
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
Let the function $f(x) = -x^3 + ax^2 + bx + c$ have a derivative $f'(x)$ that satisfies $f'(-1) = 0$ and $f'(2) = 9$.  
(1) Determine the intervals of monotonicity for $f(x)$.  
(2) If the maximum value of $f(x)$ on the interval $[-2, 2]$ is 20, find the value of $c$.  
(3) If the graph of the function $f(x)$ intersects the x-axis at three points, determine the range of $c$.

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