
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
A regular polygon is drawn on the board. Volodya wants to mark $k$ points on its perimeter such that there is no other regular polygon (not necessarily with the same number of sides) that also contains the marked points on its perimeter.

Find the smallest $k$ that is sufficient for any given initial polygon.

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