
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 parabola \( y = k x^2 \), determine the center of the circle \( x^2 - 2px + y^2 - 2qy = 0 \) such that the abscissas of its intersection points with the given parabola are the roots of the equation \( x^3 + ax + b = 0 \).

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