There exist positive integers \(A,B,C,D,E,F\) such that \[ \sum_{p=1}^n \cot^6 \left(\tfrac{p \pi}{2n+1}\right) = \tfrac{1}{F}n(2n-1)\big(An^4 + Bn^3 + Cn^2 - Dn + E\big) \] for all positive integers \(n\), where \(F\) is as small as possible.

What is \(A + B + C + D + E + F?\)

**Bonus:** Use this result to prove that
\[ \sum_{p=1}^\infty \tfrac{1}{p^6} = \tfrac{1}{945}\pi^6. \]

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