
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 coordinates of the center of the circle given by the equation

\[ 
x^{2} + y^{2} - 2px - 2qy + 2q - 1 = 0 
\]

such that the abscissas of the intersection points of the circle with the $X$-axis are equal to the roots of the quadratic equation

\[ 
x^{2} + bx + c = 0 
\].

How can we use the result to construct the roots of the quadratic equation?

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