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Abstract A search for$${\text {Z}{}{}} {\text {Z}{}{}} $$ $\text{Z}\text{Z}$and$${\text {Z}{}{}} {\text {H}{}{}} $$ $\text{Z}\text{H}$production in the$${\text {b}{}{}} {\bar{{\text {b}{}{}}}{}{}} {\text {b}{}{}} {\bar{{\text {b}{}{}}}{}{}} $$ $\text{b}\overline{\text{b}}\text{b}\overline{\text{b}}$final state is presented, where H is the standard model (SM) Higgs boson. The search uses an event sample of proton-proton collisions corresponding to an integrated luminosity of 133$$\,\text {fb}^{-1}$$ ${\text{fb}}^{-1}$collected at a center-of-mass energy of 13$$\,\text {Te}\hspace{-.08em}\text {V}$$ $\text{Te}\text{V}$with the CMS detector at the CERN LHC. The analysis introduces several novel techniques for deriving and validating a multi-dimensional background model based on control samples in data. A multiclass multivariate classifier customized for the$${\text {b}{}{}} {\bar{{\text {b}{}{}}}{}{}} {\text {b}{}{}} {\bar{{\text {b}{}{}}}{}{}} $$ $\text{b}\overline{\text{b}}\text{b}\overline{\text{b}}$final state is developed to derive the background model and extract the signal. The data are found to be consistent, within uncertainties, with the SM predictions. The observed (expected) upper limits at 95% confidence level are found to be 3.8 (3.8) and 5.0 (2.9) times the SM prediction for the$${\text {Z}{}{}} {\text {Z}{}{}} $$ $\text{Z}\text{Z}$and$${\text {Z}{}{}} {\text {H}{}{}} $$ $\text{Z}\text{H}$production cross sections, respectively.