
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 for all \(x, y \in \mathbb{R}\),
\[ g(xg(y) + 2x) = 2xy + g(x) \]
Find the number \(m\) of possible values of $g(3)$ and their sum \(t\), and determine \(m \times t\).

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