
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 integers $n \geq 2$ are there such that whenever $z_1, z_2, ..., z_n$ are complex numbers with $|z_1| = |z_2| = ... = |z_n| = 1$, and any selected $k$ numbers among them sum to $0$ where $k \mid n$, then the $k$ numbers $z_1, z_2, ..., z_k$ are equally spaced on the unit circle in the complex plane?
**A) n-2**
**B) n-1**
**C) n**
**D) n+1**

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