
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
Let  $\phi(n)$  be the number of positive integers less than  $n$  that are relatively prime to  $n$ , where  $n$  is a positive integer. Find all pairs of positive integers  $(m,n)$  such that \[2^n + (n-\phi(n)-1)! = n^m+1.\]

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