
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
If the graph of the function ${y}_{1}=\frac{k}{x}(k≠0)$ intersects the graph of $y_{2}=3x+k$ at a point $A$ with an x-coordinate of $-2$.  
$(1)$ Find the value of $k$;  
$(2)$ If the coordinates of another intersection point $B$ of the graphs of $y_{1}$ and $y_{2}$ are $\left(m,n\right)$, find the value of $\frac{3}{n}-\frac{1}{m}$.

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