
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 center at $(3,4)$, it is intercepted by the line $x=1$ forming a chord of length $2 \sqrt {5}$.
(1) Find the equation of circle $N$;
(2) If a line $l$ passing through point $D(3,6)$ is intercepted by circle $N$ forming a chord of length $4 \sqrt {2}$, find the slope of line $l$.

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