
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 $ABC$ has an area of 144 square inches and is cut by $\overline{DE}$ parallel to the base $BC$, resulting in an isosceles trapezoid and a smaller isosceles triangle above it. If the area of the trapezoid is 96 square inches, and the altitude of triangle $ABC$ from vertex $A$ (to base $BC$) is 24 inches, find the length of $\overline{DE}$.

[asy]
draw((-12,0)--(0,28)--(12,0)--cycle);
draw((-8,9)--(8,9));
label("$A$",(0,28),N);
label("$B$",(-12,0),W);
label("$D$",(-8,9),NW);
label("$E$",(8,9),NE);
label("$C$",(12,0),E);
[/asy]

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