
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 \( g : \mathbb{R} \to \mathbb{R} \) be a function such that \( g(2) = 2 \) and
\[ g(xy + g(x)) = xg(y) + g(x) \]
for all real numbers \( x \) and \( y \). Determine the number of possible values of \( g \left( \frac{1}{3} \right) \), denote it by \( m \), and the sum of all possible values of \( g \left( \frac{1}{3} \right) \), denote it by \( t \). Find \( m \times t \).

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