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This article describes an algorithm for factorizing a finitely ambiguous finite-state transducer (FST) into two FSTs, T1 and T2, such that T1 is functional and T2 retains the ambiguity of the original FST. The application of T2 to the output of T1 never leads to a state that does not provide a transition for the next input symbol, and always terminates in a final state. In other words, T2 contains no "failing paths" whereas T1 in general does. Since T1 is functional, it can be factorized into a left-sequential and a right-sequential FST that jointly constitute a bimachine. The described factorization can accelerate the processing of input because no failing paths are ever followed.