
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
We call a point where the ordinate is twice the abscissa a bilingual point, such as point $A(2,4)$, point $B(-3,-6)$.  
$(1)$ The bilingual point of the function $y=3x+1$ is ______;  
$(2)$ Does the function $y=\frac{k}{x}$ (where $k$ is a constant and $k\neq 0$) have a bilingual point? If so, find the coordinates of the bilingual point. If not, please explain why;  
$(3)$ The graph of the function $y=\frac{1}{4}x^2+(n-k-1)x+m+k+2$ has only one bilingual point, and when $1\leqslant n\leqslant 3$, the minimum value of $m$ is $k$. Find the value of the real number $k$.

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