
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 $f : \mathbb{R} \to \mathbb{R}$ be a function such that $f(1) = 2$ and
\[ f(x^2 + y^2) = (x + y)(f(x) - f(y)) \]
for all real numbers $x$ and $y$. Determine $n$, the number of possible values of $f(2)$, and $s$, the sum of all possible values of $f(2)$. Finally, compute $n \times s$.

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