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It is noted that the appearance of these black holes with scalar hair is closely related to the instability of Reissner-Nordstr\\\"om black holes without scalar hair in the EMS theory. The scalarized black hole solutions are classified by the node number of $n=0,1,2,\\cdots$, where $n=0$ is called the fundamental branch and $n=1,2,\\cdots$ denote the $n$ excited branches. 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