
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
Triangle $DEF$ has side lengths $DE = 15$, $EF = 39$, and $FD = 36$. Trapezoid $WXYZ$ has vertex $W$ on $\overline{DE}$, vertex $X$ on $\overline{DF}$, and vertices $Y$ and $Z$ on $\overline{EF}$. Base $WZ$ of the trapezoid is parallel to $XY$. If the length of $WZ = \omega$, express the area of $WXYZ$ as a quadratic polynomial \[Area(WXYZ) = \gamma \omega - \delta \omega^2.\] Find $\delta$ expressed as the fraction $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, and determine $p+q$.

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