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$\begin{aligned} & \frac{1}{\log _a a b c}+\frac{1}{\log _b a b c}+\frac{1}{\log _c a b c} \\ = \; & \log _{a b c} a+\log _{a b c} b+\log _{a b c} c \\ =\; & \log _{a b c} a b c \\ = \;& 1\end{aligned}$
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