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A unique decoding algorithm for general AG codes, namely multipoint evaluation codes on algebraic curves, is presented. It is a natural generalization of the previous decoding algorithm which was only for one-point AG codes. As such, it retains the same advantages of fast speed, regular structure, and direct message recovery. Upon this generalization, we add a technique from the Guruswami-Sudan list decoding that boosts the decoding speed significantly. Compared with other known decoding algorithms for general AG codes, it has a similar decoding performance and allows streamlined practical implementation by its simple and regular structure.