Products of Distances 2500 pts

Let $n\ge2$ be an integer and let $x_1<x_2<\cdots<x_n$ be real numbers satisfying

$$\prod_{\substack{1\le j\le n\\j\ne i}}|x_i-x_j|=x_i^2+1 \qquad\text{for every }1\le i\le n.$$

Determine all possible values of $n$ and all corresponding tuples $(x_1,x_2,\ldots,x_n)$.