
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 one-third of the area of $WXYZ,$ what is the length of $AP?$

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