
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
Find all polynomials  $P_n(x)$  of the form  $$ P_n(x)=n!x^n+a_{n-1}x^{n-1}+\ldots+a_1x+(-1)^nn(n+1), $$  with integer coefficients, such that its roots  $x_1,x_2,\ldots,x_n$  satisfy  $k\le x_k\le k+1$  for  $k=1,2,\ldots,n$ .

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