Effect of the LMC-induced reflex motion on the MW–M31 orbital tracks for two representative M31 PMs (top: values from H. Wu et al. 2025, used in our fiducial model; bottom: values from J. B. Salomon et al. 2021, used in S25’s fiducial model). Tracks are projected face-on (left) and edge-on (right) relative to the MW–M31 orbital plane, which is defined by the initial positions and velocities of the MW and M31. Curves compare the two-body configuration (MW and M31 only) with the corresponding three-body case including the LMC. In the H. Wu et al. (2025)–like cases, the LMC-induced reflex motion makes the trajectory more radial and promotes the MW-M31 merger, whereas in the J. B. Salomon et al. (2021)–like case, it increases the tangential velocity component and suppresses the merger.