
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 with a radius of 3, the center C is below the x-axis and on the line $y=x$, and the chord cut by circle C on the x-axis is $2\sqrt{5}$.  
(Ⅰ) Find the equation of circle C;  
(Ⅱ) Is there a line $l$ with a slope of 1 such that the circle with diameter AB, where AB is the chord cut by circle C, passes through the origin? If it exists, find the equation of $l$; if not, explain why.

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