2021 MPFG Problem 19

Problem

Let $T$ be a regular tetrahedron. Let $t$ be the regular tetrahedron whose vertices are the centers of the faces of $T$ . Let $O$ be the circumcenter of either tetrahedron. Given a point $P$ different from $O$ , let $m(P)$ be the midpoint of the points of intersection of the ray $\overrightarrow{OP}$ with $t$ and $T$ . Let $S$ be the set of eight points m(P) where P is a vertex of either $t$ or $T$ . What is the volume of the convex hull of $S$ divided by the volume of $t$ ? Express your answer as a fraction in simplest form.