
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 $f(x)$ and $g(x)$ be nonzero polynomials, and introduce a nonzero polynomial $h(x)$ such that
\[f(g(x)) = f(x) + g(x)h(x).\]
Given $g(2) = 12$, determine $g(x)$ assuming all polynomials are quadratic.

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