
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
Consider an isosceles right triangle $ABC$, where $AB = AC = 24$ and $\angle BAC = 90^\circ$. If a point $P$ within the triangle is such that when $A$, $B$, and $C$ are folded onto $P$, the resulting creases (which are three unless $P$ coincides with a vertex) do not intersect within the triangle. Calculate the area of the set of all such points $P$ in the triangle, expressed as $q \pi - r \sqrt{s}$, where $q, r, s$ are positive integers and $s$ is not divisible by the square of any prime. Determine $q + r + s$.

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