
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 functions $f(x)=\frac{{x}^{3}}{3}-\frac{{x}^{2}}{2}+\frac{{a}^{2}}{2}-\frac{1}{3}$ and $g(x)=aln(x+1)-\frac{{x}^{2}}{2}-ax$, where $a\in \left(-\infty ,-1\right)$. Find:<br/>$(1)$ The monotonic intervals of $g\left(x\right)$;<br/>$(2)$ If $g\left(x\right)_{max}=f\left(x\right)_{min}+b$, find the range of real number $b$.

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