
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
n identical coins are lying on the table, forming a closed chain. The centers of the coins form a convex polygon. How many revolutions will a coin of the same size make as it rolls once along the outer side of the entire chain?

How will the answer change if the radius of this coin is k times larger than the radius of each coin in the chain?

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