Let \(\omega\) be a nonreal root of \(z^3 = 1.\) Let \(a_1,\) \(a_2,\) \(\dots,\) \(a_n\) be real numbers such that \(\frac{1}{a_1 + \omega} + \frac{1}{a_2 + \omega} + \dots + \frac{1}{a_n + \omega} = 2 + 5i.\) Compute \(\frac{2a_1 - 1}{a_1^2 - a_1 + 1} + \frac{2a_2 - 1}{a_2^2 - a_2 + 1} + \dots + \frac{2a_n - 1}{a_n^2 - a_n + 1}.\)