2018 IMO Shortlist N4

Let $a_{1}, a_{2}, \ldots, a_{n}, \ldots$ be a sequence of positive integers such that $$\frac{a_{1}}{a_{2}}+\frac{a_{2}}{a_{3}}+\cdots+\frac{a_{n-1}}{a_{n}}+\frac{a_{n}}{a_{1}}$$ is an integer for all $n \geq k$, where $k$ is some positive integer. Prove that there exists a positive integer $m$ such that $a_{n}=a_{n+1}$ for all $n \geq m$. (Mongolia)

设 $a_{1}, a_{2}, \ldots, a_{n}, \ldots$ 为正整数序列，使得 $$\frac{a_{1}}{a_{2}}+\frac{a_{2}}{a_{3}}+\cdots+\frac{a_{n-1}}{a_{n}}+\frac{a_{n}}{a_{1}}$$ 为整数所有 $n \geq k$，其中 $k$ 是某个正整数。证明对于所有$n \geq m$，存在一个正整数$m$，使得$a_{n}=a_{n+1}$。 (蒙古)