
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
Given an even positive integer  $m$ , find all positive integers  $n$  for which there exists a bijection  $f:[n]\to [n]$  so that, for all  $x,y\in [n]$  for which  $n\mid mx-y$ ,   $$ (n+1)\mid f(x)^m-f(y). $$  Note: For a positive integer  $n$ , we let  $[n] = \{1,2,\dots, n\}$ . 

*Proposed by Milan Haiman and Carl Schildkraut*

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