
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 $C$ passes through points $A(1,3)$ and $B(-1,1)$, and the center of the circle is on the line $y=x$.  
(Ⅰ) Find the equation of circle $C$;  
(Ⅱ) Suppose a line $l$ passes through the point $(2,-2)$, and $l$ intersects circle $C$ with a chord length of $2\sqrt{3}$, find the equation of line $l$.

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