
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 $LMNO$, $P$ is a point on $MN$ such that $\angle LPO=90^{\circ}$. $TS$ is perpendicular to $MN$ with $MP=PT$, as shown. $PO$ intersects $TS$ at $Q$. Point $R$ is on $NO$ such that $RA$ passes through $Q$. In $\triangle PQO$, $PO=24$, $OQ=20$ and $QP=16$. Find the lengths of $MP$ and $QT$.

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