
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 $f(x)=x^{3}+ax^{2}+b$ ($a,b\in\mathbb{R}$).

$(1)$ Discuss the monotonicity of $f(x)$;

$(2)$ If $b=c-a$ (where real number $c$ is a constant unrelated to $a$), and when the function $f(x)$ has three distinct zeros, the range of $a$ is exactly $(-\infty,-3)\cup\left(1,\frac{3}{2}\right)\cup\left(\frac{3}{2},+\infty\right)$, find the value of $c$.

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