We report the first determination of the relative strong-phase difference between D^0 -> K^0_S,L K^+ K^- and D^0-bar -> K^0_S,L K^+ K^-. In addition, we present updated measurements of the relative strong-phase difference between D^0 -> K^0_S,L pi^+ pi^- and D^0-bar -> K^0_S,L pi^+ pi^-. Both measurements exploit the quantum coherence between a pair of D^0 and D^0-bar mesons produced from psi(3770) decays. The strong-phase differences measured are important for determining the Cabibbo-Kobayashi-Maskawa angle gamma/phi_3 in B^- -> K^- D^0-tilde decays, where D^0-tilde is a D^0 or D^0-bar meson decaying to K^0_S h^+ h^- (h=pi,K), in a manner independent of the model assumed to describe the D^0 -> K^0_S h^+ h^- decay. Using our results, the uncertainty in gamma/phi_3 due to the error on the strong-phase difference is expected to be between 1.7 and 3.9 degrees for an analysis using B^- K^- D^0-tilde D^0-tilde -> K^0_S pi^+ pi^- decays, and between 3.2 and 3.9 degrees for an analysis based on B^- -> K^- D^0-tilde, D^0-tilde -> K^0_S K^+ K^- decays. A measurement is also presented of the CP-odd fraction, F_-, of the decay D^0 -> K^0_S K^+ K^- in the region of the phi -> K^+ K^- resonance. We find that in a region within 0.01 GeV^2/c^4 of the nominal phi mass squared F_- > 0.91 at the 90% confidence level.
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