/* Palamedes code for absorbing Markov model of simplified Chainmail combat */
/* 3 LF vs 3LF */
/* Transient states */
T ↜ [1,2,3,5,6,7,9,10,11]
/* Absorption states */
A ↜ [4,8,12,13,14,15,16]
/* All states in canonical form */
S ↜ T ⊎ A
/* Number of states */
n ↜ count S
/* In each turn, the system transitions from s₀ to s₁ */
/* In state s₀, troop strengths are A₀ and B₀ */
A₀ ← 4 - ceil(s₀/4)
B₀ ← (4 - s₀ mod 4) mod 4
/* In state s₁, troop strengths are A₁ and B₁ */
A₁ ← 4 - ceil(s₁/4)
B₁ ← (4 - s₁ mod 4) mod 4
/* Mapping of states 1 to 16 to troop strengths for opposing sides: [state,A,B] */
map [s₀,A₀,B₀],s₀,[1..16]
/* Later, these states will be the row r and column c in the transition matrix P. */
/* But rows and cols start at 0, so add 1. */
s₀ ← r+1; s₁ ← c+1
/* Probability A scores Ka kills on B */
Pa ← { /* if A scores more kills on B than there are B */ ...
if [ A₀>B₀ && B₀>0 && B₁=0 ⇒ Ka ← [B₀ .. A₀],⊤ ⇒ Ka ← B₀-B₁]; ...
sum(prob(binom(A₀, 1/6)) ∘ Ka) }
/* Probability B scores Kb kills on A */
Pb ← { /* if B scores more kills on A than there are A */ ...
if[B₀>A₀ && A₀>0 && A₁=0 ⇒ Kb ← [A₀ .. B₀],⊤ ⇒ Kb ← A₀-A₁]; ...
sum(prob(binom(B₀, 1/6)) ∘ Kb) }
/* State transition probabilities */
Pt ← if [ ...
/* If system starts in an absorbing state, it ends in same state with prob 1 */ ...
s₀ ∈ A && s₀ = s₁ ⇒ 1, ...
s₀ ∈ A && s₀ ≠ s₁ ⇒ 0, ...
/* If system starts in a transient state, troop strengths are non-increasing */ ...
/* and starting strengths on each side are ≥ kills on the opposing side */ ...
A₀≥A₁ && B₀≥B₁ && A₀≥B₀-B₁ && B₀≥A₀-A₁ ⇒ Pa * Pb, ...
⊤ ⇒ 0 ]
/* The state transition matrix P is an n-by-n matrix calculated from Pt */
P ↜ matrix(Pt, n, n)
/* Check that rows of P sum to 1 */
sum P
/* In Palamedes, matrix rows and cols are numbered from 0.*/
/* So adjust state arrays before calculating submatrices. */
T ↜ T - 1; A ↜ A - 1
/* Transition submatrix Q from a transient state to another */
Q ← submatrix(P, T, T)
/* Transition submatrix R from a transient state to absorption state */
R ← submatrix(P, T, A)
/* Zero submatrix */
ζ ← submatrix(P, A, T)
/* Identity submatrix */
Ι ← submatrix(P, A, A)
/* Column vector whose entries are all 1 */
ones ↜ trans[(count T)#1]
/* The t-by-t identity matrix where t is the number of transient states */
Iₜ ↜ I(count T)
/* Fundamental Matrix N */
/* The expected number of visits to a transient state before being absorbed. */
N ↜ (Iₜ - Q)^-1
/* Main properties of the Markov chain are now derived from Fundamental Matrix */
/* Variance on number of visits */
N₂ ← N × (2*diag(N) - Iₜ) - N*N
/* Expected number of steps */
t ← N × ones
/* Variance on number of steps */
t₂ ← (2*N - Iₜ) × t - t*t
/* Transient probabilities */
H ← (N - Iₜ) × diag(N)^-1
/* Absorbing probabilities */
B ← N × R
/* Show intermediate results */
P; Q; R; ζ; Ι; ones; Iₜ
/* Show main results */
N; N₂; t; t₂; H; B
T ↜ [1, 2, 3, 5, 6, 7, 9, 10, 11] → [1, 2, 3, 5, 6, 7, 9, 10, 11]
A ↜ [4, 8, 12, 13, 14, 15, 16] → [4, 8, 12, 13, 14, 15, 16]
S ↜ T ⊎ A → [1, 2, 3, 5, 6, 7, 9, 10, 11, 4, 8, 12, 13, 14, 15, 16]
n ↜ count(S) → 16
A₀ ← 4 - ceil(s₀ / 4)
B₀ ← (4 - s₀ mod 4) mod 4
A₁ ← 4 - ceil(s₁ / 4)
B₁ ← (4 - s₁ mod 4) mod 4
map([_, 4 - ceil(_ / 4), (4 - _ mod 4) mod 4], [1 .. 16]) →
[[ 1, 3, 3 ], ...
[ 2, 3, 2 ], ...
[ 3, 3, 1 ], ...
[ 4, 3, 0 ], ...
[ 5, 2, 3 ], ...
[ 6, 2, 2 ], ...
[ 7, 2, 1 ], ...
[ 8, 2, 0 ], ...
[ 9, 1, 3 ], ...
[ 10, 1, 2 ], ...
[ 11, 1, 1 ], ...
[ 12, 1, 0 ], ...
[ 13, 0, 3 ], ...
[ 14, 0, 2 ], ...
[ 15, 0, 1 ], ...
[ 16, 0, 0 ]]
s₀ ← r + 1
s₁ ← c + 1
Pa ← if [A₀ > B₀ and B₀ > 0 and B₁ = 0, {Ka ← [B₀ .. A₀]; sum(comb(A₀,Ka) * 0.16666666666666666 ^ Ka * 0.8333333333333334 ^ (A₀ - Ka))}, true, {Ka ← B₀ - B₁; comb(A₀,Ka) * 0.16666666666666666 ^ Ka * 0.8333333333333334 ^ (A₀ - Ka)}]
Pb ← if [B₀ > A₀ and A₀ > 0 and A₁ = 0, {Kb ← [A₀ .. B₀]; sum(comb(B₀,Kb) * 0.16666666666666666 ^ Kb * 0.8333333333333334 ^ (B₀ - Kb))}, true, {Kb ← A₀ - A₁; comb(B₀,Kb) * 0.16666666666666666 ^ Kb * 0.8333333333333334 ^ (B₀ - Kb)}]
Pr ← if [s₀ ∈ A and s₀ = s₁, 1, s₀ ∈ A and s₀ ≠ s₁, 0, A₀ ≥ A₁ and B₀ ≥ B₁ and A₀ ≥ B₀ - B₁ and B₀ ≥ A₀ - A₁, Pa * Pb, true, 0]
P ↜ matrix(Pr,n,n) →
[[ 0.33489797668038424, 0.20093878600823054, 0.0401877572016461, 0.0026791838134430732, 0.20093878600823054, 0.12056327160493831, 0.024112654320987657, 0.0016075102880658437, 0.0401877572016461, 0.024112654320987657, 0.0048225308641975315, 0.0003215020576131687, 0.0026791838134430732, 0.0016075102880658437, 0.0003215020576131687, 0.00002143347050754458 ], ...
[ 0, 0.40187757201646107, 0.24112654320987661, 0.051440329218107, 0, 0.1607510288065844, 0.09645061728395063, 0.0205761316872428, 0, 0.01607510288065844, 0.009645061728395063, 0.0020576131687242796, 0, 0, 0, 0 ], ...
[ 0, 0, 0.48225308641975323, 0.35108024691358036, 0, 0, 0.09645061728395063, 0.07021604938271606, 0, 0, 0, 0, 0, 0, 0, 0 ], ...
[ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], ...
[ 0, 0, 0, 0, 0.40187757201646107, 0.1607510288065844, 0.01607510288065844, 0, 0.24112654320987661, 0.09645061728395063, 0.009645061728395063, 0, 0.051440329218107, 0.0205761316872428, 0.0020576131687242796, 0 ], ...
[ 0, 0, 0, 0, 0, 0.48225308641975323, 0.19290123456790126, 0.019290123456790126, 0, 0.19290123456790126, 0.0771604938271605, 0.007716049382716049, 0, 0.019290123456790126, 0.007716049382716049, 0.0007716049382716049 ], ...
[ 0, 0, 0, 0, 0, 0, 0.5787037037037038, 0.25462962962962965, 0, 0, 0.11574074074074076, 0.05092592592592593, 0, 0, 0, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0.48225308641975323, 0.09645061728395063, 0, 0, 0.35108024691358036, 0.07021604938271606, 0, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.5787037037037038, 0.11574074074074076, 0, 0, 0.25462962962962965, 0.05092592592592593, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.6944444444444445, 0.1388888888888889, 0, 0, 0.1388888888888889, 0.027777777777777776 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ]]
sum(P) → [1.0000000000000004, 1.0000000000000004, 1.0000000000000002, 1, 1.0000000000000002, 1.0000000000000004, 1, 1, 1.0000000000000002, 1.0000000000000002, 1, 1, 1, 1, 1, 1]
T ↜ T - 1 → [0, 1, 2, 4, 5, 6, 8, 9, 10]
A ↜ A - 1 → [3, 7, 11, 12, 13, 14, 15]
Q ← submatrix(P,T,T)
R ← submatrix(P,T,A)
ζ ← submatrix(P,A,T)
Ι ← submatrix(P,A,A)
ones ↜ trans([(count(T)) # 1]) →
[[ 1 ], ...
[ 1 ], ...
[ 1 ], ...
[ 1 ], ...
[ 1 ], ...
[ 1 ], ...
[ 1 ], ...
[ 1 ], ...
[ 1 ]]
Iₜ ↜ I(count(T)) →
[[ 1, 0, 0, 0, 0, 0, 0, 0, 0 ], ...
[ 0, 1, 0, 0, 0, 0, 0, 0, 0 ], ...
[ 0, 0, 1, 0, 0, 0, 0, 0, 0 ], ...
[ 0, 0, 0, 1, 0, 0, 0, 0, 0 ], ...
[ 0, 0, 0, 0, 1, 0, 0, 0, 0 ], ...
[ 0, 0, 0, 0, 0, 1, 0, 0, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 1, 0, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 1, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 1 ]]
N ↜ (Iₜ - Q) ^ -1 →
[[ 1.5035287293351813, 0.5051093613386848, 0.3519455490380722, 0.5051093613386848, 0.6637684021605676, 0.6054621773324489, 0.3519455490380722, 0.6054621773324489, 0.6819199040596049 ], ...
[ 0, 1.6718985164480764, 0.778641261386027, 0, 0.5190941742573513, 0.7987014941274742, 0, 0.30147392428023106, 0.6005920748547777 ], ...
[ 0, 0, 1.9314456035767515, 0, 0, 0.4421807700496228, 0, 0, 0.1674927159278875 ], ...
[ 0, 0, 0, 1.6718985164480764, 0.5190941742573513, 0.30147392428023106, 0.778641261386027, 0.7987014941274744, 0.6005920748547777 ], ...
[ 0, 0, 0, 0, 1.9314456035767515, 0.8843615400992456, 0, 0.8843615400992456, 1.1577096524935582 ], ...
[ 0, 0, 0, 0, 0, 2.3736263736263745, 0, 0, 0.8991008991008999 ], ...
[ 0, 0, 0, 0, 0, 0, 1.9314456035767515, 0.4421807700496228, 0.1674927159278875 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 2.3736263736263745, 0.8991008991008999 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 3.272727272727274 ]]
N₂ ← N × (2 * diag(N) - Iₜ) - N * N
t ← N × ones
t₂ ← (2 * N - Iₜ) × t - t * t
H ← (N - Iₜ) × diag(N) ^ -1
B ← N × R
P →
[[ 0.33489797668038424, 0.20093878600823054, 0.0401877572016461, 0.0026791838134430732, 0.20093878600823054, 0.12056327160493831, 0.024112654320987657, 0.0016075102880658437, 0.0401877572016461, 0.024112654320987657, 0.0048225308641975315, 0.0003215020576131687, 0.0026791838134430732, 0.0016075102880658437, 0.0003215020576131687, 0.00002143347050754458 ], ...
[ 0, 0.40187757201646107, 0.24112654320987661, 0.051440329218107, 0, 0.1607510288065844, 0.09645061728395063, 0.0205761316872428, 0, 0.01607510288065844, 0.009645061728395063, 0.0020576131687242796, 0, 0, 0, 0 ], ...
[ 0, 0, 0.48225308641975323, 0.35108024691358036, 0, 0, 0.09645061728395063, 0.07021604938271606, 0, 0, 0, 0, 0, 0, 0, 0 ], ...
[ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], ...
[ 0, 0, 0, 0, 0.40187757201646107, 0.1607510288065844, 0.01607510288065844, 0, 0.24112654320987661, 0.09645061728395063, 0.009645061728395063, 0, 0.051440329218107, 0.0205761316872428, 0.0020576131687242796, 0 ], ...
[ 0, 0, 0, 0, 0, 0.48225308641975323, 0.19290123456790126, 0.019290123456790126, 0, 0.19290123456790126, 0.0771604938271605, 0.007716049382716049, 0, 0.019290123456790126, 0.007716049382716049, 0.0007716049382716049 ], ...
[ 0, 0, 0, 0, 0, 0, 0.5787037037037038, 0.25462962962962965, 0, 0, 0.11574074074074076, 0.05092592592592593, 0, 0, 0, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0.48225308641975323, 0.09645061728395063, 0, 0, 0.35108024691358036, 0.07021604938271606, 0, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.5787037037037038, 0.11574074074074076, 0, 0, 0.25462962962962965, 0.05092592592592593, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.6944444444444445, 0.1388888888888889, 0, 0, 0.1388888888888889, 0.027777777777777776 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ]]
Q →
[[ 0.33489797668038424, 0.20093878600823054, 0.0401877572016461, 0.20093878600823054, 0.12056327160493831, 0.024112654320987657, 0.0401877572016461, 0.024112654320987657, 0.0048225308641975315 ], ...
[ 0, 0.40187757201646107, 0.24112654320987661, 0, 0.1607510288065844, 0.09645061728395063, 0, 0.01607510288065844, 0.009645061728395063 ], ...
[ 0, 0, 0.48225308641975323, 0, 0, 0.09645061728395063, 0, 0, 0 ], ...
[ 0, 0, 0, 0.40187757201646107, 0.1607510288065844, 0.01607510288065844, 0.24112654320987661, 0.09645061728395063, 0.009645061728395063 ], ...
[ 0, 0, 0, 0, 0.48225308641975323, 0.19290123456790126, 0, 0.19290123456790126, 0.0771604938271605 ], ...
[ 0, 0, 0, 0, 0, 0.5787037037037038, 0, 0, 0.11574074074074076 ], ...
[ 0, 0, 0, 0, 0, 0, 0.48225308641975323, 0.09645061728395063, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 0.5787037037037038, 0.11574074074074076 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0.6944444444444445 ]]
R →
[[ 0.0026791838134430732, 0.0016075102880658437, 0.0003215020576131687, 0.0026791838134430732, 0.0016075102880658437, 0.0003215020576131687, 0.00002143347050754458 ], ...
[ 0.051440329218107, 0.0205761316872428, 0.0020576131687242796, 0, 0, 0, 0 ], ...
[ 0.35108024691358036, 0.07021604938271606, 0, 0, 0, 0, 0 ], ...
[ 0, 0, 0, 0.051440329218107, 0.0205761316872428, 0.0020576131687242796, 0 ], ...
[ 0, 0.019290123456790126, 0.007716049382716049, 0, 0.019290123456790126, 0.007716049382716049, 0.0007716049382716049 ], ...
[ 0, 0.25462962962962965, 0.05092592592592593, 0, 0, 0, 0 ], ...
[ 0, 0, 0, 0.35108024691358036, 0.07021604938271606, 0, 0 ], ...
[ 0, 0, 0, 0, 0.25462962962962965, 0.05092592592592593, 0 ], ...
[ 0, 0, 0.1388888888888889, 0, 0, 0.1388888888888889, 0.027777777777777776 ]]
ζ →
[[ 0, 0, 0, 0, 0, 0, 0, 0, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 0 ]]
Ι →
[[ 1, 0, 0, 0, 0, 0, 0 ], ...
[ 0, 1, 0, 0, 0, 0, 0 ], ...
[ 0, 0, 1, 0, 0, 0, 0 ], ...
[ 0, 0, 0, 1, 0, 0, 0 ], ...
[ 0, 0, 0, 0, 1, 0, 0 ], ...
[ 0, 0, 0, 0, 0, 1, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 1 ]]
ones →
[[ 1 ], ...
[ 1 ], ...
[ 1 ], ...
[ 1 ], ...
[ 1 ], ...
[ 1 ], ...
[ 1 ], ...
[ 1 ], ...
[ 1 ]]
Iₜ →
[[ 1, 0, 0, 0, 0, 0, 0, 0, 0 ], ...
[ 0, 1, 0, 0, 0, 0, 0, 0, 0 ], ...
[ 0, 0, 1, 0, 0, 0, 0, 0, 0 ], ...
[ 0, 0, 0, 1, 0, 0, 0, 0, 0 ], ...
[ 0, 0, 0, 0, 1, 0, 0, 0, 0 ], ...
[ 0, 0, 0, 0, 0, 1, 0, 0, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 1, 0, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 1, 0 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 1 ]]
N →
[[ 1.5035287293351813, 0.5051093613386848, 0.3519455490380722, 0.5051093613386848, 0.6637684021605676, 0.6054621773324489, 0.3519455490380722, 0.6054621773324489, 0.6819199040596049 ], ...
[ 0, 1.6718985164480764, 0.778641261386027, 0, 0.5190941742573513, 0.7987014941274742, 0, 0.30147392428023106, 0.6005920748547777 ], ...
[ 0, 0, 1.9314456035767515, 0, 0, 0.4421807700496228, 0, 0, 0.1674927159278875 ], ...
[ 0, 0, 0, 1.6718985164480764, 0.5190941742573513, 0.30147392428023106, 0.778641261386027, 0.7987014941274744, 0.6005920748547777 ], ...
[ 0, 0, 0, 0, 1.9314456035767515, 0.8843615400992456, 0, 0.8843615400992456, 1.1577096524935582 ], ...
[ 0, 0, 0, 0, 0, 2.3736263736263745, 0, 0, 0.8991008991008999 ], ...
[ 0, 0, 0, 0, 0, 0, 1.9314456035767515, 0.4421807700496228, 0.1674927159278875 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 2.3736263736263745, 0.8991008991008999 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 3.272727272727274 ]]
N₂ →
[[ 0.7570699106010834, 0.9287383554817061, 0.8837161482501987, 0.9287383554817061, 1.4597082304250264, 1.9022353591865002, 0.8837161482501987, 1.9022353591865002, 3.316541076050604 ], ...
[ 0, 1.1233461328532024, 1.6228830068161457, 0, 1.2166513854180578, 2.355012291382593, 0, 1.0388124639632406, 2.969845211089038 ], ...
[ 0, 0, 1.7990365159992106, 0, 0, 1.4614392719491078, 0, 0, 0.900769432983931 ], ...
[ 0, 0, 0, 1.1233461328532024, 1.2166513854180578, 1.0388124639632406, 1.6228830068161457, 2.355012291382594, 2.969845211089038 ], ...
[ 0, 0, 0, 0, 1.7990365159992106, 2.531830877094861, 0, 2.531830877094861, 5.079734615260252 ], ...
[ 0, 0, 0, 0, 0, 3.2604757879483177, 0, 0, 4.177540740977309 ], ...
[ 0, 0, 0, 0, 0, 0, 1.7990365159992106, 1.4614392719491078, 0.900769432983931 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 3.2604757879483177, 4.177540740977309 ], ...
[ 0, 0, 0, 0, 0, 0, 0, 0, 7.438016528925626 ]]
t →
[[ 5.774251210973766 ], ...
[ 4.670401445353937 ], ...
[ 2.541119089554262 ], ...
[ 4.670401445353937 ], ...
[ 4.857878336268801 ], ...
[ 3.2727272727272743 ], ...
[ 2.541119089554262 ], ...
[ 3.2727272727272743 ], ...
[ 3.272727272727274 ]]
t₂ →
[[ 10.099411430815458 ], ...
[ 9.266752742488116 ], ...
[ 4.808251360552346 ], ...
[ 9.266752742488116 ], ...
[ 9.463427676501855 ], ...
[ 7.43801652892563 ], ...
[ 4.808251360552346 ], ...
[ 7.43801652892563 ], ...
[ 7.438016528925626 ]]
H →
[[ 0.3348979766803843, 0.3021172376011088, 0.18221872176276727, 0.3021172376011088, 0.34366404155072594, 0.25507897285765196, 0.18221872176276727, 0.25507897285765196, 0.20836441512932366 ], ...
[ 0, 0.4018775720164612, 0.40313910986884566, 0, 0.2687594065792304, 0.33648998132222274, 0, 0.12700984772917137, 0.18351424509451536 ], ...
[ 0, 0, 0.48225308641975323, 0, 0, 0.18628912071535028, 0, 0, 0.0511783298668545 ], ...
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B →
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Last update: Fri Sep 23 2016