Let $\omega$ be a root of $z^3=1$


Let $\exists~\{a_k\} \ni \sum \limits_{k=1}^n \dfrac{1}{a_k + \omega} = 2+5i$

Find $\sum \limits_{k=1}^n \dfrac{2a_k-1}{a_k^2-a_k+1}$

This can be rewritten as

$\sum \limits_{k=1}^n \dfrac{(2a_k-1)(a_k+1)}{(a_k+1)^3}$

But other than that I don't see any way forward.

One of you number theory/abstract algebra gurus can make short work of this I'm sure.

Thanks