BG-MO 2022 Problem 4

Let $n\geq 4$ be a positive integer and $x_{1},x_{2},\ldots ,x_{n},x_{n+1},x_{n+2}$ be real numbers such that $x_{n+1}=x_{1}$ and $x_{n+2}=x_{2}$ . If there exists an $a>0$ such that $$x_{i}^2=a+x_{i+1}x_{i+2} \forall 1\leq i\leq n$$ then prove that at least $2$ of the numbers $x_{1},x_{2},\ldots ,x_{n}$ are negative.