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 √𝑠 =13 and 13.6 TeV, corresponding to an integrated luminosity of 200  fb−1. Events with four charged leptons (electrons or muons) in the final state are selected. Both nonresonant 𝑊⁢𝑊⁢𝑍 production and 𝑍⁢𝐻 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+0.34−0.29 and 1.74+0.71−0.60 are measured for √𝑠 =13 and 13.6 TeV, respectively. The observed (expected) significance for the triboson signal is 3.8 (2.5) standard deviations for √𝑠=13.6  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+0.31−0.28, with an observed (expected) significance of 4.5 (5.0) standard deviations.