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We report an experimental investigation of the reactions n¯p→2π+π-, n¯p→2π+π-π0, and n¯p→2π+π-2π0 using (10.087±0.044)×109 J/ψ events collected with the BESIII detector at the BEPCII storage ring. The antineutron (n¯) is produced in the decay J/ψ→pπ-n¯ with studied momentum from 200 to 1174 MeV/c, while the target proton originates from the hydrogen nuclei in the cooling oil of the beam pipe. This novel method pioneers the study of n¯-nucleon interactions at an e+e- collider, providing the first experimental data for n¯ momenta exceeding 800 MeV/c.