
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 triangle $ABC$, $AB = 24$, $BC = 48$, and $\angle B = 30^\circ$. Let's consider a point $P$ on the plane such that when $A, B,$ and $C$ are folded onto $P$, the creases do not intersect inside the triangle. Determine the area of the region where $P$ can be placed, expressed 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. Find $q + r + s$.

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