
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 $b_1, b_2, \dots, b_n$ be real numbers such that
\[
\frac{1}{b_1 + \omega} + \frac{1}{b_2 + \omega} + \dots + \frac{1}{b_n + \omega} = 3 - 4i.
\]
Compute
\[
\frac{3b_1 + 2}{b_1^2 - b_1 + 1} + \frac{3b_2 + 2}{b_2^2 - b_2 + 1} + \dots + \frac{3b_n + 2}{b_n^2 - b_n + 1}.
\]

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