
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
Determine which positive integers  $n$  have the following property: For all integers  $m$  that are relatively prime to  $n$ , there exists a permutation  $\pi:\{1,2, \ldots, n\} \rightarrow\{1,2, \ldots, n\}$  such that  $\pi(\pi(k)) \equiv m k(\bmod n)$  for all  $k \in\{1,2, \ldots, n\}$ .

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