The results of a search for Higgs boson pair ($\textrm{HH}$) production in the $\textrm{WWWW}$, $\textrm{WW}\tau\tau$, and $\tau\tau\tau\tau$ decay modes are presented. The search uses proton-proton collision data recorded by the CMS experiment at a center-of-mass energy of 13~\mathrm{TeV}, corresponding to an integrated luminosity of 138~\mathrm{fb}^{-1}. The analyzed events contain two, three, or four reconstructed leptons, including electrons, muons, and hadronically decaying tau leptons. No evidence for a signal is found in the data, and upper limits are set on the cross section for nonresonant $\textrm{HH}$ production, as well as resonant production in which a new heavy particle decays to a pair of Higgs bosons. For nonresonant $\textrm{HH}$ production, the observed (expected) upper limit on the cross section at $95\%$ confidence level (CL) is $21.8$ ($19.6$) times the standard model (SM) prediction. The ratio of the trilinear Higgs boson self-coupling to its value in the SM is constrained to be between $-7.0$ and $11.2$ at $95\%$ CL, and limits are set on a variety of new-physics models using an effective field theory approach. The observed (expected) limits on the cross section for resonant $\textrm{HH}$ production range from $0.18$ to $0.90$ ($0.08$ to $1.07$)~\mathrm{pb} at $95\%$ CL for new heavy-particle masses from $250$ to 1000~\mathrm{GeV}.

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