
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
How many odd integers $n\geq2$ are there such that for complex numbers $z_1,z_2,...,z_n$ with $|z_1|=|z_2|=\dots=|z_n|=1$, $z_1+z_2+\dots+z_n=0$, and the angle between consecutive $z_i$ is a multiple of $\frac{\pi}{4}$, the numbers $z_1,z_2,...,z_n$ are equally spaced on the unit circle in the complex plane?

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