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Ingredients in our construction are {\\it admissible pairs}, which were dealt with by Kovalev in \\cite{K03} and further studied by Kovalev and Lee in \\cite{KL11}. An admissible pair $(\\overline{X},D)$ consists of a three-dimensional compact K\\\"{a}hler manifold $\\overline{X}$ and a smooth anticanonical $K3$ divisor $D$ on $\\overline{X}$. 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