
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
Determine all positive integer pairs $(m, n)$ such that
$$
1 + x^n + x^{2n} + \cdots + x^{mn}
$$
is divisible by $1 + x + x^2 + \cdots + x^m$. 
(6th United States of America Mathematical Olympiad, 1977)

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