
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
 $p$  is a prime. Let  $K_p$  be the set of all polynomials with coefficients from the set  $\{0,1,\dots ,p-1\}$  and degree less than  $p$ . Assume that for all pairs of polynomials  $P,Q\in K_p$  such that  $P(Q(n))\equiv n\pmod p$  for all integers  $n$ , the degrees of  $P$  and  $Q$  are equal. Determine all primes  $p$  with this condition.

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