
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
An isosceles triangle $PQR$ has an area of 144 square inches. A horizontal line $\overline{ST}$ cuts $PQR$ into an isosceles trapezoid and a smaller isosceles triangle at the top. The area of the trapezoid is 108 square inches. If the altitude of triangle $PQR$ from $P$ is 24 inches, find the length of $\overline{ST}$.

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