International Mathematical Olympiad (IMO) $2018$ , Day $1$ , Problem $2$ of $6$

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

$a_i a_{i+1} + 1 = a_{i + 2}$

for $i = 1, 2, ..., n$

The person who answers it correctly and gives the official solution first - there is $2$ - go to International Mathematical Olympiad (IMO) Hall of Fame