Problem 5. Let $n \ge 4$ be a positive integer. Find all positive real numbers $x_1, x_2, \dots, x_n$ such that
$$\begin{cases}x_1 + x_2 = x_2x_3 + 1 \\ x_2 + x_3 = x_3x_4 + 1 \\ \vdots \\ x_{n - 1} + x_n = x_nx_1 + 1 \\ x_n + x_1 = x_1x_2 + 1.\end{cases}$$
