
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
In triangle $ABC$, where $AB = 45, AC = 81,$ and $\angle B = 90^\circ,$ a point $P$ inside the triangle is considered a fold point if the creases formed by folding $A, B,$ and $C$ onto $P$ do not intersect within the triangle. Determine the area of the region consisting of all such fold points in the triangle, and express it 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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