
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 $g(1) = 1$ and
\[ g(x^2 - y^2) = (x - y) \left(g(x)^3 + g(y)^3 \right) \]
for all real numbers $x$ and $y.$ 

Let $m$ be the number of possible values of $g(3)$, and let $u$ be the sum of all possible values of $g(3)$. Find $m \times u$.

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