
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 trapezoid $ABCD$ with $AB \parallel CD$, $P$ is a point on $AB$ such that $\angle APD=90^\circ$. $TS$ is perpendicular to $AB$ with $AP=PT$, and $PD$ intersects $TS$ at $Q$. Point $R$ is on $CD$ such that $RA$ passes through $Q$. In $\triangle PQA$, $PA=24$, $AQ=30$, and $QP=18$. Find the length of $SD$. Assume that $TQ$ and $TS$ remain perpendicular to the bases of the trapezoid.

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