
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 $A(3,0)$ and $B(5,2)$, and the center of the circle lies on the line $2x-y-4=0$.
$(1)$ Find the equation of circle $C$;
$(2)$ If a ray is emitted from point $M(-4,-3)$, reflects off the line $x-y-4=0$, and just touches circle $C$, find the equation of the line on which the reflected ray lies.

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