
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:
\[f(x^2 + 2f(y)) = xf(x) + yf(z)\]
for all real numbers $x,$ $y,$ and $z.$

Find the number $n$ of possible functions $f$, and determine the sum $s$ of $f(3)$ among all possible functions. Calculate $n \times s.$

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