Question 25
(VIE 1) Solve the following system of equations, in which $a$ is a given number satisfying $|a|\gt 1$:
$$\begin{aligned}x_1^2&=ax_2+1,\\x_2^2&=ax_3+1,\\&\cdots\qquad\cdots\\x_{999}^2&=ax_{1000}+1,\\x_{1000}^2&=ax_1+1.\end{aligned}$$
$$\begin{aligned}x_1^2&=ax_2+1,\\x_2^2&=ax_3+1,\\&\cdots\qquad\cdots\\x_{999}^2&=ax_{1000}+1,\\x_{1000}^2&=ax_1+1.\end{aligned}$$