
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 the three vertices of $\triangle ABC$, $A(0,2)$, $B(0,4)$, and $C(1,3)$, and its circumcircle is circle $M$:

$(1)$ Find the equation of circle $M$;  
$(2)$ If line $l$ passes through point $D\left(\frac{1}{2}, 2\right)$ and the chord cut by circle $M$ has a length of $\sqrt{3}$, find the equation of line $l$;  
$(3)$ Suppose point $P$ is any point on circle $M$ other than $A$ and $B$, line $PA$ intersects the $x$-axis at point $E$, and line $PB$ intersects the $x$-axis at point $F$. Is the circle $N$ with diameter $EF$ passing through a fixed point? If it passes through a fixed point, find the coordinates of the fixed point; if not, explain why.

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