
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 the circle C: $x^2 + y^2 - 2x + 4y - 4 = 0$, is there a line $l$ with a slope of 1 that intersects circle C at chord AB, and the circle with AB as its diameter passes through the origin O? If it exists, write the equation of line $l$; if not, explain why.

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