
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 diagram, $ABCD$ is a square with side length $8$, and $WXYZ$ is a rectangle with $ZY = 12$ and $XY = 8$. Also, $AD$ and $WX$ are perpendicular. If the shaded area is equal to half of the area of $WXYZ$, what is the length of $AP$?

[asy]
draw((0,0)--(12,0)--(12,8)--(0,8)--cycle,black+linewidth(1));
draw((2,1)--(10,1)--(10,9)--(2,9)--cycle,black+linewidth(1));
filldraw((2,1)--(10,1)--(10,8)--(2,8)--cycle,gray,black+linewidth(1));
label("$W$",(0,8),NW);
label("$X$",(12,8),NE);
label("$Y$",(12,0),SE);
label("$Z$",(0,0),SW);
label("$A$",(2,9),NW);
label("$B$",(10,9),NE);
label("$C$",(10,1),SE);
label("$D$",(2,1),SW);
label("$P$",(2,8),W);
label("8",(2,9)--(10,9),N);
label("8",(12,0)--(12,8),E);
label("12",(0,0)--(12,0),S);
[/asy]

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