Panels (a)–(d): time evolution of the 3D separation between the MW and M31 for two-, three-, and four-body configurations. Each panel shows 100 representative tracks randomly drawn from the 10,000 MC realizations. The MW–M31 merger fraction within 10 Gyr (f_m), computed from all 10,000 samples, is indicated in the lower-left corner. White symbols along the bottom mark the corresponding merger times, and the white curves show the evolution integrated using the central values of the fiducial priors for all 20 observables listed in Table 1. Panels (e) and (f): similar to panels (a)–(d), but showing the LMC–MW and M33–M31 separations in the fiducial four-body model. The merger fractions of each satellite with its host, before and after the MW–M31 merger, are indicated in white and red text, respectively. The corresponding merger times are marked by white and red symbols along the bottom axis of each panel.