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Autonomous exploration is one of the important parts to achieve the autonomous operation of Unmanned Aerial Vehicles (UAVs). To improve the efficiency of the exploration process, a fast and autonomous exploration planner (FAEP) is proposed in this paper. We firstly design a novel frontiers exploration sequence generation method to obtain a more reasonable exploration path, which considers not only the flight-level but frontier-level factors into TSP. According to the exploration sequence and the distribution of frontiers, a two-stage heading planning strategy is proposed to cover more frontiers by heading change during an exploration journey. To improve the stability of path searching, a guided kinodynamic path searching based on a guiding path is devised. In addition, a dynamic start point selection method for replanning is also adopted to increase the fluency of flight. We present sufficient benchmark and real-world experiments. Experimental results show the superiority of the proposed exploration planner compared with typical and state-of-the-art methods.