
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 convex quadrilateral $PQRS$, side $\overline{RS}$ is perpendicular to diagonal $\overline{PQ}$, and side $\overline{PQ}$ is perpendicular to diagonal $\overline{RS}$. Side lengths are $RS = 75$, and $PQ = 36$. A line through $Q$ perpendicular to side $\overline{PR}$ intersects diagonal $\overline{PQ}$ at a point $T$ such that $PT = 12$. Find the length of segment $QT$.

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