
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
Does there exist a natural number $k$ such that from any 180 vertices of a regular 360-gon centered at point $O$, one can choose two vertices $A$ and $B$ such that the angle $AOB$ is equal to $k$ degrees?

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