
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
(Full score: 12 points)  
It is known that a circle passes through points $A(2, -3)$ and $B(-2, -5)$.  
(1) If the center of the circle is on the line $x - 2y - 3 = 0$, find the equation of the circle. (2) If the area of the circle is minimized, find the equation of the circle.

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