
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 all pairs of numbers \( m, n \in \mathbf{N} \) for which the polynomial \( 1 + x^n + x^{2n} + \ldots + x^{mn} \) is divisible by the polynomial \( 1 + x + x^2 + \ldots + x^m \).

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