
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 $\triangle ABC$ is a triangle where $AB = 36, AC = 36$, and $\angle B = 60^\circ$. A point $P$ is considered a fold point if the creases formed when vertices $A, B,$ and $C$ are folded onto point $P$ do not intersect inside $\triangle ABC$. Find the area of the set of all fold points of $\triangle ABC$, given in the form $q\pi-r\sqrt{s}$, where $q, r,$ and $s$ 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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