Middle European Mathematical Olympiad 2024 Problem T-8

Let $k$ be a positive integer and $a_1, a_2, \ldots$ be an infinite sequence of positive integers such that $$a_i a_{i+1} \mid k - a_i^2$$ for all integers $i \geq 1$. Prove that there exists a positive integer $M$ such that $a_n = a_{n+1}$ for all integers $n \geq M$.