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

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