We propose a technique to efficiently search a large family of abstractions in order to prove a query using a parametric dataflow analysis. Our technique either finds the cheapest such abstraction or shows that none exists. It is based on counterexample-guided abstraction refinement but applies a novel meta-analysis on abstract counterexample traces to efficiently find abstractions that are incapable of proving the query. We formalize the technique in a generic framework and apply it to two analyses: a type-state analysis and a thread-escape analysis. We demonstrate the effectiveness of the technique on a suite of Java benchmark programs.