
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
\[ f(xy + f(x)) = 2xf(y) + f(x) \]
for all real numbers \( x \) and \( y \).

Let \( n \) be the number of possible values of \( f \left( \frac{1}{2} \right) \), and let \( s \) be the sum of all possible values of \( f \left( \frac{1}{2} \right) \). Find \( n \times s \).

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