
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 integers \( n \) for which there exist integers \( k_1, k_2, \ldots, k_n \) that are pairwise distinct, and a polynomial \( P \) of degree at most \( n \) such that \( P(k_i) = n \) for all integers \( i \) with \( 1 \leq i \leq n \), and \( P \) has an integer root.

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