
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 points $A(\sqrt{6},1)$, $B(1,0)$, $C(3,2)$ lie on circle $M$.
$(1)$ Find the standard equation of circle $M$.
$(2)$ If a line $l$ passes through point $C$ and intersects circle $M$ with a chord length of $2$, find the equation of line $l$.

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