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One of the most promising possible applications of such new powerful laser facilities as ELI Beamlines is particle acceleration by longitudinal electric fields generated by nonlinear propagation of intense laser pulses through plasma (LWFA scheme). However, since for modern facilities Lorentz factor of electron quiver oscillations in such a huge electromagnetic field reaches the order of hundreds, radiation reaction effects should also come into play. We included classical radiation reaction in EPOCH code and studied propagation of a high intensity electromagnetic pulse through a cold plasma with heavy immobile ions. Our simulations demonstrate that in ultrarelativistic regime radiation reaction dramatically enhances longitudinal electric field in comparison with ordinary ponderomotive mechanism. We provide both numerical simulations and an analytical model, which can be used to optimize the conditions for electron acceleration in a regime when radiation reaction becomes essential