
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 point $A(2,-1)$ and is tangent to the line $x+y-1=0$.

$(1)$ Find the equation of circle $C$;  
$(2)$ The circle $C$ is divided by line $l$: $y=k(x-2)$ into two arcs with the length ratio of $\dfrac{1}{2}$. Find the equation of line $l$.

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