
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]


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Given integers $n \geq 2$ such that for non-conjugate complex numbers $z_1, z_2, ..., z_n$ with $|z_1| = |z_2| = ... = |z_n| = 1$ and $z_1 + z_2 + ... + z_n = 0$, determine the number of possible values of $n$.

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