
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 $C$: $(x-1)^2 + y^2 = 9$, there is a point $P(2,2)$ inside it. A line $l$ is drawn through point $P$ and intersects circle $C$ at points $A$ and $B$.
(1) Find the equation of line $l$ when it passes through the center of the circle $C$.
(2) Find the length of chord $AB$ when the angle of inclination of line $l$ is $45^{\circ}$.

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