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A mega-constellation of low-altitude earth orbit (LEO) satellites (SATs) are envisaged to provide a global coverage SAT network in beyond fifth-generation (5G) cellular systems. However, such wide coverage rather makes it difficult to apply existing multiple access protocols, such as random access channel (RACH). To overcome this issue, in this paper, we propose a novel random access solution for LEO SAT networks, called as S-RACH. In contrast to existing standardized protocols, S-RACH is a model-free approach using deep reinforcement learning (DRL). Compared to RACH, we show from various simulations that our proposed S-RACH yields around 2x lower average access delay.