
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 $xOy$, the following definitions are given: the smaller value of the distances from point $A$ to the $x$-axis and the $y$-axis is called the "short distance" of point $A". When the "short distance" of point $P$ is equal to the "short distance" of point $Q$, points $P$ and $Q$ are called "equidistant points".  
$(1)$ The "short distance" of point $A(-5, -2)$ is ______;  
$(2)$ The "short distance" of point $B(-2, -2m+1)$ is $1$, find the value of $m$;  
$(3)$ If points $C(-1, k+3)$ and $D(4, 2k-3)$ are "equidistant points", find the value of $k$.

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