
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 the center of circle $C$ lies on the line $y=-2x$, and circle $C$ passes through point $A(2,-1)$ and is tangent to the line $x+y-1=0$.  
$(1)$ Find the equation of circle $C$.  
$(2)$ Find the slope of the tangent line to circle $C$ passing through point $B(1,2)$.

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