
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
Find the equations of the circles that satisfy the following conditions:

(1) The circle is circumscribed about $\triangle ABC$, whose vertex coordinates are $A(4,1)$, $B(6,-3)$, $C(-3,0)$. Find the equation of this circle.

(2) The circle passes through points $A(5, 2)$ and $B(3, -2)$, and its center lies on the line $2x-y=3$. Find the equation of this circle.

(3) The circle is tangent to the $x$-axis, its center lies on the line $3x-y=0$, and the length of the chord intercepted by the line $x-y=0$ is $2\sqrt{7}$. Find the equation of this circle.

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