
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 \( T \) be the set of all nonzero real numbers. Define \( g : T \to T \) such that
\[ g(x) + g(y) = g\left(\frac{xyg(x+y)}{2}\right) \]
for all \( x, y \in T \) with \( x + y \neq 0 \).

Let \( m \) be the number of possible values of \( g(2) \), and let \( t \) be the sum of all possible values of \( g(2) \). Find \( m \times t \).

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