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Calculus Level 5

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

Multiple polylogarihtms are multiply nested sums of the form above. Given that

\[\displaystyle \text{Li}_{1,2} \left(\dfrac12,1\right)=\dfrac{\ln A}{B} - \dfrac{\zeta(C)}{D} - \dfrac{(\ln F)^E}{G} ,\]

where \(A,B,C,D,E,F\) and \(G\) are integers, find the minimum value of \(A+B+C+D+E+F+G\).

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