
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 the triplets $(p, q, n)$, where $p$ and $q$ are odd prime numbers, and $n$ is an integer greater than 1, such that:

$$
\begin{aligned}
q^{n+2} & \equiv 3^{n+2} \pmod{p^n}, \\
p^{n+2} & \equiv 3^{n+2} \pmod{q_n}
\end{aligned}
$$

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