
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  $P$  with integer coefficients such that  $P (0)\ne  0$  and   $$ P^n(m)\cdot P^m(n) $$   is a square of an integer for all nonnegative integers  $n, m$ .

*Remark:* For a nonnegative integer  $k$  and an integer  $n$ ,  $P^k(n)$  is defined as follows:  $P^k(n) = n$  if  $k = 0$  and  $P^k(n)=P(P(^{k-1}(n))$  if  $k >0$ .

Proposed by Adrian Beker.

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