
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 positive integers  $n$  with the following property: 
there exists a polynomial  $P_n(x)$  of degree  $n$ , with integer coefficients, such that  $P_n(0)=0$  and  $P_n(x)=n$  for  $n$  distinct integer solutions.

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