
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
Prime  $p>2$  and a polynomial  $Q$  with integer coefficients are such that there are no integers  $1 \le i < j \le p-1$  for which  $(Q(j)-Q(i))(jQ(j)-iQ(i))$  is divisible by  $p$ . What is the smallest possible degree of  $Q$ ?

*(Proposed by Anton Trygub)*

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