
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 symmetrical ring of \( m \) identical regular \( n \)-sided polygons is formed according to the rules:
(i) each polygon in the ring meets exactly two others;
(ii) two adjacent polygons have only an edge in common; and
(iii) the perimeter of the inner region enclosed by the ring consists of exactly two edges of each polygon.
For how many different values of \( n \) is such a ring possible?

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