
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 $f(x) = x^3 + ax^2 + bx + c$, where $x \in [-1, 2]$, and the function $f(x)$ has extreme values at $x = 1$ and $x = -\frac{2}{3}$.  
(I) Find the values of the real numbers $a$ and $b$;  
(II) For any $x \in [-1, 2]$, the equation $f(x) = 2c$ has three real roots, find the range of the real number $c$.

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