
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 \\(N\\) with its center at \\((3,4)\\) and its radius equal to the distance between two parallel lines \\((a-2)x+y+ \sqrt {2}=0\\) and \\(ax+3y+2 \sqrt {2}a=0\\).  
\\((1)\\) Find the equation of circle \\(N\\);  
\\((2)\\) Point \\(B(3,-2)\\) and point \\(C\\) are symmetric about the line \\(x=-1\\). Find the equation of the circle with center \\(C\\) that is tangent to circle \\(N\\) externally.

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