
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
\[ f(f(x) + y) = f(x^2 - y) + kf(x)y \]
for all real numbers \( x \) and \( y \), and for some constant \( k \). Determine the number of possible values of \( f(2) \), and let \( s \) be the sum of all possible values of \( f(2) \). Find the product of the number of possible values and the sum, i.e., \( n \times s \), where \( n \) is the number of possibilities for \( f(2) \).

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