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We adopt flavor-dependent quark imaginary chemical potentials, namely $(\\mu_{u},\\mu_{d},\\mu_{s})/iT=(\\theta-2\\pi{C}/3,\\,\\theta,\\,\\theta+2\\pi{C}/3)$ with $C\\in[0,1]$. The RW periodicity is guaranteed and the center symmetry of $\\mathbb{Z}_{3}$-QCD is explicitly broken when $C\\neq{1}$ or/and quark masses are non-degenerate. 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