Question 2
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 $i=1,2, \ldots, n$.
$$
a_{i} a_{i+1}+1=a_{i+2}
$$
for all $i=1,2, \ldots, n$.