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The Askaryan Radio Array (ARA) is an ultrahigh energy (UHE, >1017 eV) neutrino detector designed to observe neutrinos by searching for the radio waves emitted by the relativistic products of neutrino-nucleon interactions in Antarctic ice. In this paper, we present constraints on the diffuse flux of ultrahigh energy neutrinos between 1016 and 1021 eV resulting from a search for neutrinos in two complementary analyses, both analyzing four years of data (2013-2016) from the two deep stations (A2, A3) operating at that time. We place a 90% CL upper limit on the diffuse all flavor neutrino flux at 1018 eV of EF(E)=5.6×10-16 cm-2 s-1 sr-1. This analysis includes four times the exposure of the previous ARA result and represents approximately 1/5th the exposure expected from operating ARA until the end of 2022.