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In this work, we investigate the properties and stability of the combustion waves propagating in a system of layered solid fuel material in which the neighbouring fuel sheets can recuperate heat via thermal electromagnetic radiation. The analysis is undertaken within the flame sheet model, which allows to determine the combustion wave speed and structure and derive the dispersion relation characterising the stability of the reaction front. The proper choice of parameters leads to superadiabatic combustion temperature and significant increase of the combustion wave speed. In addition, the stable steady regimes of combustion can be substantially extended in the space of parameters. Such an enhancement of stability and ability to control the characteristics of the combustion wave propagation can be very attractive for the combustion synthesis of new layered materials, similarly to the chemical furnace technology proposed earlier. [ABSTRACT FROM AUTHOR]