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Computational limitations and Big Data analysis pose challenges in seeking efficient techniques to predict trajectories in three-body dynamics. Thus, a reduced-complexity classical algorithm is proposed utilizing predefined spacecraft's position and velocity data to achieve precise and accurate orbital trajectories of the spacecraft within three-body dynamics. The proposed algorithm seamlessly solves polynomial interpolation along with the boundary and interior conditions without the need for the spacecraft's acceleration data. Once the algorithm is derived, it will be tested across a diverse variety of periodic trajectories in the Earth-Moon system. Moreover, a comparative analysis is performed to evaluate the time complexity of the proposed algorithm compared with conventional orbit propagators. Finally, the proposed algorithm will be utilized and extended to learn and update distant retrograde orbits (DRO) while training a neural network with several initial conditions composing minimum predefined data. After the training is done, the neural network is used to accurately predict DRO trajectories for a given initial condition, demonstrating the exceptional accuracy and effectiveness of the proposed learning process.