ISL 2018 A2

Find all positive integers $n \geqslant 3$ 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)