
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 a circle $C$ that passes through point $A(2,-1)$ and is tangent to the line $x+y=1$, with its center located on the line $y=-2x$.
(I) Find the equation of circle $C$;
(II) If a line $l$ passes through the internal point of circle $C$, $P(\frac{1}{2},-3)$, and intersects circle $C$ at points $A$ and $B$ such that chord $AB$ is bisected by point $P$, find the equation of line $l$.

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