\[\frac{1^3}{1^5}+\frac{1^3+2^3}{1^5+2^5}+\frac{1^3+2^3+3^3}{1^5+2^5+3^5}+\cdots\] If the given sum equals

\[\frac{L+\sqrt{M}\pi\tan \left(\pi\sin \left( \frac{\pi}{M}\right) \right)}{N}\]

for square free positive integers \(L,M,N\). Find \(L+M+N\).

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