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The same bound holds for every sequence of length at least $|G|$. In cyclic groups we combine this elementary support bound with the Savchev--Chen structure theorem for long zero-sumfree sequences and obtain the sharper estimate $MZ(S)\\le n-t(t-1)/2$, where $t=|\\supp(S)|$, whenever $S$ has length $n$ over $C_n$ and $MZ(S)-1>n/2$. 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