Directed greybox fuzzing (DGF) focuses on efficiently reaching specific program locations or triggering particular behaviors, making it essential for tasks like vulnerability detection and crash reproduction. However, existing methods often suffer from path explosion and randomness in input mutation, leading to inefficiencies in exploring and exploiting target paths. In this paper, we propose HGFuzzer, an automatic framework that leverages the large language model (LLM) to address these challenges. HGFuzzer transforms path constraint problems into targeted code generation tasks, systematically generating test harnesses and reachable inputs to reduce unnecessary exploration paths significantly. Additionally, we implement custom mutators designed specifically for target functions, minimizing randomness and improving the precision of directed fuzzing. We evaluated HGFuzzer on 20 real-world vulnerabilities, successfully triggering 17, including 11 within the first minute, achieving a speedup of at least 24.8x compared to state-of-the-art directed fuzzers. Furthermore, HGFuzzer discovered 9 previously unknown vulnerabilities, all of which were assigned CVE IDs, demonstrating the effectiveness of our approach in identifying real-world vulnerabilities.