
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 all nonzero real numbers. Let $g : S \to S$ be a function such that
\[g(x) + g(y) = g\left(\frac{xy}{g(x+y)}\right)\]
for all $x, y \in S$ where $x + y \neq 0$. Find the number $m$ of possible values of $g(2)$, and let $t$ be the sum of all possible values of $g(2)$. Calculate $m \times t$.

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