
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 $g: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that
\[ g((x - y)^2) = g(x)^2 - 3xg(y) + y^2 \]
for all real numbers $x$ and $y.$ Determine all possible values of $g(1)$, the sum of those values, and compute $n \times s$ where $n$ is the number of possible values of $g(1)$ and $s$ is their sum.

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