
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 rectangle $ABCD$, $P$ is a point on $BC$ so that $\angle APD=90^{\circ}$. $TS$ is perpendicular to $BC$ with $BP = PT = PS$, where $S$ is on the line $TS$. $PD$ intersects $TS$ at $Q$.  Point $R$ is on $CD$ such that $RA$ passes through $Q$.  In $\triangle PQA$, $PA=24$, $AQ=18$ and $QP=30$. Find the lengths of $BP$ and $QT$.

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