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To artificially reprogram cell fate, experimentalists manipulate the gene regulatory networks (GRNs) that maintain a cell's phenotype. In practice, reprogramming is often performed by constant overexpression of specific transcription factors (TFs). This process can be unreliable and inefficient. Here, we address this problem by introducing a new approach to reprogramming based on mathematical analysis. We demonstrate that reprogramming GRNs using constant overexpression may not succeed in general. Instead, we propose an alternative reprogramming strategy: a synthetic genetic feedback controller that dynamically steers the concentration of a GRN's key TFs to any desired value. The controller works by adjusting TF expression based on the discrepancy between desired and actual TF concentrations. Theory predicts that this reprogramming strategy is guaranteed to succeed, and its performance is independent of the GRN's structure and parameters, provided that feedback gain is sufficiently high. As a case study, we apply the controller to a model of induced pluripotency in stem cells.