Trigo(sum)mation

Geometry Level 4

\[\displaystyle \sum^{\infty}_{n=0} \tan^{-1} \left( \frac{ \cot^{-1}(n^2+3n+3) }{ 1+\cot^{-1}(n+1)\cot^{-1}(n+2) } \right) \]

If the value of the above expression is in the form \(\tan^{-1}\left(\dfrac{p}{4}\right)\), then find the value of \(\dfrac{2p}{\pi}\).

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