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tokens: We thank B. Lembège for valuable discussions. The numerical simulations were performed on the Fujitsu PRIMERGY CX400 at Kyushu University. M.N. is supported by “Boursier du Gouvernement Français.” S.M. is supported by a Grant-in-Aid for Scientific Research (C) 25400479 from JSPS. We are also grateful to ISSI (Bern) for their support of the team “Physics of the injection of particle acceleration at astrophysical, heliospheric, and laboratory collisionless shocks.”