
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 $z,$
\[f(x^2 + yf(z)) = xg(x) + zg(y)\]
where $g : \mathbb{R} \to \mathbb{R}$ is a known function given by $g(x) = 2x$. Determine all possible values of $f(4)$.

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