
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
Let  $p$  be a prime number. Find all polynomials  $P$  with integer coefficients with the following properties: $(a)$   $P(x)>x$  for all positive integers  $x$ . $(b)$  The sequence defined by  $p_0:=p$ ,  $p_{n+1}:=P(p_n)$  for all positive integers  $n$ , satisfies the property that for all positive integers  $m$  there exists some  $l\geq 0$  such that  $m\mid p_l$ . 

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