
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 $f(x)=ax^{3}+bx^{2}-2x+c$ has a local maximum of $6$ at $x=-2$ and a local minimum at $x=1$,  
$(1)$ find the values of $a$, $b$, $c$;  
$(2)$ find the maximum and minimum values of $f(x)$ in the interval $[-3,2]$.

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