
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
Let \( S \) be the set of positive real numbers. Define a function \( f: S \to \mathbb{R} \) such that
\[ f(x)f(y) = f(xy) + 1001 \left( \frac{1}{x} + \frac{1}{y} + 1000 \right) \]
for all \( x, y > 0 \).
Determine \( n \), the number of possible values of \( f(2) \), and \( s \), the sum of all possible values of \( f(2) \). Compute \( n \times s \).

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