
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 the circle $M: x^{2}+y^{2}-4x-2y=0$ and the line $l: y=kx-1$.  
$(Ⅰ)$ Write down the center and radius of circle $M$;  
$(Ⅱ)$ If there exist two points $A$ and $B$ on circle $M$ symmetric with respect to line $l$, and a circle with diameter $AB$ passes through the origin, find the equation of line $AB$.

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