More fun with quadratic forms, Part 2

Algebra Level 5

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.

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