
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
Find all pairs  $\left(m,n\right)$  of positive integers, with  $m,n\geq2$ , such that  $a^n-1$  is divisible by  $m$  for each  $a\in \left\{1,2,3,\ldots,n\right\}$ .

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