
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 $ABC$ with hypotenuse $AB = 36$ units, where $\angle C = 90^\circ$. Define a fold point $P$ such that when $A, B,$ and $C$ are folded onto $P$, the resultant creases do not overlap. Calculate the area of the set of all such fold points for $\triangle ABC$. Express your answer as $q \pi - r \sqrt{s}$, where $q, r, s$ are integers, and $s$ is 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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