
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 a circle M: $x^{2}+(y-4)^{2}=4$, and a point P on the line $l: x-2y=0$ moving along it. Tangents PA and PB are drawn from P to circle M, with A and B as the points of tangency.

(Ⅰ) When the length of the tangent PA is $2\sqrt{3}$, find the coordinates of point P;  
(Ⅱ) If the circumcircle of $\triangle PAM$ is circle N, does circle N pass through a fixed point as P moves? If such points exist, find their coordinates; if not, explain why;  
(Ⅲ) Find the minimum length of segment AB.

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