
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 that circle M passes through the intersection points of the circles $x^2+y^2+6x-4=0$ and $x^2+y^2+6y-28=0$,

(I) If the center of the circle is on the line $x-2y-3=0$, find the equation of circle M.  
(II) If the area of the circle is minimal, find the equation of circle M.

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