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We present optical and infrared (IR) light curves of the enshrouded massive binary NaSt1 (WR 122) with observations from Palomar Gattini-IR (PGIR), the Zwicky Transient Facility (ZTF), the Katzman Automatic Imaging Telescope (KAIT), and the All-Sky Automated Survey for Supernovae (ASAS-SN). The optical and IR light curves span between 2014 July and 2020 Oct., revealing periodic, sinusoidal variability from NaSt1 with a P=305.2±1.0 d period. We also present historical IR light curves taken between 1983 July and 1989 May that also indicate NaSt1 exhibits long-term IR variability on timescales of ∼decades. Fixed-period sinusoidal fits to the recent optical and IR light curves show that amplitude of NaSt1's variability is different at different wavelengths and also reveal significant phase offsets of ∼18 d between the ZTF r and PGIR J light curves.We interpret the ∼300 d period of the observed variability as the orbital period of a binary system in NaSt1. Assuming a circular orbit and adopting a range of combined stellar mass values in the range 20-100 M⊙ in NaSt1, we estimate orbital separations of ∼2-4 au. We suggest that the sinusoidal photometric variability of NaSt1 may arise from variations in the line-of-sight optical depth toward circumstellar optical/IR emitting regions throughout its orbit due to colliding-wind dust formation. We provide an interpretation on the nature of NaSt1 and speculate that the mass-transfer process may have been triggered by Roche-lobe overflow (RLOF) during an eruptive phase of a Ofpe/WN9 star. Lastly, we claim that NaSt1 ceased RLOF mass transfer ≲3400 yr ago.