HMIC 2023 Problem 5

Let $a_{1}, a_{2}, \ldots$ be an infinite sequence of positive integers such that, for all positive integers $m$ and $n$, we have that $a_{m+n}$ divides $a_{m} a_{n}-1$. Prove that there exists an integer $C$ such that, for all positive integers $k>C$, we have $a_{k}=1$.