
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)  
Given that the function $f(x) = ax^3 + bx + c$ (where $a > 0$) is an odd function, and its graph at the point $(1, f(1))$ has a tangent line perpendicular to the line $x - 6y - 7 = 0$. The minimum value of the derivative $f'(x)$ is $-12$. Find the values of $a$, $b$, and $c$.

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