
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 $DE = 15$, $EF = 36$, and $FD = 39$. Rectangle UVWX has vertex U on $\overline{DE}$, vertex V on $\overline{DF}$, and vertices W and X on $\overline{EF}$. In terms of the side length $UV = \omega$, the area of UVWX can be expressed as the quadratic polynomial \[ \text{Area}(UVWX) = \alpha \omega - \beta \omega^2. \]
Then the coefficient $\beta = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

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