IMO 2023 Problem 3

For each integer $k\geq 2$ , determine all infinite sequences of positive integers $a_1$ , $a_2$ , $\ldots$ for which there exists a polynomial $P$ of the form $$P(x)=x^k+c_{k-1}x^{k-1}+\dots + c_1 x+c_0,$$ where $c_0$ , $c_1$ , \dots, $c_{k-1}$ are non-negative integers, such that $$P(a_n)=a_{n+1}a_{n+2}\cdots a_{n+k}$$ for every integer $n\geq 1$ .