
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 \( f : S \to S \) be a function such that
\[
f(x) + f(y) + xy = f(x y f(x + y) + xy)
\]
for all \( x, y \in S \) such that \( x + y \neq 0 \).

Let \( n \) be the number of possible values of \( f(5) \), and let \( s \) be the sum of all possible values of \( f(5) \). Find \( n \times s \).

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