
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 $P$ on a triangular piece of paper $ABC$, suppose that $AB = 48$, $AC = 96$, and $\angle B = 90^\circ$. Consider that $P$ is a fold point on $\triangle ABC$ and additionally lies on the altitude from $B$ to $AC$. The creases formed when vertices $A, B$, and $C$ are folded onto $P$ do not intersect at the fold point. If these conditions are met, what is the area of the set of all possible fold points of $\triangle ABC$, and can it be expressed as $q\pi - r\sqrt{s}$, where $q, r,$ and $s$ are positive integers, with $s$ having no prime square factors?

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