
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 $m, n \in \mathbf{N}^{+}$, with $\gcd(m, n)=1$, and let $p, q$ be two distinct odd prime numbers. Find the greatest common divisor of $\left(p^{m}+q^{m}\right)$ and $\left(p^{n}+q^{n}\right)$, denoted as $\gcd\left(p^{m}+q^{m}, p^{n}+q^{n}\right)$.

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