
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
Let $x=-3$ be an extreme point of the function $f\left(x\right)=ax^{3}+bx^{2}-3x+c$, and the slope of the tangent line to the curve $y=f\left(x\right)$ at $x=1$ is $8$.  
$(1)$ Find the monotonic intervals of $f\left(x\right)$;  
$(2)$ If the maximum value of $f\left(x\right)$ on the closed interval $\left[-1,1\right]$ is $10$, find the value of $c$.

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