
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 triangle $PQR$, $PQ = QR$, and $\overline{PS}$ is an altitude. Point $T$ is on the extension of $\overline{PR}$ such that $ST = 12$. The values of $\tan \angle RST$, $\tan \angle PST$, and $\tan \angle QST$ form a geometric progression, and the values of $\cot \angle PST$, $\cot \angle RST$, $\cot \angle PSR$ form an arithmetic progression. Find the area of triangle $PQR$.

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