Search-based program synthesis systematically explores a space of programs to find one that satisfies a given specification. While effective for small programs, it struggles with scalability due to the combinatorial explosion of the search space. In contrast, large language models (LLMs) can generate large programs but often produce solutions that are incorrect or fail to meet the specification. We propose a novel distance-guided search algorithm that leverages imperfect LLM-generated programs to guide both top-down and bottom-up synthesis. Using an anti-unification-based distance metric, we prioritize candidates in the top-down search that are structurally similar to the LLM output. For bottom-up synthesis, we generate components close to subexpressions of the LLM solution while preserving completeness and pruning efficiency. We implement our approach atop Trio, a bidirectional synthesizer for recursive functional programs, and evaluate it on 80 synthesis tasks. Our results show that distance-guided search effectively combines the strengths of LLMs and search-based methods, solving tasks beyond the reach of either technique alone.