A diffuse flux of astrophysical neutrinos above $100,mathrm{TeV}$ has been observed at the IceCube Neutrino Observatory. Here we extend this analysis to probe the astrophysical flux down to $35,mathrm{TeV}$ and analyze its flavor composition by classifying events as showers or tracks. Taking advantage of lower atmospheric backgrounds for shower-like events, we obtain a shower-biased sample containing 129 showers and 8 tracks collected in three years from 2010 to 2013. We demonstrate consistency with the $(f_e:f_{mu}:f_tau)_oplusapprox(1:1:1)_oplus$ flavor ratio at Earth commonly expected from the averaged oscillations of neutrinos produced by pion decay in distant astrophysical sources. Limits are placed on non-standard flavor compositions that cannot be produced by averaged neutrino oscillations but could arise in exotic physics scenarios. A maximally track-like composition of $(0:1:0)_oplus$ is excluded at $3.3sigma$, and a purely shower-like composition of $(1:0:0)_oplus$ is excluded at $2.3sigma$.
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