
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 $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[ g(x^2 + y^2 + yg(z)) = xg(x) + z^2g(y) \]
for all real numbers $x, y, z.$ Determine if $g$ can only be a zero function or any other specific linear function, like $g(x) = cx$ for some constant $c$.

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