
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 point $P(2,0)$ and circle $⊙C$: $x^{2}+y^{2}-6x+4y+4=0$.
(1) Find the equation of line $l$ that passes through point $P$ and is at a distance of $1$ from the circle's center $C$.
(2) Let line $l$ pass through point $P$ and intersect circle $⊙C$ at points $A$ and $B$. When $|AB|=4$, find the equation of the circle with line segment $AB$ as its diameter.

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