Trigo(sum)mation

Geometry Level 4

n=0tan1(cot1(n2+3n+3)1+cot1(n+1)cot1(n+2))\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 tan1(p4)\tan^{-1}\left(\dfrac{p}{4}\right), then find the value of 2pπ\dfrac{2p}{\pi}.

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