
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
In acute triangle $DEF$, points $R$ and $S$ are the feet of the perpendiculars from $F$ to $\overline{DE}$ and from $E$ to $\overline{DF}$, respectively. Line $RS$ intersects the circumcircle of $\triangle DEF$ in two distinct points, $Z$ and $W$. Suppose $ZR=15$, $RS=30$, and $SW=20$. Find the product $DE\cdot DF$ in the form $p\sqrt q$, where $p$ and $q$ are positive integers, and $q$ is not divisible by the square of any prime. Find $p+q$.

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