
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 the center of circle C lies on the line $y=x-2$,  
(Ⅰ) If the circle passes through points A(3,-2) and B(0,-5),  
(i) find the equation of circle C;  
(ii) let the other intersection point of circle C with the y-axis be P, and let line l pass through point P and be tangent to circle C. Let D be a moving point on circle C, different from points P and B, and the line through BD intersects line l at point R. Determine the positional relationship between the circle with PR as its diameter and line CD, and justify your answer;  
(Ⅱ) Let point M(0,3), if the radius of circle C is 3, and there exists a point N on circle C such that $|MN|=2|NO|$, find the range of the x-coordinate of the center of circle C.

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