
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, the line segment $AB$ is translated to obtain the line segment ${A'}{B'}$.  
$(1)$ If the coordinates of points $A$, $B$, and ${A'}$ are $A(-2,-1)$, $B(1,-3)$, and ${A'}(2,3)$, respectively, write down the coordinates of point ${B'}$ directly.  
$(2)$ Given that the coordinates of points $A$, $B$, ${A'}$, and ${B'}$ are $A(m,n)$, $B(2n,m)$, ${A'}(3m,n)$, and ${B'}(6n,m)$, what kind of quantitative relationship exists between $m$ and $n$? Explain the reason.  
$(3)$ Given that the coordinates of points $A$, $B$, ${A'}$, and ${B'}$ are $A(m,n+1)$, $B(n-1,n-2)$, ${A'}(2n-5,2m+3)$, and ${B'}(2m+3,n+3)$, find the coordinates of points $A$ and $B$.

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