
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 non-zero real numbers. Define a function $f : S \to S$ such that for all $x, y \in S$ with $x + y \neq 0$, the following holds:
\[f(x) + f(y) = f\left(\frac{xy}{x+y}f(x+y)\right).\]
Determine the number of possible values of $f(3)$, denoted as $n$, and the sum of all possible values of $f(3)$, denoted as $s$. Finally, compute $n \times s$.

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