A geometry problem by Akshat Sharda

Geometry Level 4

\[\displaystyle \sum^{\infty}_{n=1}\left \{ \dfrac{1}{\pi} \sum^{\infty}_{k=1} \cot^{-1} \left ( 1+2 \sqrt{ \sum^{k}_{r=1} r^3 } \right ) \right \}^{n} = \dfrac{1}{A}\]

Find the value of \(A\).

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