AIME 2025b Problem 13

Let the sequence of rationals $x_1,x_2,\dots$ be defined such that $x_1=\frac{25}{11}$ and $$x_{k+1}=\frac{1}{3}\left(x_k+\frac{1}{x_k}-1\right).$$$x_{2025}$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find the remainder when $m+n$ is divided by $1000$.