
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 line $l$ passing through point $P(2, 2)$ and intersecting with the circle $C: (x-1)^2 + y^2 = 6$ at points $A$ and $B$,
(Ⅰ) If point $P$ is exactly the midpoint of line segment $AB$, find the equation of line $l$;
(Ⅱ) If $|AB| = 2\sqrt{5}$, find the equation of line $l$.

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