
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 of the form   $$ P_n(x)=n!x^n+a_{n-1}x^{n-1}+\dots+a_1x+(-1)^n(n+1) $$   with integer coefficients, having  $n$  real roots  $x_1,\dots,x_n$  satisfying  $k \leq x_k \leq k+1$  for  $k=1, \dots,n$ .

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