
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 positive integers \( k \) such that the polynomial \( x^{2k+1} + x + 1 \) is divisible by \( x^k + x + 1 \). For each \( k \) that satisfies this condition, find the positive integers \( n \) such that \( x^n + x + 1 \) is divisible by \( x^k + x + 1 \).

(British Mathematical Olympiad, 1991)

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