\[\large \sum _{k=1}^{\infty }\left[\text{Li}_{2k}\left(\frac 14 \right) - \frac 14\right]=\frac{A}{B}-\frac{C}{D}\ln 2+\frac{C}{E}\ln 3\]

The equation above holds true for positive integers \(A,B,C,D\) and \(E\), where \(\gcd(A,B)=\gcd(C,D)=\gcd(C,E)=1\).

Find \(A+B+C+D+E\).

**Notation**: \({ \text{Li} }_{ n }(a) \) denotes the polylogarithm function, \({ \text{Li} }_{ n }(a)=\displaystyle\sum _{ k=1 }^{ \infty }{ \dfrac { { a }^{ k } }{ { k }^{ n } } } \).

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