
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 that the center of circle $N$ is at $(3,4)$, and its radius is equal to the distance between two parallel lines $(a-2)x+y+\sqrt{2}=0$, $ax+3y+2\sqrt{2}a=0$.

(I) Find the equation of circle $N$;

(II) Point $B(3,-2)$ and point $C$ are symmetrical about the line $x=-1$, find the equation of the circle with center at $C$ and externally tangent to circle $N$.

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