
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 $n$ such that there exists a polynomial $P$ of degree $n$ with integer coefficients and $n$ pairwise distinct integers $k_{1}, k_{2}, \ldots, k_{n}$ satisfying $P(k_{i}) = n$ for all integers $1 \leq i \leq n$ and $P(0) = 0$.

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