
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
\[g(g(x)^2 - y^2) = g(x)^2 + g(g(y)^2 - g(-x)^2) + x^2\] for all real numbers $x$ and $y.$

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

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