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Solution #1

Body / surface

Here, we add one additional body/surface.

Regions / cells

OUTSIDE, zone #1

OUTSIDE, zone #2

OUTSIDE, zone #3

INSIDE, zone #1

INSIDE, zone #2

PIPE zone #1

PIPE zone #2

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Combinatorial Geometry / Constructive Solid Geometry (CSG)
Second attempt
Combinatorial Geometry / Constructive Solid Geometry (CSG)
Solution #2