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We present results of a nonperturbative Floquet study of above-threshold detachment (ATD) of the H– and F– negative ions. Pronounced enhancements of ATD rates in the high-energy plateau region are found as the laser field intensity passes across ponderomotively shifted multiphoton thresholds. Depending on the symmetry of the initial state (s or p symmetry), the enhancement is found to be most pronounced for even- or odd-channel closures, which is consistent with threshold laws applicable at the closing of particular multiphoton channels. Variations of ATD photoelectron angular distributions as functions of laser intensity near multiphoton thresholds are also investigated.