
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let \( \omega \) be a nonreal root of \( z^3 = 1 \). Let \( a_1, a_2, \dots, a_n \) be real numbers such that
\[
\frac{1}{a_1 + \omega} + \frac{1}{a_2 + \omega} + \dots + \frac{1}{a_n + \omega} = 3 + 4i.
\]
Compute
\[
\frac{2a_1 - 2}{a_1^2 - a_1 + 1} + \frac{2a_2 - 2}{a_2^2 - a_2 + 1} + \dots + \frac{2a_n - 2}{a_n^2 - a_n + 1}.
\]

Present the answer in LaTex format: \boxed{Your answer}
user