
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 plane Cartesian coordinate system $xOy$, the circle $C$ passes through the points $O(0,0)$, $A(1,1)$, and $B(4,2)$.  
$(1)$ Find the equation of circle $C$;  
$(2)$ If the line $l$ passing through the point $(\frac{3}{2}, \frac{9}{2})$ intersects circle $C$ at points $M$ and $N$, and $\angle MCN = 120^{\circ}$, find the equation of the line $l$.

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