
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 \( f(x) = x^{3} + a x^{2} + x + 1 \) (with \( a \in \mathbf{R} \)) is decreasing in the interval \(\left(-\frac{2}{3}, -\frac{1}{3}\right)\) and increasing in the interval \(\left(-\frac{1}{3}, +\infty\right)\), determine the value of \( a \).

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