
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
In the Cartesian coordinate system, we call points like $\left(1,-1\right)$, $\left(-2.1,2.1\right)$, where the ordinate and abscissa are opposite numbers, "complementary points."  
$(1)$ The coordinates of the "complementary point" on the line $y=2x-3$ are ______;  
$(2)$ Does the line $y=kx+2\left(k\neq 0\right)$ have "complementary points"? If so, find the coordinates of the points; if not, please explain why;  
$(3)$ If there exists a unique "complementary point" on the graph of the function $y=\frac{1}{4}x^{2}+\left(n-k-1\right)x+m+k-2$, and when $-1\leqslant n\leqslant 2$, the minimum value of $m$ is $k$, find the value of $k$.

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