
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 $C: x^2+y^2-2x+2y-4=0$ and a line $l$ with a slope of 1 intersecting at two distinct points $A$ and $B$.

(1) Find the range of values for the y-intercept $b$ of line $l$;  
(2) Determine whether there exists a line $l$ such that the circle with diameter $AB$ passes through the origin. If it exists, find the equation of line $l$; if not, explain why.

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