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

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