2018 IMO Shortlist A2

Find all positive integers $n \geq 3$ for which there exist real numbers $a_{1}, a_{2}, \ldots, a_{n}$, $a_{n+1}=a_{1}, a_{n+2}=a_{2}$ such that $$a_{i} a_{i+1}+1=a_{i+2}$$ for all $i=1,2, \ldots, n$. (Slovakia)

找到存在实数 $a_{1}, a_{2}, \ldots, a_{n}$, $a_{n+1}=a_{1}, a_{n+2}=a_{2}$ 的所有正整数 $n \geq 3$ 使得 $$a_{i} a_{i+1}+1=a_{i+2}$$ 对于所有 $i=1,2, \ldots, n$。 (斯洛伐克)