A search for long-lived particles using final states including a displaced vertex with low-momentum tracks, large missing transverse momentum, and a jet from initial-state radiation is presented. This search uses proton-proton collision data at a center-of-mass energy of 13 TeV collected by the CMS experiment at the CERN LHC in 2017 and 2018, with a total integrated luminosity of 100 fb$^{-1}$. This analysis adopts specific supersymmetric (SUSY) coannihilation scenarios as benchmark signal models, characterized by a next-to-lightest SUSY particle (NLSP) with a mass difference of less than 25GeV relative to the lightest SUSY particle, assumed to be a bino-like neutralino. In the top squark ($\tilde{\mathrm{t}}$) NLSP model, the NLSP is a long-lived $\tilde{\mathrm{t}}$, while in the bino-wino NLSP scenario, the mass-degenerate NLSPs are a wino-like long-lived neutralino and a short-lived chargino. The search excludes top squarks with masses less than 400$-$1100 GeV and wino-like neutralinos with masses less than 220$-$550 GeV, depending on the signal parameters, including the mass difference, mass, and lifetime of the long-lived particle. It sets the most stringent limits to date for the $\tilde{\mathrm{t}}$ and bino-wino NLSP models.
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