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An empty \(n\times n\times n\) cube is given, and a \(n\times n\) grid of square unit cells is drawn on each of its six faces. A beam is a \(1 \times 1 \times n\) rectangular prism.
Several beams are placed inside the cube subject to the following conditions:
  • The two \(1 \times 1\) faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are \(3n\) possible positions for a beam.)
  • No two beams have intersecting interiors.
  • The interiors of each of the four \(1 \times n\) faces of each beam touch either a face of the cube or the interior of the face of another beam.
What is the smallest positive number of beams that can be placed to satisfy these conditions?

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