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We show that there exists a sequence $(R_n, \\psi_n)$ of fibers and monodromies contained in the fibered cone of $(S,\\psi)$ such that the asymptotic translation length of $\\psi_n$ on the curve complex $\\mathcal{C}(R_n)$ behaves asymptotically like $1/|\\chi(R_n)|^2$. As applications, we can reprove the previous result by Gadre--Tsai that the minimal asymptotic translation length of a closed surface of genus $g$ asymptotically behaves like $1/g^2$. 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