
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 a circle with center $M$ passing through point $A(2,4)$, $B(6,2)$, and the perpendicular bisector of segment $AB$ intersecting the circle at points $C$ and $D$, where $|CD|=2\sqrt{10}$, <br/>$(1)$ Find the equation of the line $CD$;<br/>$(2)$ Find the equation of the circle $M$.

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