
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: $x^2+y^2-2x-7=0$.  
(1) Find the equation of the line that passes through point P(3, 4) and is intersected by circle C, producing a chord of length 4.  
(2) Determine whether there exists a line $l$ with a slope of 1, such that the midpoint D of the chord AB (which is cut by circle C on line $l$) has a distance to the origin O exactly equal to the radius of circle C. If such a line exists, find its equation; if not, explain why.

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