
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 that circle $C$ passes through points $A(3,3)$ and $B(-1,7)$, and the center of circle $C$ lies on the line $l: y=3x$.  
$(1)$ Find the equation of circle $C$;  
$(2)$ If a ray emitted from point $P(-2,4)$ is reflected by the line $2x-y-7=0$ and the reflected ray exactly bisects the circumference of circle $C$, find the equation of the line containing the reflected ray.

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