
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 $T$ be the set of positive real numbers. Define a function $g : T \to \mathbb{R}$ such that
\[g(x) g(y) = g(xy) + 2010 \left( \frac{1}{x} + \frac{1}{y} + 2010 \right)\]for all $x,$ $y > 0.$

Determine the number of possible values for $g(3)$ and the sum of all possible values for $g(3).$

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