
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
Determine the value of $x$ at which the function $x^3+x^2+px+q$ has a local minimum or maximum, where $p$ and $q$ are positive real numbers.
A) $\frac{-1 + \sqrt{1-3p}}{3}$
B) $\frac{-1 - \sqrt{1-3p}}{3}$
C) $\frac{1 - \sqrt{1+3p}}{3}$
D) $\frac{-1 + \sqrt{1-2p}}{3}$
E) $\frac{-2 + \sqrt{1-3p}}{6}$

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