
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
Circle $O\_1$ has the equation $x^{2}+(y+1)^{2}=4$, and the coordinates of the center of circle $O\_2$ are $(2,1)$.

(1) If circle $O\_1$ and circle $O\_2$ are externally tangent, find the equation of circle $O\_2$;

(2) If circle $O\_1$ and circle $O\_2$ intersect at points $A$ and $B$, and $|AB|=2 \sqrt{2}$, find the equation of circle $O\_2$.

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