
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 $l: x - y + 3 = 0$ and a circle $C: (x - 1)^2 + y^2 = 1$, where $P$ is a moving point on line $l$. A line $m$ passes through $P$ and is tangent to circle $C$ at points $A$ and $B$.

(Ⅰ) Find the minimum value of $|PA|$;  
(Ⅱ) When $|PA|$ is at its minimum, find the equation of line $AB$.

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