This paper introduces a new idea for enhancing constraint solvers and decision procedures that are backend engines for many analysis and synthesis techniques that are powerful but have high complexity. Our insight is that in many application scenarios the engines are run repeatedly against input formulas that encode problems that are related but of increasing complexity, and domain-specific knowledge can help mitigate the increase in complexity. Moreover, even for one formula the engine may perform multiple expensive tasks with commonalities that can be estimated and exploited likewise. We believe these relationships lay a foundation for making the engines more effective and their applications more scalable. We illustrate the viability of our idea by applying it in the context of a well-known constraint solver for imperative constraints that has been used for automated testing and quantitative analysis, and discuss how the idea generalizes to more general purpose methods.