
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
There are 20 chairs arranged in a circle. There are \(n\) people sitting in \(n\) different chairs. These \(n\) people stand, move \(k\) chairs clockwise, and then sit again. After this happens, exactly the same set of chairs is occupied. For how many pairs \((n, k)\) with \(1 \leq n \leq 20\) and \(1 \leq k \leq 20\) is this possible?

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