcca33333uW \\\]\\{Z{nno33333uWcccdccakjjjhCCCCCCCCSgARRRSRRvPNNNONNrLcccdccaYYYZYYx}Wx BBBBBBBBBRf@kjjjhǷcccdcca ´NNNONNrL33333uW kjjjhRRRSRRvP ,,,RRRSRRvPLLLMLLpJ33333uW stCCCCCCCCSgAkjjjh@@@@@@@@Pd>RRRSRRvPnnokjjjh ± stRRRSRRvPaaabaa_GGGGGGHWkECCCCCCCCSgANNNONNrL<<<<<<<~`: cccdccaRRRSRRvPIIIIIJIYmGnnokjjjhCCCCCCCCSgARRRSRRvPNNNONNrL... $$nnokjjjh kjjjhRRRSRRvP <<<<<<<~`:33333uWkjjjh kjjjh@@@@@@@@Pd>33333uW LLLMLLpJRRRSRRvPcccdccakjjjh RRRSRRvPnnokjjjh RRRSRRvP8888886z\6 kjjjhnnokjjjhRRRSRRvPaaabaa_CCCCCCCCSgA±IIIIIJIYmG xRRRSRRvPnno cccdcca@@@@@@@@Pd>RRRSRRvPnnoIIIIIJIYmG0000rT ±kjjjhkjjjh33333uWLLLMLLpJUUUVUUtySkjjjh kjjjh@@@@@@@@Pd>33333uWLLLMLLpJ cccdccaRRRSRRvP kjjjh@@@@@@@@Pd>±kjjjh xRRRSRRvPnno @@@@@@@@Pd>±qr33333uW UUUVUUtySaaabaa_±,,,kjjjhCCCCCCCCSgA,,,33333uW $$33333uW8888886z\6RRRSRRvPaaabaa_33333uW cccdccaRRRSRRvPIIIIIJIYmGnnokjjjhCCCCCCCCSgARRRSRRvPNNNONNrLcccdcca ,,,±NNNONNrL $$33333uW cccdcca@@@@@@@@Pd>±aaabaa_33333uW0000rTYYYZYYx}Wx 2222tVuvkjjjh33333uWNNNONNrLcccdccaCCCCCCCCSgARRRSRRvPNNNONNrL 33333uWuv33333uWaaabaa_,,,CCCCCCCCSgAcccdcca33333uWcccdcca LLLMLLpJ±aaabaa_GGGGGGHWkE33333uW0000rT stCCCCCCCCSgAkjjjh@@@@@@@@Pd> ± Ǵ!!ǵ RRRSRRvPaaabaa_ Ǵ!!!ǵ ±aaabaa_33333uW cccdccaIIIIIJIYmGCCCCCCCCSgA<<<<<<<~`:@@@@@@@@Pd>kjjjhIIIIIJIYmGx LLLMLLpJRRRSRRvPaaabaa_33333uW ±0000rTqr±NNNONNrL,,,33333uW0000rT ±NNNONNrL0000rT xRRRSRRvPnno VVVWVVuzT±NNNONNrL0000rT xRRRSRRvPnnoaaabaa_ kjjjhnnokjjjhRRRSRRvPaaabaa_CCCCCCCCSgA±IIIIIJIYmG IIIIIJIYmG33333uW±0000rT33333uWaaabaa_WWWXWWv{Uv LLLMLLpJ±x NNNONNrLRRRSRRvPkjjjh @@@@@@@@Pd>±qr33333uW kjjjhCCCCCCCCSgALLLMLLpJ33333uW kjjjhRRRSRRvP <<<<<<<~`:RRRSRRvP RRRSRRvPqr33333uWaaabaa_ kjjjh@@@@@@@@Pd>33333uWLLLMLLpJYYYZYYx}Wx SSSTSSrwQYYYZYYx}Wx M@?QDCE>1>1A4\OPOQ YLML+obcbdaUHGI`STSUqtr YLML`STSU>1A4WJKJdedfc ++RRRSRRvPNNNONNrLcccdccaCCCCCCCCSgA0000rT33333uWaaabaa_ kjjjh@@@@@@@@Pd>33333uW 8888886z\6RRRSRRvPIIIIIJIYmGIIIIIJIYmGRRRSRRvPstCCCCCCCCSgANNNONNrL<<<<<<<~`: ????????Oc=KKKLK[oIKKKLK[oI--- 2 3. Let H(X) , H(X,Y) , and H(Y|X) denote the entropy, joint entropy, and conditional entropy of discrete random variables X and Y , defined as follows:         − = − = − = X x Y y X x Y y X x x y p y x p X Y H y x p y x p Y X H x p x p X H ) | ( log ) , ( ) | ( ) , ( log ) , ( ) , ( ) ( log ) ( ) ( Consider the following statement: ) | ( ) ( ) , ( X Y H X H Y X H + = Is the statement true? If it is true, give a formal proof and clearly justify each step of your proof. If it is false, provide a concrete counter-example. 4. Consider the following HMM: Suppose this HMM has the following set of parameters: P(T1|<s>) = 1/5 P(T1|T1) = 0 P(T1|T2) = 1/8 P(T1|T3) = 3/5 P(T2|<s>) = 0 P(T2|T1) = 5/6 P(T2|T2) = 1/2 P(T2|T3) = 1/5 P(T3|<s>) = 4/5 P(T3|T1) = 0 P(T3|T2) = 1/4 P(T3|T3) = 1/5 P(</s>|T1) = 1/6 P(</s>|T2) = 1/8 P(</s>|T3) = 0 P(w1|T1) = 1/20 P(w2|T1) = 1/10 P(w1|T2) = 1/5 P(w2|T2) = 1/10 P(w1|T3) = 1/10 P(w2|T3) = 1/10 T1, T2, T3 are part-of-speech tags. Consider the input sentence “w1 w2”, where w1 and w2 are words. Trace the Viterbi algorithm, by providing the values of the cells v(T, w) where T  { T1, T2, T3 }, and w  { w1, w2 }, and determine the optimal sequence of part-of-speech tags. </s> T3 T3 <s> w1 T2 T2 T1 T1 w2 bbbcbb`nnoUUUVUUtySUUUVUUtySRRRSRRvPcccdcca33333uW kjjjh@@@@@@@@Pd>CCCCCCCCSgAcccdcca ????????Oc=KKKLK[oIKKKLK[oI @@@@@@@@Pd>±cccdcca kjjjh@@@@@@@@Pd>33333uW 8888886z\6RRRSRRvPIIIIIJIYmGIIIIIJIYmGRRRSRRvPstCCCCCCCCSgANNNONNrL<<<<<<<~`: cccdcca33333uWkjjjh RRRSRRvP8888886z\6 UUUVUUtyS±aaabaa_±LLLMLLpJ33333uWkjjjh33333uWaaabaa_cccdcca--- jiigSSSTSSrwQ jiiglllj jiiglkkki JJJJKJZnHghggecccdcca>>>>>>>Nb< JJJJKJZnHcccdcca>>>>>>>Nb< SSSTSSrwQghgge9999997{]7 { ::::::8|^8ghgge9999997{]7 { jiigSSSTSSrwQ { 9999997{]7ghggeffgffd { SSSTSSrwQghggeffgffd jiiglllj SSSTSSrwQghgge44444vX2 SSSTSSrwQghggelllj SSSTSSrwQghgge::::::8|^8 SSSTSSrwQghgge44444vX2 jiiglkkki lkkkighgge9999997{]7 SSSTSSrwQghgge9999997{]7 SSSTSSrwQghgge9999997{]7 { jiig±$$IIIIIJIYmG33333uW SSSTSSrwQ--- jiigaaabaa_±NNNONNrLcccdccaCCCCCCCCSgAkjjjhCCCCCCCCSgARRRSRRvPNNNONNrL UUUVUUtySaaabaa_RRRSRRvP$$±$$CCCCCCCCSgAIIIIIJIYmGCCCCCCCCSgAkjjjhx RRRSRRvP8888886z\6 kjjjh@@@@@@@@Pd>33333uW ????????Oc=KKKLK[oIKKKLK[oI... 8888886z\6RRRSRRvPaaabaa_ 33333uWuv±LLLMLLpJUUUVUUtySIIIIIJIYmG33333uW... P ( T 2 | T 1 ) = 5/6 YYYZYYx}Wx stSSSTSSrwQ stlllj jiigSSSTSSrwQ SSSTSSrwQghggelllj{ SSSTSSrwQghggeSSSTSSrwQ{ jiiglllj SSSTSSrwQghgge9999997{]7 SSSTSSrwQghggeSSSTSSrwQ{ jiiglkkki SSSTSSrwQghggeSSSTSSrwQ{ SSSTSSrwQghggeSSSTSSrwQ{ jiig±$$IIIIIJIYmG33333uW lllj--- 2222tVLLLMLLpJCCCCCCCCSgAcccdccacccdccaCCCCCCCCSgARRRSRRvPNNNONNrL UUUVUUtySaaabaa_RRRSRRvP$$±$$CCCCCCCCSgAIIIIIJIYmGCCCCCCCCSgAkjjjhx RRRSRRvP8888886z\6 kjjjh@@@@@@@@Pd>33333uW ????????Oc=KKKLK[oIKKKLK[oI... 8888886z\6RRRSRRvPaaabaa_ 33333uWuv±LLLMLLpJUUUVUUtySIIIIIJIYmG33333uW... P ( w 1 | T 1 ) = 1/20 YYYZYYx}Wx jiigSSSTSSrwQ... jiiglllj... jiiglkkki ±aaabaa_33333uW UUUVUUtyS±aaabaa_kjjjhAAAAAAAAQe?RRRSRRvP8888886z\6AAAAAAAAQe?cccdccaUUUVUUtyS33333uW33333uW,,,@@@@@@@@Pd> kjjjh±<<<<<<<~`:cccdccaYYYZYYx}Wx ++RRRSRRvPNNNONNrLcccdccaCCCCCCCCSgA0000rT33333uWaaabaa_ kjjjh@@@@@@@@Pd>33333uW CCCCCCCCSgANNNONNrLUUUVUUtySnnokjjjh cccdcca33333uWNNNONNrLkjjjh33333uWNNNONNrL,,,33333uW ǴstSSSTSSrwQ stllljǵ... st@@@@@@@@Pd>33333uWaaabaa_33333uW stSSSTSSrwQ ±NNNONNrL0000rT stlllj ±aaabaa_33333uW stRRRSRRvPaaabaa_0000rTcccdccaYYYZYYx}Wx jiigaaabaa_±,,,33333uW kjjjh@@@@@@@@Pd>33333uW pqCCCCCCCCSgAkjjjh33333uWaaabaa_$$CCCCCCCCSgA ±IIIIIJIYmG<<<<<<<~`:RRRSRRvPaaabaa_CCCCCCCCSgAkjjjh@@@@@@@@Pd>LLLMLLpJ $$x UUUVUUtySaaabaa_RRRSRRvPqrCCCCCCCCSgA0000rTCCCCCCCCSgANNNONNrL<<<<<<<~`: kjjjh@@@@@@@@Pd>33333uW qr±IIIIIJIYmGnno33333uWcccdcca RRRSRRvP8888886z\6 kjjjh@@@@@@@@Pd>33333uW pqCCCCCCCCSgAkjjjh33333uWaaabaa_$$CCCCCCCCSgA cccdcca,,,RRRSRRvPaaabaa_33333uW qrVVVWVVuzTjiig... stWWWXWWv{Uv st@@@@@@@@Pd>33333uWaaabaa_33333uW T ∈ { T 1, T 2, T 3 } ... ±NNNONNrL0000rT w ∈ { w 1, w 2 } ... ±NNNONNrL0000rT 0000rT33333uWkjjjh33333uWaaabaa_LLLMLLpJCCCCCCCCSgANNNONNrL33333uW kjjjh@@@@@@@@Pd>33333uW RRRSRRvPUUUVUUtySkjjjhCCCCCCCCSgALLLMLLpJ±IIIIIJIYmG cccdcca33333uW\\\]\\{Z{nno33333uWNNNONNrL,,,33333uW RRRSRRvP8888886z\6 UUUVUUtyS±aaabaa_kjjjhAAAAAAAAQe?RRRSRRvP8888886z\6AAAAAAAAQe?cccdccaUUUVUUtyS33333uW33333uW,,,@@@@@@@@Pd> kjjjh±<<<<<<<~`:cccdccaYYYZYYx}Wx TTTUTTsxR±<<<<<<<~`:33333uW SSSTSSrwQ RRRSRRvP8888886z\6 lllj

llljYYYZYYx}Wx nabacA4:qnqobcbdaobcbdaA4\OPOQkmjm [NONA4qnqobcbda+WJKJ [NONA4kmjmsvt`STSUobcbdaUHGIdedfc dWXWYunabac[NON[NONeXYXZv bpR`|ao,3- F@{~ljlj[UWUcec3/giggigt95nq*_Y[Yceclj#TNP>:;svXRTRgaca62k_Y[Y95nq62kngig2OIK#ceccecgig#gaca2F@{~IC~r95nqgig2cecgig2cecnlnl@<uxTNPgig2cecnlnl@<uxTNP_Y[YYSUSYSUS:67or@<uxgig[UWUgaca#2ljIC~295nqoRLNIC~62klj95nqgacaRLNcecgig2cecnlnl@<uxTNPGA|[UWUgaca#2ljIC~295nqGA|RLNIC~62klj95nqgacaRLNGA|mkGA|cecXRTROIK_Y[Y[UWU62k>:;sv QQQRQQuONNNONNrL33333uW LLLMLLpJRRRSRRvP0000rT33333uWIIIIIJIYmG ,,,RRRSRRvPLLLMLLpJLLLMLLpJRRRSRRvPNNNONNrLIIIIIJIYmGx nnocccdcca33333uW0000rT 8888886z\6RRRSRRvPaaabaa_ cccdcca33333uWNNNONNrLkjjjhCCCCCCCCSgALLLMLLpJ33333uWNNNONNrLkjjjh ,,,IIIIIJIYmG±cccdccacccdccaCCCCCCCCSgA´,,,±kjjjhCCCCCCCCSgARRRSRRvPNNNONNrL CCCCCCCCSgAcccdcca kjjjh@@@@@@@@Pd>33333uW ```a``^MMMNMMqKMMMNMMqKYYYZYYx}Wx jiig@@@@@@@@Pd>CCCCCCCCSgAcccdcca LLLMLLpJRRRSRRvP0000rT33333uWIIIIIJIYmG CCCCCCCCSgANNNONNrLUUUVUUtySnnokjjjhcccdcca stRRRSRRvPaaabaa_0000rT qr33333uW,,,kjjjhRRRSRRvPaaabaa_cccdcca kjjjhRRRSRRvP kjjjh@@@@@@@@Pd>33333uW LLLMLLpJRRRSRRvP0000rT33333uWIIIIIJIYmG cccdcca33333uW\\\]\\{Z{nno33333uWNNNONNrLkjjjhCCCCCCCCSgA±IIIIIJIYmGIIIIIJIYmGx... RRRSRRvPNNNONNrL33333uW ±kjjjh 33333uW±,,,@@@@@@@@Pd> kjjjhCCCCCCCCSgALLLMLLpJ33333uW cccdccakjjjh33333uWUUUVUUtyS... ±NNNONNrL0000rT nnocccdcca33333uWcccdcca kjjjh@@@@@@@@Pd>33333uW IIIIIJIYmG±cccdccakjjjh @@@@@@@@Pd>CCCCCCCCSgA0000rT0000rT33333uWNNNONNrL cccdccakjjjh±kjjjh33333uW qr33333uW,,,kjjjhRRRSRRvPaaabaa_ kjjjhRRRSRRvP UUUVUUtySaaabaa_33333uW0000rTCCCCCCCCSgA,,,kjjjh kjjjh@@@@@@@@Pd>33333uW cccdcca33333uWNNNONNrLkjjjhCCCCCCCCSgALLLMLLpJ33333uWNNNONNrLkjjjh IIIIIJIYmG±$$33333uWIIIIIJIYmG VVVWVVuzT ˆ y WWWXWWv{UvYYYZYYx}Wx VVVWVVuzT±WWWXWWv{Uv ²cccdccacccdccannoLLLMLLpJ33333uW kjjjh@@@@@@@@Pd>±kjjjh st33333uW @@@@@@@@Pd>±qr33333uW ± Ǵ$$±<<<<<<<~`:AAAAAAAAQe?RRRSRRvP8888886z\6AAAAAAAAQe?qr33333uW,,,kjjjhRRRSRRvPaaabaa_cccdccaǵ LLLMLLpJRRRSRRvP0000rT33333uWIIIIIJIYmG 8888886z\6RRRSRRvPaaabaa_ cccdcca33333uWNNNONNrLkjjjhCCCCCCCCSgALLLMLLpJ33333uWNNNONNrLkjjjh ,,,IIIIIJIYmG±cccdccacccdccaCCCCCCCCSgA´,,,±kjjjhCCCCCCCCSgARRRSRRvPNNNONNrL--- st33333uW nnocccdcca33333uW kjjjh@@@@@@@@Pd>33333uW ±qr33333uWaaabaa_±<<<<<<<~`:33333uW RRRSRRvP8888886z\6 ±IIIIIJIYmGIIIIIJIYmG kjjjh@@@@@@@@Pd>33333uW stRRRSRRvPaaabaa_0000rTcccdcca qr33333uW,,,kjjjhRRRSRRvPaaabaa_cccdcca CCCCCCCCSgANNNONNrL ± cccdcca33333uWNNNONNrLkjjjh33333uWNNNONNrL,,,33333uW kjjjhRRRSRRvP UUUVUUtySaaabaa_33333uW0000rTCCCCCCCCSgA,,,kjjjh cccdcca33333uWNNNONNrLkjjjhCCCCCCCCSgALLLMLLpJ33333uWNNNONNrLkjjjh IIIIIJIYmG±$$33333uWIIIIIJIYmGYYYZYYx}Wx rs@@@@@@@@Pd>±kjjjh CCCCCCCCSgAcccdcca kjjjh@@@@@@@@Pd>33333uW $$33333uWNNNONNrL33333uW´kjjjh RRRSRRvP8888886z\6 kjjjh@@@@@@@@Pd>33333uW ```a``^MMMNMMqKMMMNMMqK LLLMLLpJRRRSRRvP0000rT33333uWIIIIIJIYmG RRRSRRvPqr33333uWaaabaa_ kjjjh@@@@@@@@Pd>33333uW $$±<<<<<<<~`:AAAAAAAAQe?RRRSRRvP8888886z\6AAAAAAAAQe?qr33333uW,,,kjjjhRRRSRRvPaaabaa_cccdcca LLLMLLpJRRRSRRvP0000rT33333uWIIIIIJIYmG]]]^]]|[| VVVWVVuzT$$WWWXWWv{Uv ²cccdccacccdccannoLLLMLLpJ33333uW kjjjh@@@@@@@@Pd>±kjjjh st33333uW CCCCCCCCSgANNNONNrLCCCCCCCCSgAkjjjhCCCCCCCCSgA±IIIIIJIYmGCCCCCCCCSgAz33333uW kjjjh@@@@@@@@Pd>33333uW stRRRSRRvPaaabaa_0000rT qr33333uW,,,kjjjhRRRSRRvPaaabaa_cccdcca RRRSRRvP8888886z\6 ```a``^MMMNMMqKMMMNMMqK LLLMLLpJRRRSRRvP0000rT33333uWIIIIIJIYmG nnocccdccaCCCCCCCCSgANNNONNrL<<<<<<<~`: stRRRSRRvPaaabaa_0000rTllljqr33333uW,,,YYYZYYx}Wx rsRRRSRRvPnnoIIIIIJIYmG0000rT xRRRSRRvPnno ,,,@@@@@@@@Pd>RRRSRRvPRRRSRRvPcccdcca33333uW kjjjhRRRSRRvP 8888886z\6aaabaa_33333uW33333uWz33333uW RRRSRRvPaaabaa_ nnoUUUVUUtyS0000rT±kjjjh33333uW kjjjh@@@@@@@@Pd>33333uW stRRRSRRvPaaabaa_0000rT qr33333uW,,,kjjjhRRRSRRvPaaabaa_cccdcca st@@@@@@@@Pd>33333uWNNNONNrL kjjjhaaabaa_±CCCCCCCCSgANNNONNrLCCCCCCCCSgANNNONNrL<<<<<<<~`: kjjjh@@@@@@@@Pd>33333uW ```a``^MMMNMMqKMMMNMMqK LLLMLLpJRRRSRRvP0000rT33333uWIIIIIJIYmG RRRSRRvPNNNONNrL cccdcca33333uWNNNONNrLkjjjhCCCCCCCCSgALLLMLLpJ33333uWNNNONNrLkjjjh ,,,IIIIIJIYmG±cccdccacccdccaCCCCCCCCSgA´,,,±kjjjhCCCCCCCCSgARRRSRRvPNNNONNrL 0000rT±kjjjh±]]]^]]|[| TTTUTTsxRIIIIIJIYmG33333uW±cccdcca33333uW <<<<<<<~`:CCCCCCCCSgAqr33333uW xRRRSRRvPnnoaaabaa_ EEEEEEEFUiCnnocccdccakjjjhCCCCCCCCSgA´,,,±kjjjhCCCCCCCCSgARRRSRRvPNNNONNrL stCCCCCCCCSgAkjjjh@@@@@@@@Pd> 33333uWuv±LLLMLLpJUUUVUUtySIIIIIJIYmG33333uWYYYZYYx}Wx lkkkiYYYZYYx}Wx ps+\OPOQQDCEdedfcNA@BQDCE\OPOQJ=< I<;obcbda+>1QDCEA4\OPOQkmjm bpR`|ao,3- F@{~ljlj[UWUcec3/giggig2cecnlnl@<ux62k_Y[Yceclj#TNP>:;svXRTRgaca62k_Y[Y95nq62kngigRLNIC~62klj95nqgacaRLNgigRLNIC~62klj95nqgacaRLNocecXRTROIKnljIC~XRTRTNPcec_Y[Y[UWU62k>:;sv pq±NNNONNrLCCCCCCCCSgAcccdcca@@@@@@@@Pd>CCCCCCCCSgANNNONNrL<<<<<<<~`: <<<<<<<~`:aaabaa_±0000rTCCCCCCCCSgA33333uWNNNONNrLkjjjh CCCCCCCCSgAcccdcca RRRSRRvPNNNONNrL33333uW ,,,RRRSRRvPLLLMLLpJLLLMLLpJRRRSRRvPNNNONNrL UUUVUUtySaaabaa_RRRSRRvP$$IIIIIJIYmG33333uWLLLMLLpJ 8888886z\6RRRSRRvPaaabaa_ kjjjh@@@@@@@@Pd>33333uW kjjjhaaabaa_±CCCCCCCCSgANNNONNrLCCCCCCCCSgANNNONNrL<<<<<<<~`: RRRSRRvP8888886z\6 0000rT33333uW33333uWUUUVUUtyS NNNONNrL33333uWnnoaaabaa_±IIIIIJIYmG NNNONNrL33333uWkjjjhstRRRSRRvPaaabaa_GGGGGGHWkEcccdcca VVVWVVuzT////qMMMNMMqKMMMNMMqKcccdccaWWWXWWv{UvYYYZYYx}Wx ////qnnoaaabaa_CCCCCCCCSgANNNONNrL<<<<<<<~`: kjjjh@@@@@@@@Pd>33333uW $$±,,,GGGGGGHWkEAAAAAAAAQe?UUUVUUtySaaabaa_RRRSRRvPUUUVUUtyS±<<<<<<<~`:±kjjjhCCCCCCCCSgARRRSRRvPNNNONNrL... kjjjh@@@@@@@@Pd>33333uW st33333uWCCCCCCCCSgA<<<<<<<~`:@@@@@@@@Pd>kjjjh RRRSRRvP8888886z\6 ////qMMMNMMqKMMMNMMqK stCCCCCCCCSgAIIIIIJIYmGIIIIIJIYmG $$33333uW nnoUUUVUUtyS0000rT±kjjjh33333uW0000rT UUUVUUtySaaabaa_RRRSRRvPUUUVUUtySRRRSRRvPaaabaa_kjjjhCCCCCCCCSgARRRSRRvPNNNONNrL±IIIIIJIYmGIIIIIJIYmGx kjjjhRRRSRRvP kjjjh@@@@@@@@Pd>33333uW UUUVUUtyS±aaabaa_kjjjhCCCCCCCCSgA±IIIIIJIYmG 0000rT33333uWaaabaa_CCCCCCCCSgAqr±kjjjhCCCCCCCCSgAqr33333uW RRRSRRvP8888886z\6 kjjjh@@@@@@@@Pd>33333uW IIIIIJIYmGRRRSRRvPcccdccacccdcca 8888886z\6nnoNNNONNrL,,,kjjjhCCCCCCCCSgARRRSRRvPNNNONNrL stCCCCCCCCSgAkjjjh@@@@@@@@Pd> aaabaa_33333uWcccdccaUUUVUUtyS33333uW,,,kjjjh kjjjhRRRSRRvP kjjjh@@@@@@@@Pd>33333uW ,,,nnoaaabaa_aaabaa_33333uWNNNONNrLkjjjh st33333uWCCCCCCCCSgA<<<<<<<~`:@@@@@@@@Pd>kjjjhYYYZYYx}Wx jiig@@@@@@@@Pd>33333uW qr±NNNONNrLCCCCCCCCSgAcccdcca@@@@@@@@Pd>CCCCCCCCSgANNNONNrL<<<<<<<~`: <<<<<<<~`:aaabaa_±0000rTCCCCCCCCSgA33333uWNNNONNrLkjjjh UUUVUUtySaaabaa_RRRSRRvP$$IIIIIJIYmG33333uWLLLMLLpJ @@@@@@@@Pd>±UUUVUUtySUUUVUUtyS33333uWNNNONNrLcccdcca st@@@@@@@@Pd>33333uWNNNONNrL kjjjh@@@@@@@@Pd>33333uW st33333uWCCCCCCCCSgA<<<<<<<~`:@@@@@@@@Pd>kjjjh CCCCCCCCSgAcccdcca qr±NNNONNrLCCCCCCCCSgAcccdcca@@@@@@@@Pd>CCCCCCCCSgANNNONNrL<<<<<<<~`:IIIIIJIYmGx cccdccaLLLMLLpJ±IIIIIJIYmGIIIIIJIYmG... UUUVUUtySaaabaa_33333uWqr33333uWNNNONNrLkjjjhCCCCCCCCSgANNNONNrL<<<<<<<~`: kjjjh@@@@@@@@Pd>33333uW st33333uWCCCCCCCCSgA<<<<<<<~`:@@@@@@@@Pd>kjjjh 8888886z\6aaabaa_RRRSRRvPLLLMLLpJ $$33333uWCCCCCCCCSgANNNONNrL<<<<<<<~`: 8888886z\6nnoaaabaa_kjjjh@@@@@@@@Pd>33333uWaaabaa_ nnoUUUVUUtyS0000rT±kjjjh33333uW0000rT YUUUS YYYZYYx}Wx VVVWVVuzT±WWWXWWv{Uv BBBBBBBBBRf@0000rT33333uWNNNONNrLkjjjhCCCCCCCCSgA8888886z\6x kjjjh@@@@@@@@Pd>33333uW LLLMLLpJRRRSRRvP0000rT33333uWIIIIIJIYmGVVVWVVuzTcccdccaWWWXWWv{Uv $$33333uWIIIIIJIYmGRRRSRRvPst kjjjh@@@@@@@@Pd>±kjjjh stRRRSRRvPnnoIIIIIJIYmG0000rT cccdccanno‰33333uWaaabaa_ 8888886z\6aaabaa_RRRSRRvPLLLMLLpJ qr±NNNONNrLCCCCCCCCSgAcccdcca@@@@@@@@Pd>CCCCCCCCSgANNNONNrL<<<<<<<~`: <<<<<<<~`:aaabaa_±0000rTCCCCCCCCSgA33333uWNNNONNrLkjjjh UUUVUUtySaaabaa_RRRSRRvP$$IIIIIJIYmG33333uWLLLMLLpJYYYZYYx}Wx DDDDDDDThBnnocccdccakjjjhCCCCCCCCSgA8888886z\6x xRRRSRRvPnnoaaabaa_ ±NNNONNrLcccdccast33333uWaaabaa_YYYZYYx}Wx VVVWVVuzTCCCCCCCCSgAWWWXWWv{Uv rsRRRSRRvPaaabaa_0000rTllljqr33333uW,,, bbbcbb`GGGGGGHWkECCCCCCCCSgAUUUVUUtySAAAAAAAAQe?;;;;;;;}_9aaabaa_±LLLMLLpJ VVVWVVuzTCCCCCCCCSgACCCCCCCCSgAWWWXWWv{Uv rsRRRSRRvPaaabaa_0000rTllljqr33333uW,,, ++##QQQRQQuOrs VVVWVVuzTCCCCCCCCSgACCCCCCCCSgACCCCCCCCSgAWWWXWWv{Uv SSSTSSrwQAAAAAAAAQe?HHHHHHIXlF±x33333uWaaabaa_ 777775y[533333uW33333uW0000rTAAAAAAAAQe?8888886z\6RRRSRRvPaaabaa_st±aaabaa_0000rT MMMNMMqKMMMNMMqK VVVWVVuzTCCCCCCCCSgAqrWWWXWWv{Uv pq33333uWaaabaa_x ////q33333uW33333uWUUUVUUtyS 777775y[533333uW33333uW0000rTAAAAAAAAQe?8888886z\6RRRSRRvPaaabaa_st±aaabaa_0000rT MMMNMMqKMMMNMMqK VVVWVVuzTqrWWWXWWv{Uv ```a``^33333uW,,,nnoaaabaa_cccdccaCCCCCCCCSgAqr33333uW MMMNMMqKMMMNMMqK VVVWVVuzTqrCCCCCCCCSgAWWWXWWv{Uv ```a``^33333uW,,,nnoaaabaa_aaabaa_33333uWNNNONNrLkjjjh MMMNMMqKMMMNMMqK VVVWVVuzT$$WWWXWWv{Uv rs@@@@@@@@Pd>CCCCCCCCSgA,,,@@@@@@@@Pd> RRRSRRvP8888886z\6 kjjjh@@@@@@@@Pd>33333uW 8888886z\6RRRSRRvPIIIIIJIYmGIIIIIJIYmGRRRSRRvPstCCCCCCCCSgANNNONNrL<<<<<<<~`:VVVWVVuzTcccdccaWWWXWWv{Uv stCCCCCCCCSgAIIIIIJIYmGIIIIIJIYmG cccdccaRRRSRRvPIIIIIJIYmGqr33333uW kjjjh@@@@@@@@Pd>33333uW qr±NNNONNrLCCCCCCCCSgAcccdcca@@@@@@@@Pd>CCCCCCCCSgANNNONNrL<<<<<<<~`: <<<<<<<~`:aaabaa_±0000rTCCCCCCCCSgA33333uWNNNONNrLkjjjh UUUVUUtySaaabaa_RRRSRRvP$$IIIIIJIYmG33333uWLLLMLLpJ 8888886z\6RRRSRRvPaaabaa_ ±NNNONNrL ```a``^MMMNMMqKMMMNMMqKAAAAAAAAQe?$$±cccdcca33333uW0000rT bbbcbb`33333uW\\\]\\{Z{llljbbbcbb`33333uW\\\]\\{Z{ LLLMLLpJRRRSRRvP0000rT33333uWIIIIIJIYmG]]]^]]|[| DDDDDDDThBnnocccdccakjjjhCCCCCCCCSgA8888886z\6x xRRRSRRvPnnoaaabaa_ ±NNNONNrLcccdccast33333uWaaabaa_YYYZYYx}Wx VVVWVVuzTCCCCCCCCSgAWWWXWWv{Uv ²0000rT0000rTCCCCCCCCSgANNNONNrL<<<<<<<~`: LLLMLLpJRRRSRRvPaaabaa_33333uW @@@@@@@@Pd>CCCCCCCCSgA0000rT0000rT33333uWNNNONNrL IIIIIJIYmG±x33333uWaaabaa_cccdcca VVVWVVuzTCCCCCCCCSgACCCCCCCCSgAWWWXWWv{Uv ²0000rT0000rTCCCCCCCCSgANNNONNrL<<<<<<<~`: HHHHHHIXlFllljAAAAAAAAQe?aaabaa_33333uW<<<<<<<~`:nnoIIIIIJIYmG±aaabaa_CCCCCCCCSgAz±kjjjhCCCCCCCCSgARRRSRRvPNNNONNrL VVVWVVuzTCCCCCCCCSgACCCCCCCCSgACCCCCCCCSgAWWWXWWv{Uv ++IIIIIJIYmGCCCCCCCCSgAUUUVUUtySUUUVUUtySCCCCCCCCSgANNNONNrL<<<<<<<~`: kjjjh@@@@@@@@Pd>33333uW <<<<<<<~`:aaabaa_±0000rTCCCCCCCCSgA33333uWNNNONNrLkjjjh VVVWVVuzTCCCCCCCCSgAqrWWWXWWv{Uv ²0000rT0000rTCCCCCCCCSgANNNONNrL<<<<<<<~`: ±kjjjhkjjjh33333uWNNNONNrLkjjjhCCCCCCCCSgARRRSRRvPNNNONNrL IIIIIJIYmG±x33333uWaaabaa_ ±kjjjh 33333uWNNNONNrL,,,RRRSRRvP0000rT33333uWaaabaa_ q FBB~ljlj[WWWUeeec3iigiig95qTPPP_[[[Y·IEEMIIIEE[WWWU95q62IEE#_[[[YXTTTRgcccaB>ziig·IEEMIIIEEiigp#TPPPIEEeeecFBB~IEETPPPB>zoB>zgccca#62IEE95qTPPPljo[WWWUgcccaXTTTR*OKKK95qRNNN TTTUTTsxR±<<<<<<<~`:33333uW lllj RRRSRRvP8888886z\6 lllj

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