
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  $Q(x) = a_{2023}x^{2023}+a_{2022}x^{2022}+\dots+a_{1}x+a_{0} \in \mathbb{Z}[x]$  be a polynomial with integer coefficients. For an odd prime number  $p$  we define the polynomial  $Q_{p}(x) = a_{2023}^{p-2}x^{2023}+a_{2022}^{p-2}x^{2022}+\dots+a_{1}^{p-2}x+a_{0}^{p-2}.$  
Assume that there exist infinitely primes  $p$  such that  $$ \frac{Q_{p}(x)-Q(x)}{p} $$  is an integer for all  $x \in \mathbb{Z}$ .  Determine the largest possible value of  $Q(2023)$  over all such polynomials  $Q$ .

*Proposed by Nikola Velov*

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