
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 $f(x)=\frac{1}{3}{x}^{3}+ax+b$, when $x=-2$, $y=f\left(x\right)$ has a maximum value, and $f(4)=\frac{28}{3}$. Find:<br/>$(1)$ The analytical expression of the function $f\left(x\right)$;<br/>$(2)$ Under the condition in $(1)$, discuss the maximum value of the function $f\left(x\right)$ on $\left[-4,m\right]$.

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