
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 $f(x)=x^{3}+ax^{2}+bx+c$, the function takes extreme values at $x=1$ and $x=-2$.

$(1)$ Find the values of $a$ and $b$;  
$(2)$ If $f(x) > \frac{1}{c} - \frac{1}{2}$ holds for all $x \in [-3,2]$, find the range of values for $c$.

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