
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 $f\left( x \right)=ax^{3}+bx+c\left( a\neq 0 \right)$ as an odd function, its graph has a tangent line at the point $\left( 1,f\left( 1 \right) \right)$ that is perpendicular to the line $x-6y-7=0$, and the minimum value of the derivative function ${f}'\left( x \right)$ is $-12$.

(1) Find the values of $a$, $b$, and $c$.
(2) Find the interval(s) where the function $f\left( x \right)$ is strictly increasing, and find the maximum and minimum values of $f\left( x \right)$ on the interval $\left[ -1,3 \right]$.

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