
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 that a circle $C$ passes through the point $A(2,-1)$ and is tangent to the line $x+y=1$, with the center of the circle lying on the line $y=-2x$.  
$(Ⅰ)$ Find the equation of the circle $C$.  
$(Ⅱ)$ Given that a line $l$ passes through the origin and cuts off a chord of length $2$ on the circle $C$, find the equation of the line $l$.

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