
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
(1) Given a circle passing through points $P(4,-2)$ and $Q(-1,3)$, and the length of the segment intercepted by the circle on the $y$-axis is $4\sqrt{3}$, find the equation of the circle;  
(2) Find the equation of the circle whose center lies on the line $x+y=0$, and passes through the intersection points of two circles $x^{2}+y^{2}-2x+10y-24=0$ and $x^{2}+y^{2}+2x+2y-8=0$.

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