
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 two circles, circle O: $x^2+y^2=4$ and circle C: $(x^2+(y-4)^2=1$.
(I) Determine the positional relationship between circle O and circle C.
(II) Draw tangent line l to circle O passing through the center C of circle C and find the equation of l.
(III) Draw a variable straight line m that passes through the center C of circle C and intersects circle O at points A and B. The question is: among all the circles with AB as their diameter, is there a circle P that passes through point M (2,0)? If it exists, find the equation of circle P; if not, provide the reason.

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