
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$  will be called **$ n$ -independent** if  $ P(n) \equal{} 0$  and we can get from any  $ Q \in M(P)$  a polynomial  $ Q_1$  such that  $ Q_1(n) \equal{} 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}
user