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Effect and coeffect tracking integrate many types of compile-time analysis, such as cost, liveness, or dataflow, directly into a language's type system. In this paper, we investigate the addition of effect and coeffect tracking to the type system of call-by-push-value (CBPV), a computational model useful in compilation for its isolation of effects and for its ability to cleanly express both call-by-name and call-by-value computations. Our main result is effect-and-coeffect soundness, which asserts that the type system accurately bounds the effects that the program may trigger during execution and accurately tracks the demands that the program may make on its environment. This result holds for two different dynamic semantics: a generic one that can be adapted for different coeffects and one that is adapted for reasoning about resource usage. In particular, the second semantics discards the evaluation of unused values and pure computations while ensuring that effectful computations are always evaluated, even if their results are not required. Our results have been mechanized using the Coq proof assistant.