
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 a right triangular piece of paper $ABC$ with $AB = 45, AC = 90$, and $\angle B = 90^\circ$. Determine the area of the set of all fold points of $\triangle ABC$, assuming that the creases formed when $A, B$, and $C$ are folded onto point $P$ do not intersect unless $P$ is one of the vertices. Express the area in the form $q\pi - r\sqrt{s}$, where $q, r$, and $s$ are integers, with $s$ not divisible by the square of any prime. What is $q+r+s$?

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