Beyond Basel

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 $\sum_{p=1}^\infty \tfrac{1}{p^6} = \tfrac{1}{945}\pi^6.$