For an arbitrary point $M$ inside a given square $ABCD$, let $T_1,T_2,T_3$ be the centroids of triangles $ABM,BCM$, and $DAM$, respectively. Let $OM$ be the circumcenter of triangle $T_1T_2T_3$. Find the locus of points $OM$ when $M$ takes all positions within the interior of the square.