
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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?

Present the answer in LaTex format: \boxed{Your answer}
user