
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 $XYZ$ of area 180 square inches is cut by $\overline{PQ}$ into an isosceles trapezoid and a smaller isosceles triangle. The area of the trapezoid is 135 square inches. If the altitude of triangle $XYZ$ from $X$ is 30 inches, determine the length of $\overline{PQ}$.

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