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This lower bound is sharp for the 3-manifolds $\\mathbb{R P}^3$, $L(3,1)$, $L(5,2)$, $S^1\\times S^1 \\times S^1$, $S^2 \\times S^1$, $S^2 \\mbox{$\\times \\hspace{-2.8mm}_{-}$} S^1$ and $S^3/Q_8$, where $Q_8$ is the quaternion group. 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