
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
For a polynomial \(P\) of degree 2000 with distinct real coefficients, let \(M(P)\) be the set of all polynomials that can be produced from \(P\) by permutation of its coefficients. A polynomial \(P\) is called \(n\)-independent if \(P(n) = 0\) and we can get from any \(Q\) in \(M(P)\) a polynomial \(Q_{1}\) such that \(Q_{1}(n) = 0\) by interchanging at most one pair of coefficients of \(Q\). Find all integers \(n\) for which \(n\)-independent polynomials exist.

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