
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 circle $C$: $(x-1)^{2}+y^{2}=9$, there is a point $P(2,2)$ inside the circle. A line $l$ passes through point $P$ and intersects circle $C$ at points $A$ and $B$.
(1) When line $l$ passes through the center of the circle, find the equation of line $l$;
(2) When the chord $AB$ has a length of $4\sqrt{2}$, write the equation of line $l$.

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