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Many problem ensembles exhibit a phase transition that is associated with a large peak in the average cost of solving the problem instances. However, this peak is not necessarily due to a lack of solutions: indeed the average number of solutions is typically exponentially large. Here, we study this situation within the context of the satisfiability transition in Random 3SAT. We find that a significant subclass of instances emerges as we cross the phase transition. These instances are characterized by having about 85-95% of their variables occurring in unary prime implicates (UPIs), with their remaining variables being subject to few constraints. In such instances the models are not randomly distributed but all lie in a cluster that is exponentially large, but still admits a simple description. Studying the effect of UPIs on the local search algorithm WSAT shows that these "single-cluster" instances are harder to solve, and we relate their appearance at the phase transition to the peak in search cost.