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We use a simplified model that maps the quantum electrodynamics processes governing pair cascades to analytically and numerically model these cascades, and show that large-amplitude oscillations of the electric field are inductively driven by the resulting plasma. A plasma instability arises in these oscillations, and particles accelerated in growing electric field perturbations can drive secondary pair bursts that damp the large-amplitude oscillations. An analytical model is proposed to describe this interplay between the pair production and kinetic collective plasma processes. 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