International Mathematical Olympiad 2018 Problem 2

Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \ldots, a_{n+2}$, such that $a_{n+1} = a_1$ and $a_{n+2} = a_2$, and $$a_i a_{i+1} + 1 = a_{i+2}$$ for $i = 1, 2, \ldots, n$.