
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
Circle $C$ passes through point $A(2,-1)$ and is tangent to the line $x+y=1$, with the center of the circle lying on the line $y=-2x$.<br/>$(1)$ Find the equation of circle $C$;<br/>$(2)$ Find the chord length intercepted by circle $C$ on the $y$-axis.

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