
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 rectangular piece of paper $ABCD$ with $AB = 48$, $BC = 96$, and $\angle B = 90^\circ$. Determine the area of all possible locations for $P$ such that when $A$, $B$, $C$, and $D$ are folded onto $P$, the resulting creases, which number four unless $P$ is one of the vertices, do not intersect within the rectangle. Express the area in the form $k\pi - m\sqrt{n}$, where $k, m, n$ are positive integers and $n$ is not divisible by the square of any prime.

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