
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 $C$ with its center on the $x$-axis and passing through points $(0,0)$ and $(-1,1)$, and the equation of circle $D$ is $(x-4)^{2}+y^{2}=4$.

(Ⅰ) Find the equation of circle $C$;

(Ⅱ) A moving point $P$ on circle $D$ draws two tangents to circle $C$, intersecting the $y$-axis at points $A$ and $B$. Find the range of $|AB|$.

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