
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
Given the function $f(x) = ax^3 + x^2 + bx$, where $a$ and $b$ are constants in $\mathbb{R}$, and $g(x) = f(x) + f'(x)$ is an odd function.

(1) Find the expression for $f(x)$;

(2) Discuss the monotonicity of $g(x)$, and find the maximum and minimum values of $g(x)$ on the interval $[1, 2]$.

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