Effect of the M33-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 R. P. van der Marel et al. 2012a, used in R. P. van der Marel et al. 2012b’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 M33. In the R. P. van der Marel et al. (2012a)–like cases, the M33-induced reflex motion makes the trajectory less radial.