
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 \( g : S \to S \) such that
\[ g(x) + g(y) = g\left(\frac{x+y}{g(xy)}\right) \]
for all \( x, y \in S \) with \( x + y \neq 0 \). Determine the number of possible values of \( g(5) \) and their sum.

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