
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}\\) and the equation of their common tangent line.

\\((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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