
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 quadrilateral $PQRS$ with right angles at $Q$ and $R$, $\triangle PQR \sim \triangle QRS$, and $PQ > QR$. There is a point $T$ inside $PQRS$ such that $\triangle PQR \sim \triangle RTQ$ and the area of $\triangle PTS$ is $9$ times the area of $\triangle RTQ$. Determine $\frac{PQ}{QR}$.

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