
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
Suppose in a triangle $ABC$, where $AB = 24$, $BC = 48$, and $\angle C = 90^\circ$, a point $P$ on the triangle serves as a fold point if, when $A, B,$ and $C$ are folded onto $P$, the resulting creases do not intersect inside the triangle. Calculate the area of all possible positions for $P$ in the triangle, if expressible in the form $q\pi - r\sqrt{s}$ where $q, r,$ and $s$ 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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