
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  $a>1, b>1$  with  $d=\gcd (a, b)$ , we have
\[f(ab)=f(d)\left(f\left(\frac{a}{d}\right)+f\left(\frac{b}{d}\right)\right) \]
Determine all possible values of  $f(2001)$ .

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