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We present the results of deep Chandra and XMM-Newton observations of a complex merging galaxy cluster Abell 2256 (A2256) that hosts a spectacular radio relic (RR). The temperature and metallicity maps show clear evidence of a merger between the western subcluster (SC) and the primary cluster (PC). We detect five X-ray surface brightness edges. Three of them near the cluster center are cold fronts (CFs): CF1 is associated with the infalling SC; CF2 is located in the east of the PC; and CF3 is to the west of the PC core. The other two edges at cluster outskirts are shock fronts (SFs): SF1 near the RR in the NW has Mach numbers derived from the temperature and the density jumps, respectively, of $M_T=1.62\pm0.12$ and $M_ρ=1.23\pm0.06$; SF2 in the SE has $M_T=1.54\pm0.05$ and $M_ρ=1.16\pm0.13$. In the region of the RR, there is no evidence for the correlation between X-ray and radio substructures, from which we estimate an upper limit for the inverse-Compton emission, and therefore set a lower limit on the magnetic field ($\sim$ 450 kpc from PC center) of $B>1.0\ μ$G for a single power-law electron spectrum or $B>0.4\ μ$G for a broken power-law electron spectrum. We propose a merger scenario including a PC, an SC, and a group. Our merger scenario accounts for the X-ray edges, diffuse radio features, and galaxy kinematics, as well as projection effects.