
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
Rectangle $PQRS$ is inscribed in triangle $XYZ$ such that side $PQ$ of the rectangle is on side $XZ$ of the triangle. The triangle's altitude from $Y$ to side $XZ$ is 8 inches, and $XZ = 12 \text{ inches}$. The length of segment $PQ$ is equal to one-third the length of segment $PS$. What is the area of rectangle $PQRS$?

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