
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,

(1) Draw a line $l$ through point $P$ intersecting circle $C$ at points $A$ and $B$. Find the equation of line $l$ when it passes through the center of the circle $C$.

(2) Write the equation of line $l$ when chord $AB$ is bisected by point $P$.

(3) When the slant angle of line $l$ is $45º$, find the length of chord $AB$.

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