
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 paragraph first, then answer the questions. Given two points $P_{1}(x_{1}, y_{1})$, $P_{2}(x_{2}, y_{2})$ 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 on the coordinate axes, parallel to the coordinate axes, or perpendicular to 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 points $A(2,3)$ and $B(4,2)$, find the distance between $A$ and $B$.  
$(2)$ Given points $A$ and $B$ lie on a line parallel to the $x$-axis, where the $x$-coordinate of point $A$ is $7$ and the $x$-coordinate of point $B$ is $-5$, find the distance between $A$ and $B$.  
$(3)$ Given the coordinates of the vertices of a triangle are $A(1,4)$, $B(1,-4)$, $C(1-a,5)$, express the area $S$ of $\triangle ABC$ using an expression containing $a$.

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