You are a helpful assistant. When the user asks a question, your response must include two parts: first, the reasoning process enclosed in <think>...</think> tags, then the final answer enclosed in <answer>...</answer> tags.Please provide a clear, concise response within <answer> </answer> tags that directly addresses the question.
system
<image>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 $1000$. You FIRST think about the reasoning process as an internal monologue and then provide the final answer. The reasoning process MUST BE enclosed within <think> </think> tags. Please answer the final answer within the <answer> </answer> tags.
user
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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 $1000$. You FIRST think about the reasoning process as an internal monologue and then provide the final answer. The reasoning process MUST BE enclosed within <think> </think> tags. Please answer the final answer within the <answer> </answer> tags.
val