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LS 5039 is one of a handful of γ -ray binary systems in the milky Way, comprising a pulsar and a massive O-type companion star with an orbital period of 3.9 days. Recently, we conducted a data analysis using approximately 16 yr of Fermi Large Area Telescope observations, spanning from 2008 August 4 to 2024 July 8. In our timing analysis, we discovered two new periodic signals with frequencies higher and lower than the known orbital period. The higher-frequency signal has a period of 3.63819 days with a 7.1 σ confidence level, while the lower-frequency signal has a period of 4.21654 days with a 6.3 σ confidence level. Additionally, in data from the High Energy Stereoscopic System of Cherenkov telescopes, two potential signals are found with periods similar to the two newly discovered ones. Considering that these two signals fall within the same frequency interval as the orbital period, we suggest the possibility of a third body orbiting the barycenter of the LS 5039 binary system, with the new periodic signals arising from specific frequency combinations of the two orbital periods. [ABSTRACT FROM AUTHOR]