
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 pairs \((n, m)\) such that \(n > 2, m > 2\), and there are infinitely many natural numbers \(k\) such that \(k^n + k^2 - 1\) divides \(k^m + k - 1\).

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