Predicted orbital evolution of the Local Group (LG) galaxies. Panels (a) and (b) show 100 randomly selected tracks from 10,000 MC realizations for the two-body configuration (MW–M31), and panels (c) and (d) show the same for the four-body configuration (MW, M31, M33, and LMC). In each case, the left panel is the face-on projection and the right panel is the edge-on projection with respect to the MW–M31 orbital plane, defined by the initial positions and velocities of the MW and M31, following S25. The green and red lines trace the orbits of the MW and M31, while the pink and silver lines trace the LMC and M33, respectively. Line brightness indicates the relative probability density at any position. For each panel, the track of the satellites will terminate if a merger between them and the hosts is flagged. Additionally, if an MW–M31 merger is flagged, a white point marks the merger location and the corresponding track terminates; otherwise, tracks are integrated to 10 Gyr.