
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 such that
\[|z_1| = |z_2| = ... = |z_n| = 1 \text{ and } z_1 + z_2 + ... + z_n = 0,\]
and also 
\[z_1z_2 + z_2z_3 + ... + z_nz_1 = -n,\]
then the numbers $z_1, z_2, ..., z_n$ must be equally spaced on the unit circle in the complex plane?
A) 1
B) 2
C) 3
D) 4
E) 5

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