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The rate for Higgs (H) bosons production in association with either one (tH) or two ($\mathrm{t\bar{t}}$H) top quarks is measured in final states containing multiple electrons, muons, or tau leptons decaying to hadrons and a neutrino, using proton-proton collisions recorded at a center-of-mass energy of 13 TeV by the CMS experiment. The analyzed data correspond to an integrated luminosity of 137 fb$^{-1}$. The analysis is aimed at events that contain H $\to$ WW, H $\to$ $ττ$, or H $\to$ ZZ decays and each of the top quark(s) decays either to lepton+jets or all-jet channels. Sensitivity to signal is maximized by including ten signatures in the analysis, depending on the lepton multiplicity. The separation among the tH, the $\mathrm{t\bar{t}}$H, and the backgrounds is enhanced through machine-learning techniques and matrix-element methods. The measured production rates for the $\mathrm{t\bar{t}}$H and tH signals correspond to 0.92 $\pm$ 0.19 (stat) $^{+0.17}_{-0.13}$ (syst) and 5.7 $\pm$ 2.7 (stat) $\pm$ 3.0 (syst) of their respective standard model (SM) expectations. The corresponding observed (expected) significance amounts to 4.7 (5.2) standard deviations for $\mathrm{t\bar{t}}$H, and to 1.4 (0.3) for tH production. Assuming that the Higgs boson coupling to the tau lepton is equal in strength to its expectation in the SM, the coupling $y_{\mathrm{t}}$ of the Higgs boson to the top quark divided by its SM expectation, $κ_\mathrm{t}$ = $y_\mathrm{t} / y_\mathrm{t}^\mathrm{SM}$, is constrained to be within $-$0.9 $\lt$ $κ_\mathrm{t}$ $\lt$ $-$0.7 or 0.7 $\lt$ $κ_\mathrm{t}$ $\lt$ 1.1, at 95% confidence level. This result is the most sensitive measurement of the $\mathrm{t\bar{t}}$H production rate to date.