
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
Given a point $P$ within triangle $DEF$ where $DE = 24$, $DF = 48$, and $\angle E = 90^\circ$, find the area of the set of all fold points of $\triangle DEF$. Fold points are defined as those points for which the creases formed by folding $D$, $E$, and $F$ onto $P$ do not intersect within the triangle. Express the area of this set as $q\pi - r\sqrt{s}$, where $q, r, s$ are positive integers and $s$ is not divisible by the square of any prime.

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