ISL 2021 A7

Let $n \geqslant 1$ be an integer, and let $x_{0}, x_{1}, \ldots, x_{n+1}$ be $n+2$ non-negative real numbers that satisfy $x_{i} x_{i+1}-x_{i-1}^{2} \geqslant 1$ for all $i=1,2, \ldots, n$. Show that $$x_{0}+x_{1}+\cdots+x_{n}+x_{n+1}>\left(\frac{2 n}{3}\right)^{3 / 2}$$