
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
Determine the smallest integer value of $n \geq 3$ such that whenever $z_1, z_2, ..., z_n$ are complex numbers with the conditions:
\[|z_1| = |z_2| = ... = |z_n| = 1,\]
\[z_1 = 1,\]
and
\[z_1 + z_2 + ... + z_n = 0,\]
then the numbers $z_1, z_2, ..., z_n$ are equally spaced on the unit circle in the complex plane.
A) 3
B) 4
C) 5
D) 6
E) 7

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