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This paper considers a heterogeneous team of cooperating unmanned aerial vehicles (UAVs) drawn from several distinct classes and engaged in a search and action mission over a spatially extended battlefield with targets of several types. During the mission, the UAVs seek to confirm and verifiably destroy suspected targets and discover, confirm, and verifiably destroy unknown targets. The locations of some (or all) targets are unknown a priori, requiring them to be located using cooperative search. In addition, the tasks to be performed at each target location by the team of cooperative UAVs need to be coordinated. The tasks must, therefore, be allocated to UAVs in real time as they arise, while ensuring that appropriate vehicles are assigned to each task. Each class of UAVs has its own sensing and attack capabilities, so the need for appropriate assignment is paramount. In this paper, an extensive dynamic model that captures the stochastic nature of the cooperative search and task assignment problems is developed, and algorithms for achieving a high level of performance are designed. The paper focuses on investigating the value of predictive task assignment as a function of the number of unknown targets and number of UAVs. In particular, it is shown that there is a tradeoff between search and task response in the context of prediction. Based on the results, a hybrid algorithm for switching the use of prediction is proposed, which balances the search and task response. The performance of the proposed algorithms is evaluated through Monte Carlo simulations.