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In this study, it has been clear for the past several years that new physics in the quark sector can only appear, in low energy observables, as a perturbation. Therefore precise theoretical predictions and precise experimental measurements have become mandatory. Here we draw attention to the significant advances that have been made in lattice QCD simulations in recent years in K→ππ, in the long-distance contribution to indirect CP violation in the Kaon system (ε) and in rare K-decays. Thus, in conjunction with experiments, the construction of a unitarity triangle purely from Kaon physics should soon become feasible. We want tomore » emphasize that in our approach to the K -unitarity triangle, the ability of lattice QCD methods to systematically improve the calculation of the direct CP-violation parameter (ε') plays a pivotal role. Along with the B-unitarity triangle, this could allow, depending on the pattern of new physics, for more stringent tests of the Standard Model and tighter constraints on new physics.« less