Using an $e^{+}e^{-}$ annihilation data sample corresponding to an integrated luminosity of $3.19~mathrm{fb}^{-1}$ and collected at a center-of-mass energy $sqrt{s} = 4.178~mathrm{GeV}$ with the BESIII detector, we measure the absolute branching fractions $mathcal{B}(D_{s}^{+} rightarrow K_{S}^{0}K^{+}) = (1.425pm0.038_{rm stat.}pm0.031_{rm syst.})%$ and $mathcal{B}(D_{s}^{+} rightarrow K_{L}^{0}K^{+}) =(1.485pm0.039_{rm stat.}pm0.046_{rm syst.})%$. The branching fraction of $D_{s}^{+} rightarrow K_{S}^{0}K^{+}$ is compatible with the world average and that of $D_{s}^{+} rightarrow K_{L}^{0}K^{+}$ is measured for the first time. We present the first measurement of the $K_{S}^{0}$-$K_{L}^{0}$ asymmetry in the decays $D_{s}^{+} rightarrow K_{S,L}^{0}K^{+}$, and $R(D_{s}^{+} rightarrow K_{S,L}^{0}K^{+})=frac{mathcal{B}(D_{s}^{+} rightarrow K_{S}^{0}K^{+}) -mathcal{B}(D_{s}^{+} rightarrow K_{L}^{0}K^{+})}{mathcal{B}(D_{s}^{+} rightarrow K_{S}^{0}K^{+}) +mathcal{B}(D_{s}^{+} rightarrow K_{L}^{0}K^{+})}= (-2.1pm1.9_{rm stat.}pm1.6_{rm syst.})%$. In addition, we measure the direct $CP$ asymmetries $A_{rm CP}(D_{s}^{pm} rightarrow K_{S}^{0}K^{pm}) = (0.6pm2.8_{rm stat.}pm0.6_{rm syst.})%$ and $A_{rm CP}(D_{s}^{pm} rightarrow K_{L}^{0}K^{pm}) = (-1.1pm2.6_{rm stat.}pm0.6_{rm syst.})%$.
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