
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 circle $M$: $(x-2)^{2}+(y-2)^{2}=2$, circle $N$: $x^{2}+(y-8)^{2}=40$, and two lines through the origin $l_{1}$ and $l_{2}$ satisfying $l_{1} \perp l_{2}$, with $l_{1}$ intersecting circle $M$ at two distinct points $A$ and $B$, and $l_{2}$ intersecting circle $N$ at two distinct points $C$ and $D$. Let the slope of $l_{1}$ be $k$.  
$(1)$ Find the range of values for $k$;  
$(2)$ If quadrilateral $ABCD$ is a trapezoid, find the value of $k$.

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