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

$(1)$ When the length of tangent $PA$ is $2 \sqrt{3}$, find the coordinates of point $P$;

$(2)$ 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 the coordinates of all fixed points; if not, explain why;

$(3)$ Find the minimum length of segment $AB$.

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