
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
Find  $ f(x): (0,\plus{}\infty) \to \mathbb R$  such that \[ f(x)\cdot f(y) \plus{} f(\frac{2008}{x})\cdot f(\frac{2008}{y})\equal{}2f(x\cdot y)\] and  $ f(2008)\equal{}1$  for  $ \forall x \in (0,\plus{}\infty)$ .

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