
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
Consider the polynomial equation 
\[z^{12} - z^9 + z^6 - z^3 + 1 = 0.\]
Among its roots, determine the maximum value of the imaginary part, expressed as \(\sin \theta\) where \(-90^\circ \leq \theta \leq 90^\circ\). Find \(\theta\).

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