
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
A polynomial \( p(x) \) of degree 2000 with distinct real coefficients satisfies condition \( n \) if:
1. \( p(n) = 0 \), and
2. If \( q(x) \) is obtained from \( p(x) \) by permuting its coefficients, then either \( q(n) = 0 \), or we can obtain a polynomial \( r(x) \) by transposing two coefficients of \( q(x) \) such that \( r(n) = 0 \).

Find all integers \( n \) for which there is a polynomial satisfying condition \( n \).

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