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Observation of $D^+\toηηπ^+$ and improved measurement of $D^{0(+)}\toηπ^+π^{-(0)}$

Using an $e^+e^-$ annihilation data sample corresponding to an integrated luminosity of $2.93\,\rm fb^{-1}$ collected at the center-of-mass energy of 3.773\,GeV with the BESIII detector, we measure the absolute branching fractions of $D^+\toηηπ^+$, $D^+\toηπ^+π^0$, and $D^0\toηπ^+π^-$ to be $(2.96 \pm 0.24 \pm 0.13)\times 10^{-3}$, $(2.23 \pm 0.15 \pm 0.11)\times 10^{-3}$, and $(1.20 \pm 0.07 \pm 0.04)\times 10^{-3}$, respectively, where the first uncertainties are statistical and the second ones systematic. The $D^+\toηηπ^+$ decay is observed for the first time and the branching fractions of $D^{+(0)}\toηπ^+π^{0(-)}$ are measured with much improved precision. In addition we test for $CP$ asymmetries in the separated charge-conjugate branching fractions; no evidence of $CP$ violation is found.