Let $x_1, x_2, x_3, \ldots$ be a sequence of rational number...

Prompt

Let $x_1, x_2, x_3, \ldots$ be a sequence of rational numbers defined by $x_1 = \frac{25}{11}$ and $$x_{k+1} = \frac{1}{3}\left(x_k + \frac{1}{x_k} - 1\right)$$ for all $k \ge 1$. Then $x_{2025}$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the remainder when $m + n$ is divided by $1000$.