
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 $C$: $(x-1)^{2}+y^{2}=9$ with a point $P(2,2)$ inside it, a line $l$ passes through point $P$ and intersects the circle $C$ at points $A$ and $B$.  
$(1)$ When $l$ passes through the center $C$, find the equation of line $l$; (Write in general form)  
$(2)$ When the slope of line $l$ is $45^{\circ}$, find the length of chord $AB$.

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