
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
A circle $C$ with its center on the line $2x+y=0$ passes through the point $A(2,-1)$ and is tangent to the line $x+y-1=0$. Find:<br/>
$(1)$ The equation of circle $C$;<br/>
$(2)$ The equation of the line $l: y=k(x-2)$ that divides circle $C$ into two arcs with a ratio of arc lengths equal to $\frac{1}{2}$.

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