
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
Read the following passage first, then answer the questions that follow. Given two points $P_{1}(x_{1}, y_{1})$ and $P_{2}(x_{2}, y_{2})$ in a plane, the distance between these two points is $P_{1}P_{2} = \sqrt{(x_{2}-x_{1})^{2} + (y_{2}-y_{1})^{2}$. Also, when the line connecting the two points is parallel to the coordinate axes or coincides with the coordinate axes, the distance formula between the two points can be simplified to $|x_{2}-x_{1}|$ or $|y_{2}-y_{1}|$.  
$(1)$ Given $A(2,4)$ and $B(-3,-8)$, find the distance between points $A$ and $B$;  
$(2)$ Given that $A$ and $B$ lie on a line parallel to the $y$-axis, with the ordinate of point $A$ being $5$ and the ordinate of point $B$ being $-1$, find the distance between points $A$ and $B$.

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