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The ratios of rates for several such decays, including $\\Gamma(B^0 \\to \\overline{K}^{*0} K^0)/\\Gamma(B^0 \\to \\phi K^0)$, $\\Gamma(B^0 \\to \\overline{K}^{*0} K^{*0})/\\Gamma(B^0 \\to \\phi K^{*0})$, $\\Gamma(B^+ \\to \\overline{K}^{*0} K^+)$ $/\\Gamma(B^+ \\to \\phi K^+)$, and $\\Gamma(B^+ \\to \\overline{K}^{*0} K^{*+})/\\Gamma(B^+ \\to \\phi K^{*+})$, can provide information on the ratio of Cabibbo-Kobayashi-Maskawa (CKM) elements $|V_{td}/V_{ts}|$ in a manner complementary to other proposed determinations. 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