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Definition:

$\displaystyle \text{Li}_{s_1,s_2,\dots,s_k}(z_1,z_2,\dots,z_k)=\sum_{n_1>n_2>\dots>n_k>0}\prod_{j=1}^{k}n_j^{-s_j}z_j^{n_j}$

Multiple polylogarihtms are multiply nested sums of the form above.

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$\displaystyle \text{Li}_{3,1,1}(-1)=\frac{\zeta(A)}{B}-\zeta(C)(\frac{\pi^D}{E}+\frac{F}{G}\log^H I)-J\text{Li}_K(\frac{L}{M})-J\text{Li}_N(\frac{L}{M})\log I+\frac{\pi^O\log^P Q}{R}-\frac{\log^S T}{U}$

In the above equation, $A,B,C,\ldots,U$ are positive integers. Hence, find the minimum value of:

$A+B+C+D+E+F+G+H+I+J \\ +K+L+M+N+O+P+Q+R+S+T+U.$

Details and Assumptions:

• $\text{Li}_{3,1,1}(-1)$ is a multi polylogarithm.