
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 that circle $C$ passes through the point $(1,1)$ and has its center at $(2,0)$.  
$(Ⅰ)$ Find the equation of circle $C$;  
$(Ⅱ)$ Determine the relationship between the line $y=x-4$ and circle $C$;  
$(Ⅲ)$ Given that the line $l$ passing through point $P(1,3)$ intersects circle $C$ at points $A$ and $B$, and $|AB|=2$, find the equation of line $l$.

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