More fun with quadratic forms, Part 2

Find the smallest integer $n\geq 3$ such that there exist real numbers $x_0,x_1,\ldots,x_n$ with $\displaystyle \sum_{k=0}^{n}x_k^2=x_0x_3+\sum_{k=1}^{n-1}x_kx_{k+1}=1$

If you come to the conclusion that no such $n$ exists, enter 666.