
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
Isosceles triangle $PQR$ has an area of 180 square inches and is divided by line $\overline{XY}$ into a smaller isosceles triangle and an isosceles trapezoid. The area of the trapezoid is 135 square inches. If the altitude of triangle $PQR$ from $P$ is 30 inches, determine the length of $\overline{XY}$.

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