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We present precision radial velocity (RV) data sets from Keck-HIRES and from Lick Observatory's new Automated Planet Finder Telescope and Levy Spectrometer on Mt. Hamilton that reveal a multiple-planet system orbiting the nearby, slightly evolved, K-type star HD 141399. Our 91 observations over 10.5 yr suggest the presence of four planets with orbital periods of 94.35, 202.08, 1070.35, and 3717.35 days and minimum masses of 0.46, 1.36, 1.22, and 0.69 MJ, respectively. The orbital eccentricities of the three inner planets are small, and the phase curves are well sampled. The inner two planets lie just outside the 2:1 resonance, suggesting that the system may have experienced dissipative evolution during the protoplanetary disk phase. The fourth companion is a Jupiter-like planet with a Jupiter-like orbital period. Its orbital eccentricity is consistent with zero, but more data will be required for an accurate eccentricity determination.