
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 $f(x)=ax^{2}+(b-8)x-a-ab$, when $x\in (-3,2)$, $f(x) \gt 0$, and when $x\in (-\infty ,-3)\cup (2,+\infty)$, $f(x) \lt 0$. Find:<br/>
$(1)$ The analytical expression of $f(x)$;<br/>
$(2)$ If the solution set of the inequality $ax^{2}+bx+c\leqslant 0$ is $R$, find the range of $c$;<br/>
$(3)$ When $x \gt -1$, find the maximum value of $y=\frac{f(x)-21}{x+1}$.

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