Numerical orbital integration backward for M33 in the galactocentric frame centered on M31. The 3D galactocentric velocity of M31 is taken from this work. Orbits are integrated backward for 6 Gyr from the present-day position and velocity. Panels (a) and (b) show the orbit of M33 using its 3D velocity measured solely in this work, while panels (c) and (d) adopt the velocity from this work combined with the VLBA measurement. Similar to Patel et al. (2017) and van der Marel et al. (2019), we consider four mass combinations for M31 and M33, with two corresponding virial radii of M31: R_vir = 299 kpc (solid gray lines) for an M31 mass of 1.5 × 10¹² M_⊙, and R_vir = 329 kpc (dashed gray lines) for an M31 mass of 2 × 10¹² M_⊙. Panels (a) and (c) display the orbit of M33 projected onto the Y − Z plane, while panels (b) and (d) show the evolution of M33’s distance from M31 as a function of lookback time.