
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 $M: x^2 + (y-4)^2 = 4$, point $P$ is a moving point on the line $l: x - 2y = 0$. A tangent line $PA$ is drawn through point $P$ to circle $M$, with $A$ and $B$ being the points of tangency.

(I) When the length of the tangent line $PA$ 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 on line $l$? If such fixed points exist, find their coordinates; if not, explain why.

(III) Find the minimum length of line segment $AB$.

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