
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
Let  $S$  be a finite set of points in the plane, such that for each  $2$  points  $A$  and  $B$  in  $S$ , the segment  $AB$  is a side of a regular polygon all of whose vertices are contained in  $S$ . Find all possible values for the number of elements of  $S$ .

Proposed by *Viktor Simjanoski, Macedonia*

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