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We analyze photometric data of the active RS CVn-type star HR 1099 for the years 1975-2006 with an inversion technique and reveal the nature of two activity cycles of 15-16 yr and 5.3 ± 0.1 yr duration. The 16 yr cycle is related to variations of the total spot area and is coupled with the differential rotation, while the 5.3 yr cycle is caused by the symmetric redistribution of the spotted area between the opposite stellar hemispheres (flip-flop cycle). We recover long-lived active regions comprising two active longitudes that migrate in the orbital reference frame with a variable rate because of the differential rotation along with changes in the mean spot latitudes. The migration pattern is periodic with the 16 yr cycle. Combining the longitudinal migration of the active regions with a previously measured differential rotation law, we recover the first stellar butterfly diagram without an assumption about spot shapes. We find that mean latitudes of active regions at opposite longitudes change antisymmetrically in the course of the 16 yr cycle: while one active region migrates to the pole, the other approaches the equator. This suggests a precession of the global magnetic field with respect to the stellar rotational axis.