
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 \( S \) be the set of all non-zero real-valued functions \( f \) defined on the set of all real numbers such that
$$
f\left(x^{2} + y f(z)\right) = x f(x) + z f(y)
$$
for all real numbers \( x, y, \) and \( z \). Find the maximum value of \( f(12345) \), where \( f \in S \).

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