
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
The equation of circle $O_1$ is $x^2+(y+1)^2=4$, and the center of circle $O_2$ is $O_2(2,1)$.
1. If circle $O_2$ is tangent to circle $O_1$, find the equation of circle $O_2$.
2. If circle $O_2$ intersects circle $O_1$ 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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