
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. Let $f : S \to \mathbb{R}$ be a function such that
\[f(x) f(y) = f(xy) + 2010 \left( \frac{1}{x} + \frac{1}{y} + 2009 \right)\]
for all $x,$ $y > 0.$

Let $n$ be the number of possible values of $f(2),$ and let $s$ be the sum of all possible values of $f(2).$ Find $n \times s.$

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