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The first search for supersymmetry in events with an experimental signature of one soft, hadronically decaying $\tau$ lepton, one energetic jet from initial-state radiation, and large transverse momentum imbalance is presented. These event signatures are consistent with direct or indirect production of scalar $\tau$ leptons ($\tilde{\tau}$) in supersymmetric models that exhibit coannihilation between the $\tilde{\tau}$ and the lightest neutralino ($\tilde{\chi}^0_1$), and that could generate the observed relic density of dark matter. The data correspond to an integrated luminosity of 77.2 fb$^{-1}$ of proton-proton collisions at $\sqrt{s}=$ 13 TeV collected with the CMS detector at the LHC in 2016 and 2017. The results are interpreted in a supersymmetric scenario with a small mass difference ($\Delta m$) between the chargino ($\tilde{\chi}^\pm_1$) or next-to-lightest neutralino ($\tilde{\chi}^0_2$), and the $\tilde{\chi}^0_1$. The mass of the $\tilde{\tau}$ is assumed to be the average of the $\tilde{\chi}^\pm_1$ and $\tilde{\chi}^0_1$ masses. The data are consistent with standard model background predictions. Upper limits at 95% confidence level are set on the sum of the $\tilde{\chi}^\pm_1$, $\tilde{\chi}^0_2$, and $\tilde{\tau}$ production cross sections for $\Delta m(\tilde{\chi}^\pm_1, \tilde{\chi}^0_1) =$ 50 GeV, resulting in a lower limit of 290 GeV on the mass of the $\tilde{\chi}^\pm_1$, which is the most stringent to date and surpasses the bounds from the LEP experiments.