
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. Define a function \(f : S \to S\) such that for all \(x, y \in S\) with \(x + y = 1\),
\[ f(x) + f(y) = f(xyf(x + y)) \]
Determine the number of possible values of \(f(3)\) and the sum of all possible values of \(f(3)\). Calculate \(n \times s\) where \(n\) is the number of possible values and \(s\) is the sum of those values.

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