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This paper describes a resilient navigation and planning system used in the Indy Autonomous Challenge (IAC) competition. The IAC is a competition where full-scale race cars run autonomously on Indianapolis Motor Speedway(IMS) up to 290 km/h (180 mph). Race cars will experience severe vibrations. Especially at high speeds. These vibrations can degrade standard localization algorithms based on precision GPS-aided inertial measurement units. Degraded localization can lead to serious problems, including collisions. Therefore, we propose a resilient navigation system that enables a race car to stay within the track in the event of localization failures. Our navigation system uses a multi-sensor fusion-based Kalman filter. We detect degradation of the navigation solution using probabilistic approaches to computing optimal measurement values for the correction step of our Kalman filter. In addition, an optimal path planning algorithm for obstacle avoidance is proposed. In this challenge, the track has static obstacles on the track. The vehicle is required to avoid them with minimal time loss. By taking the original optimal racing line, obstacles, and vehicle dynamics into account, we propose a road-graph-based path planning algorithm to ensure that our race car can perform efficient obstacle avoidance. The proposed localization system was successfully validated to show its capability to prevent localization failures in the event of faulty GPS measurements during the historic world's first autonomous racing at IMS. Owing to our robust navigation and planning algorithm, we were able to finish the race as one of the top four teams while the remaining five teams failed to finish due to collisions or out-of-track violations.