
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 $C$ is tangent to the $y$-axis, the center of the circle $C$ lies on the line $x-3y=0$, and the line $x=y$ intersects the circle $C$ to form a segment of length $2\sqrt{7}$.  
$(1)$ Find the equation of circle $C$;  
$(2)$ If circle $C$ is tangent to the positive $y$-axis and a ray emitted from point $M(-1,-2)$ reflects off the line $y=x+4$ and passes through the center 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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