
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 for all real numbers \( x \), \( y \), and \( z \):
\[ f(x^2 + yf(z) + z) = xf(x) + zf(y) + f(z). \]
Find the number of possible values of \( f(5) \), and let \( s \) be the sum of all possible values of \( f(5) \). Compute \( n \times s \).

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