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In next-generation wireless networks, reconfigurable intelligent surface (RIS)-assisted multiple-input multiple-output (MIMO) systems are foreseeable to support a large number of antennas at the transceiver as well as a large number of reflecting elements at the RIS. To fully unleash the potential of RIS, the phase shifts of RIS elements should be carefully designed, resulting in a high-dimensional non-convex optimization problem that is hard to solve. In this paper, we address this scalability issue by partitioning RIS into sub-surfaces, so as to optimize the phase shifts in sub-surface levels to reduce complexity. Specifically, each subsurface employs a linear phase variation structure to anomalously reflect the incident signal to a desired direction, and the sizes of sub-surfaces can be adaptively adjusted according to channel conditions. We formulate the achievable rate maximization problem by jointly optimizing the transmit covariance matrix and the RIS phase shifts. Under the RIS partitioning framework, the RIS phase shifts optimization reduces to the manipulation of the sub-surface sizes, the phase gradients of sub-surfaces, and the common phase shifts of sub-surfaces. Then, we characterize the asymptotic behavior of the system with an infinitely large number of transceiver antennas and RIS elements. The asymptotic analysis provides useful insights on the understanding of the fundamental performance-complexity tradeoff in RIS partitioning design. We show that in the asymptotic domain, the achievable rate maximization problem has a rather simple form. We develop an efficient algorithm to find an approximate optimal solution via a 1D grid search. By applying the asymptotic result to a finite-size system with necessary modifications, we show by numerical results that the proposed design achieves a favorable tradeoff between system performance and computational complexity.