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Our method is relevant especially for automata-based string solvers, which have recently achieved the best results in terms of practical efficiency, generality, and completeness guarantees. This work allows them to excel also on position constraints, which used to be their weakness. Besides the efficiency gains, we show that our framework may be extended to solve a large fragment of \u00accontains (in NExpTime), for which decidability has been long open, and gives a hope to solve the general problem. 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