
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  $n$  satisfying the following condition: for every monic polynomial  $P$  of degree at most  $n$  with integer coefficients, there exists a positive integer  $k\le n$  and  $k+1$  distinct integers  $x_1,x_2,\cdots ,x_{k+1}$  such that \[P(x_1)+P(x_2)+\cdots +P(x_k)=P(x_{k+1})\].

*Note.* A polynomial is *monic* if the coefficient of the highest power is one.

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