
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  $p>13$  be a prime of the form  $2q+1$ , where  $q$  is prime. Find the number of ordered pairs of integers  $(m,n)$  such that  $0\le m<n<p-1$  and
\[3^m+(-12)^m\equiv 3^n+(-12)^n\pmod{p}.\]

*Alex Zhu.*

<details><summary>Note</summary>The original version asked for the number of solutions to  $2^m+3^n\equiv 2^n+3^n\pmod{p}$  (still  $0\le m<n<p-1$ ), where  $p$  is a Fermat prime.</details>

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