
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
Fine all positive integers  $m,n\geq 2$ , such that 
(1)  $m+1$  is a prime number of type  $4k-1$ ;
(2) there is a (positive) prime number  $p$  and nonnegative integer  $a$ , such that
\[\frac{m^{2^n-1}-1}{m-1}=m^n+p^a.\]

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