
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 triangle $\triangle ABC$ with vertices $A(0, 2)$, $B(0, 4)$, $C(1, 3)$, and its circumcircle as 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 intercepted by circle $M$ has a length of $\sqrt{3}$, find the equation of line $l$;  
(3) Suppose point $P$ is any point on circle $M$ different from $A$ and $B$, line $PA$ intersects the x-axis at point $E$, and line $PB$ intersects the x-axis at point $F$. Does the circle $N$ with diameter $EF$ pass through a fixed point? If it does, find the coordinates of the fixed point; if not, explain why.

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