
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 $f\left( x \right)=a{{x}^{3}}+{{x}^{2}}+bx$, where $a,b$ are constants, and $g\left(x\right)=f\left(x\right)+{f}^{,}\left(x\right)$ is an odd function.

(1) Find the analytic expression of $f\left( x \right)$;

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

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