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A search for the production of a single top quark in association with invisible particles is performed using proton-proton collision data collected with the CMS detector at the LHC at TeV, corresponding to an integrated luminosity of 138 fb−1. In this search, a flavor-changing neutral current produces a single top quark or antiquark and an invisible state nonresonantly. The invisible state consists of a hypothetical spin-1 particle acting as a new mediator and decaying to two spin-1/2 dark matter candidates. The analysis searches for events in which the top quark or antiquark decays hadronically. No significant excess of events compatible with that signature is observed. Exclusion limits at 95% confidence level are placed on the masses of the spin-1 mediator and the dark matter candidates, and are compared to constraints from the dark matter relic density measurements. In a vector (axial-vector) coupling scenario, masses of the spin-1 mediator are excluded up to 1.85 (1.85) TeV with an expectation of 2.0 (2.0) TeV, whereas masses of the dark matter candidates are excluded up to 0.75 (0.55) TeV with an expectation of 0.85 (0.65) TeV.