2023 IMO Shortlist A6

Let $k \geq 2$ be an integer. Determine all sequences of positive integers $a_{1}, a_{2}, \ldots$ for which there exists a monic polynomial $P$ of degree $k$ with non-negative integer coefficients such that $$P\left(a_{n}\right)=a_{n+1} a_{n+2} \cdots a_{n+k}$$ for every integer $n \geq 1$. (Malaysia)

设 $k \geq 2$ 为整数。确定所有正整数序列 $a_{1}, a_{2}, \ldots$，其中存在一个具有非负整数系数的 $k$ 次一元多项式 $P$，使得对于每个整数 $n \geq 1$ $$P\left(a_{n}\right)=a_{n+1} a_{n+2} \cdots a_{n+k}$$。 (马来西亚)