
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 \( 0 \leq m < n < p - 1 \) and
\[ 3^m + (-12)^m \equiv 3^n + (-12)^n \pmod{p} \]

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