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How to quickly determine if A – (A ∩ (ΣBi)) set exists and pick a random value from it?

I am trying to make my code randomly and quickly find a point (x,y) in the A disc and that is not part of any other present discs, if that point does exist. However, the only method I have found is a long and labourous method since it randomly picks a point and checks if it corresponds to or not. However, in the event where over 99% of the A disc is intersecting with other Bi discs, it evidently is near impossible.

How to quickly determine if A – (A ∩ (ΣBi)) set exists and all values it’s comprised of?

I am trying to make my code randomly and quickly find a point (x,y) in the A disc and that is not part of any other present discs, if that point does exist. However, the only method I have found is a long and labourous method since it randomly picks a point and checks if it corresponds to or not. However, in the event where over 99% of the A disc is intersecting with other Bi discs, it evidently is near impossible.