
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 $A(0,2)$ and $B(6,4)$, and the line $x-3y-4=0$ bisects the circumference of circle $C$.  
$(1)$ Find the equation of circle $C$;  
$(2)$ If $P(-6,0)$, $Q(6,0)$, point $M$ is a point on circle $C$ and satisfies $\frac{|MP|}{|MQ|}=2$, find the coordinates of point $M$.

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