We present time-series $BVI$ photometry for the Galactic globular cluster NGC 6402 (M14). The data consists of $sim$137 images per filter, obtained using the 0.9m and 1.0m SMARTS telescopes at the Cerro Tololo Inter-American Observatory. The images were obtained during two observing runs in 2006-2007. The image-subtraction package ISIS, along with DAOPHOT II/ALLFRAME, were used to perform crowded-field photometry and search for variable stars. We identified 130 variables, 8 of which are new discoveries. The variable star population is comprised of 56 ab-type RR Lyrae stars, 54 c-type RR Lyrae, 6 type II Cepheids, 1 W UMa star, 1 detached eclipsing binary, and 12 long-period variables. We provide Fourier decomposition parameters for the RR Lyrae, and discuss the physical parameters and photometric metallicity derived therefrom. The M14 distance modulus is also discussed, based on different approaches for the calibration of the absolute magnitudes of RR Lyrae stars. The possible presence of second-overtone RR Lyrae in M14 is critically addressed, with our results arguing against this possibility. By considering all of the RR Lyrae stars as members of the cluster, we derive $langle P_{ab}rangle = 0.589$ d. This, together with the position of the RR Lyrae stars of both Bailey types in the period-amplitude diagram, suggests an Oosterhoff-intermediate classification for the cluster. Such an intermediate Oosterhoff type is much more commonly found in nearby extragalactic systems, and we critically discuss several other possible indications that may point to an extragalactic origin for this cluster.
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