
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
The real-valued function \( f \) is defined for all positive integers. For any integers \( a > 1 \) and \( b > 1 \) with \( d = \operatorname{gcd}(a, b) \), we have

\[ f(ab) = f(d) \cdot \left( f\left( \frac{a}{d} \right) + f\left( \frac{b}{d} \right) \right). \]

Determine all possible values of \( f(2001) \).

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