<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 A function \( f \) from \( \mathbb{Z} \) to \(\{1, \ldots, 1000\}\) is said to be interesting if for all integers \( x \) and \( y \), we have

\[ \operatorname{gcd}(f(x), f(y))=\operatorname{gcd}(f(x), x-y). \]

How many interesting functions exist? Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
500500
A function \( f \) from \( \mathbb{Z} \) to \(\{1, \ldots, 1000\}\) is said to be interesting if for all integers \( x \) and \( y \), we have

\[ \operatorname{gcd}(f(x), f(y))=\operatorname{gcd}(f(x), x-y). \]

How many interesting functions exist?
test