
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 $WZ=12$ and $XY=8.$ Also, $AD$ and $WX$ are perpendicular. If the shaded area is equal to one-third 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((2,2)--(10,2)--(10,10)--(2,10)--cycle,black+linewidth(1));
filldraw((2,2)--(10,2)--(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("$A$",(2,10),NW);
label("$B$",(10,10),NE);
label("$C$",(10,2),E);
label("$D$",(2,2),W);
label("$P$",(2,8),SW);
label("8",(2,10)--(10,10),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