Trig + summation = Easy

Geometry Level 4

\[\large3\sum_{n=1}^{\infty}\left({\dfrac{1}{\pi}\displaystyle\sum_{k=1}^{\infty}\cot^{-1}\left(1+2\sqrt{\sum_{r=1}^k r^3}\right)}\right)^n = \, ? \]

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