Search for $CP$ violation in $D^{+}_{(s)}\rightarrow K^{+}K^{0}_{S}h^{+}h^{-}$ $(h=K,π)$ decays and observation of the Cabibbo-suppressed decay $D^{+}_{s}\rightarrow K^{+}K^{-}K^{0}_{S}π^{+}$

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We search for $CP$ violation by measuring a $T$-odd asymmetry in the Cabibbo-suppressed $D^{+}\rightarrow K^{+}K^{0}_{S}π^{+}π^{-} $ decay, and in the Cabibbo-favored $D^{+}_{s}\rightarrow K^{+}K^{0}_{S}π^{+}π^{-}$ and $D^{+}\rightarrow K^{+}K^{-}K^{0}_{S}π^{+}$ decays. We use 980 ${\rm fb}^{-1}$ of data collected by the Belle detector running at the KEKB asymmetric-energy $e^{+}e^{-}$ collider. The $C\!P$-violating $T$-odd parameter ${a}^{T\text{-}\rm{odd}}_{CP}$ is measured to be ${a}^{T\text{-}\rm{odd}}_{CP}(D^{+}\rightarrow K^{+}K^{0}_{S}π^{+}π^{-})=(0.34\pm0.87\pm0.32)\%,$ ${a}^{T\text{-}\rm{odd}}_{CP}(D^{+}_{s}\rightarrow K^{+}K^{0}_{S}π^{+}π^{-})=(-0.46\pm0.63\pm0.38)\%,$ and ${a}^{T\text{-}\rm{odd}}_{CP}(D^{+}\rightarrow K^{+}K^{-}K^{0}_{S}π^{+})=(-3.34\pm2.66\pm0.35)\%,$ where the first uncertainty is statistical and the second is systematic. We also report the first observation of the Cabibbo-suppressed decay $D^{+}_{s}\rightarrow K^{+}K^{-}K^{0}_{S}π^{+}$. The branching fraction is measured relative to that of the analogous Cabibbo-favored decay : $B(D^{+}_{s}\rightarrow K^{+}K^{-}K^{0}_{S}π^{+}) / B(D^{+}_{s}\rightarrow K^{+}K^{0}_{S}π^{+}π^{-}) = (1.36\pm 0.15\pm 0.04)\%$.

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