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The standard model (SM) production of four top quarks ($\mathrm{t\bar{t}t\bar{t}}$) in proton-proton collision is studied by the CMS Collaboration. The data sample, collected during the 2016-2018 data taking of the LHC, corresponds to an integrated luminosity of 137 fb$^{-a}$ at a center-of-mass energy of 13 TeV. The events are required to contain two same-sign charged leptons (electrons or muons) or at least three leptons, and jets. The observed and expected significances for the $\mathrm{t\bar{t}t\bar{t}}$ signal are respectively 2.6 and 2.7 standard deviations, and the $\mathrm{t\bar{t}t\bar{t}}$ cross section is measured to be 12.6$^{+5.8}_{-5.2}$ fb. The results are used to constrain the Yukawa coupling of the top quark to the Higgs boson, $y_{\mathrm{t}}$, yielding a limit of $|{y_{\mathrm{t}}/y_{\mathrm{t}}^{\mathrm{SM}}}| <$ 1.7 at 95% confidence level, where $y_{\mathrm{t}}^{\mathrm{SM}}$ is the SM value of $y_{\mathrm{t}}$. They are also used to constrain the oblique parameter of the Higgs boson in an effective field theory framework, $\hat{H}<0.12$. Limits are set on the production of a heavy scalar or pseudoscalar boson in Type-II two-Higgs-doublet and simplified dark matter models, with exclusion limits reaching 350-470 GeV and 350-550 GeV for scalar and pseudoscalar bosons, respectively. Upper bounds are also set on couplings of the top quark to new light particles.