
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
\[ z^6 - z^5 + z^4 - z^3 + z^2 - z + 1 = 0, \]
the maximum imaginary part of a root can be expressed as \( \sin \phi \), where \( -90^\circ \le \phi \le 90^\circ \). Find \( \phi \).

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