
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

Consider the real numbers \( p \) and \( q \) and the cubic equation:

\[ x^{3}+px+q=0 \]

a) If \( x_{0} \) is a real root of the equation, then

\[ x^{3}+px+q=(x-x_{0})(x^{2}+ax+b) \]

Verify that \( a=x_{0} \).

b) Verify that \( p^{2} \geq 4x_{0}q \).

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