
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 $f(x) = x^3 + ax^2 + bx + c$ has extreme values at $x = -\frac{2}{3}$ and $x = 1$.

$(1)$ Find the values of $a$ and $b$;  
$(2)$ Determine the intervals of monotonicity for the function $f(x)$.

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