Let's try again

$\large\sum_{k=1}^{n-2}x_kx_{k+1}+x_nx_{n-3}>\sum_{k=1}^{n}x_k^2$

Find the smallest positive integer $n\geq4$ such that the above inequality holds for some real numbers $x_1,\ldots,x_n$ .

I have posted variants of this problem before, but I'm still waiting for a solution.