An analysis of the decays of $B^\mp \rightarrow D K^\mp$ and $B^\mp \rightarrow D \pi^\mp $ is presented in which the $D$ meson is reconstructed in the threebody final states $K^\mp \pi^\pm \pi^0$, $\pi^+ \pi^ \pi^0$ and $K^+ K^ \pi^0$. Using data from LHCb corresponding to an integrated luminosity of 3.0 fb$^{1}$ of $pp$ collisions, measurements of several $CP$ observables are performed. First observations are obtained of the suppressed ADS decay $B^\mp \rightarrow [\pi^\mp K^\pm \pi^0]_D \pi^\mp$ and the quasiGLW decay $B^\mp \rightarrow [K^+ K^ \pi^0]_D \pi^\mp$. The results are interpreted in the context of the unitarity triangle angle $\gamma$ and related parameters.
Invariant mass distributions of selected $ B ^\mp \rightarrow [ K ^\mp \pi ^\pm \pi ^0 ]_Dh^\mp$ candidates, separated by $ B $ hadron charge. $ B ^\mp \rightarrow D K ^\mp $ signal events are in the upper plots and $ B ^\mp \rightarrow D \pi ^\mp $ events are in the lower plots. The solid dark (red) curve represents $ B ^\mp \rightarrow D K ^\mp $ events and the solid light (green) curve represents $ B ^\mp \rightarrow D \pi ^\mp $ events. The solid (grey) shape indicates partially reconstructed $ B ^\mp$ decays and the heavy dotted (red) curve indicates wrongly reconstructed $ D $ decays. The solid (blue) line represents the total PDF and includes the combinatorial component. 
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Invariant mass distributions of selected $ B ^\mp \rightarrow [\pi ^\mp K ^\pm \pi ^0 ]_Dh^{\mp}$ candidates. See the caption of Fig. 1 for a full description. The lightly dotted (blue) line represents the combinatorial component and the longdashed (magenta) line indicates contributions from partially reconstructed $ B ^0_ s \rightarrow D K ^\mp \pi ^\pm $ decays where the pion is not reconstructed. 
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Invariant mass distributions of selected $ B ^\mp \rightarrow [\pi ^+ \pi ^ \pi ^0 ]_Dh^{\mp}$ candidates. See the caption of Fig. 1 for a full description. The lightly dotted (blue) line represents the combinatorial component. 
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Invariant mass distributions of selected $ B ^\mp \rightarrow [ K ^+ K ^ \pi ^0 ]_Dh^{\mp}$ candidates. See the caption of Fig. 1 for a full description. The lightly dotted (blue) line represents the combinatorial component. 
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Scan of the $\chi^2$ probabilities over the $\gamma $ $r_B$ parameter space. Shown are the $n\sigma$ profile likelihood contours, where $\Delta\chi^2 = n^2$, with $n=1$ being the light (blue) shaded region, $n=2$ the dark (blue) shaded region and $n=3$ corresponding to the white area. The result is seen to be compatible with the current LHCb measurement of $\gamma $ and $r_B$, indicated by the point with error bars. 
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Scan of the $\chi^2$ probabilities over the $\gamma $ $\delta_B$ parameter space. Shown are the $n\sigma$ profile likelihood contours, where $\Delta\chi^2 = n^2$, with $n=1$ being the light (blue) shaded region, $n=2$ the dark (blue) shaded region and $n=3$ corresponding to the white area. The result is compatible with the current LHCb measurement of $\gamma $ and $\delta_B$, indicated by the point with error bars. 
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Animated gif made out of all figures. 
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The final signal yields, split in categories based on the charges of the $ B $ hadron (only statistical uncertainties are shown). 
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Systematic uncertainties on the observables, multiplied by a factor of $10^{3}$. `PID' refers to the fixed PID efficiency attributed to the bachelor tracks. `PDFs' refers to the uncertainties based on fixed parameters in the PDF shapes that are used in the invariant mass fit. `Sim' refers to the use of simulation to calculate relative efficiencies between the $ B ^\mp \rightarrow D K ^\mp $ and $ B ^\mp \rightarrow D \pi ^\mp $ modes, in addition to the estimated charmless background contributions and the fixed $DK$ to $D\pi$ ratio on the ADS modes. `$A_{\rm instr}$' refers to the interaction and detection asymmetries. The `Total' column represents the sum in quadrature of all of the categories of systematic uncertainties. 
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Supplementary material full pdf 
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This ZIP file contains supplementary material for the publication LHCbPAPER2015014. The files are: supplementary.pdf : An overview of the extra figures *.pdf, *.png, *.eps, *.C : The figures in various formats 
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Created on 14 January 2021.