
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
Determine all functions  $ f$  mapping the set of positive integers to the set of non-negative integers satisfying the following conditions:
(1)  $ f(mn) \equal{} f(m)\plus{}f(n)$ ,
(2)  $ f(2008) \equal{} 0$ , and
(3)  $ f(n) \equal{} 0$  for all  $ n \equiv 39\pmod {2008}$ .

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