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In recent years, the number of Low Earth Orbit (LEO) satellites in orbit has drastically increased. Internet service providers are exploring the hybridization of LEO and Geosynchronous Equatorial Orbit (GEO) satellite links to enhance user connectivity. However, efficient hybridization requires intelligent routing management due to the distinct characteristics of these links: while LEO offers lower latency, it also experiences greater variability in performance, whereas GEO provides more stable but higher-latency connections. Additionally, accurately gauging traffic flows to their impact on user-perceived Quality of Experience (QoE) remains a complex challenge. To address this, we investigate the use of reinforcement learning to optimize routing over dual satellite links (LEO and GEO). We train a Deep Q-Learning (DQL) agent, based on a LSTM model, to manage routing decisions with the objective of maximizing users' QoE for VoIP calls, YouTube streaming, web browsing, and data transfers. The performance of our approach is compared against the Minimum Sending Delay Scheduler algorithm, and its real-time feasibility is evaluated. Our results demonstrate that the proposed method provides a scalable routing solution, capable of efficiently handling a large number of flows while significantly improving user experience.