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A measurement is presented of the cross section in proton-proton collisions for the production of two $W$bosons and one $Z$boson. It is based on data recorded by the CMS experiment at the CERN LHC at center-of-mass energies $\sqrt{s}=13$and 13.6 TeV, corresponding to an integrated luminosity of $200{\mathrm{fb}}^{-1}$. Events with four charged leptons (electrons or muons) in the final state are selected. Both nonresonant $WWZ$production and $ZH$production, with the Higgs boson decaying into two $W$bosons, are reported. For the first time, the two processes are measured separately in a simultaneous fit. Combining the two modes, signal strengths relative to the standard model (SM) predictions of ${0.75}_{-0.29}^{+0.34}$and ${1.74}_{-0.60}^{+0.71}$are measured for $\sqrt{s}=13$and 13.6 TeV, respectively. The observed (expected) significance for the triboson signal is 3.8 (2.5) standard deviations for $\sqrt{s}=13.6\mathrm{TeV}$, thus providing the first evidence for triboson production at this center-of-mass energy. Combining the two modes and the two center-of-mass energies, the inclusive signal strength relative to the SM prediction is measured to be ${1.03}_{-0.28}^{+0.31}$, with an observed (expected) significance of 4.5 (5.0) standard deviations.