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We show that $G$ satisfies the Tutte-type condition \\[ o(G-S)\\le f(S)\\qquad\\text{for all vertex subsets $S$}, \\] if and only if it contains a colored $J_f^*$-factor for any $2$-end-coloring, where $J_f^*(v)$ is the union of all odd integers smaller than $f(v)$ and the integer $f(v)$ itself. This is a generalization of the $(1,f)$-odd factor characterization theorem, and answers a problem of Cui and Kano. 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