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Nonterrestrial networks have been attracting much interest from the industry and academia. Satellites and high-altitude platform station (HAPS) systems are expected to be the key enablers of next-generation wireless networks. In this article, we introduce a novel downlink satellite communication (SatCom) model, where free-space optical (FSO) communication is adopted between a satellite and a HAPS node. A hybrid FSO/radio-frequency transmission model is used between the HAPS node and the ground station (GS). In the first phase of transmission, the satellite selects the HAPS node that provides the highest signal-to-noise ratio. In the second phase, the selected HAPS decodes and forwards the signal to the GS. To evaluate the performance of the proposed system, outage probability expressions are derived for exponentiated Weibull and shadowed-Rician fading models while considering the atmospheric turbulence, stratospheric attenuation, and attenuation due to scattering, path loss, and pointing errors. Additionally, asymptotic analysis is carried out, and diversity gain is provided. Furthermore, the impacts of the aperture averaging technique, temperature, and wind speed are investigated. We also provide some important guidelines that can be helpful for the design of practical HAPS-aided SatCom. Finally, the results show that the use of HAPS improves the system performance and that the proposed model performs better than all other existing models. [ABSTRACT FROM AUTHOR]