Given a regular tetrahedron with edge $a$, its edges are divided into $n$ equal segments, thus obtaining $n + 1$ points: $2$ at the ends and $n - 1$ inside. The following set of planes is considered:
- Those that contain the faces of the tetrahedron, and
- Each of the planes parallel to a face of the tetrahedron and containing at least one of the points determined above.

Now all those points $P$ that belong (simultaneously) to four planes of that set are considered. Determine the smallest positive natural $n$ so that among those points $P$ the eight vertices of a square-based rectangular parallelepiped can be chosen.

Let's think step by step and output the final answer within \boxed{}.
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