
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: $x^2+y^2-2x+4y-4=0$, and a line $l: y=kx-1$.  
(1) When circle C is bisected by line $l$, find the value of $k$.  
(2) On circle C, do there exist points A and B such that they are symmetric with respect to the line $y=kx-1$ and $OA \perp OB$? If they exist, find the equation of line AB; if not, explain why.

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