
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
In the Cartesian coordinate system $xOy$, given four points $A(0,1)$, $B(0,3)$, $C(4,1)$, $D(3,0)$.  
$(1)$ Find the equation of the circle $M$ passing through points $A$, $B$, and $C$, and determine the relationship between point $D$ and circle $M$;  
$(2)$ If the chord intercepted by circle $M$ passing through point $D$ has a length of $4$, find the equation of the line $l$ passing through point $D$.

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