
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 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 a tangent line \\(l\\) from the center \\(C\\) of circle \\(C\\) to circle \\(O\\), and find the equation of the tangent line \\(l\\);
\\((\\)Ⅲ\\()\\) Draw a moving line \\(m\\) through the center \\(C\\) of circle \\(C\\) intersecting circle \\(O\\) at points \\(A\\) and \\(B\\). Question: Among all the circles with \\(AB\\) as the diameter, does there exist such a circle \\(P\\) that passes through point \\(M(2,0)\\)? If it exists, find the equation of circle \\(P\\); if not, please explain why.

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