
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
A rectangle $PQRS$ is inscribed in triangle $XYZ$ such that side $PQ$ of the rectangle lies on side $XZ$ of the triangle, as shown. The altitude from vertex $Y$ to side $XZ$ is 9 cm, and $XZ = 15 \text{ cm}$. If the length of segment $PQ$ is one-third the length of segment $PS$, calculate the area of rectangle $PQRS$. Express your answer as a common fraction.

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