
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
(The full score for this question is 12 points) Let the function $f(x) = ax^3 + bx + c$ ($a \neq 0$) be an odd function. The tangent line at the point $(1, f(1))$ is perpendicular to the line $x - 6y - 7 = 0$, and the minimum value of the derivative $f'(x)$ is $-12$.
(1) Find the values of $a$, $b$, and $c$;
(2) Find the intervals of monotonic increase for the function $f(x)$, and find the maximum and minimum values of the function $f(x)$ on the interval $[-1,3]$.

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