
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
Among all the roots of
\[z^{12} - z^9 + z^6 - z^3 + 1 = 0,\]
find the maximum imaginary part of a root which can be expressed as $\sin \theta$, where $-90^\circ \le \theta \le 90^\circ$. Determine $\theta.$

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