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The dynamics of an electron in a strong laser field can be significantly altered by radiation reaction. This usually results in a strongly damped motion, with the electron losing a large fraction of its initial energy. Here we show that the electron dynamics in a bichromatic laser pulse can be indirectly controlled by a comparatively small radiation reaction force through its interplay with the Lorentz force. By changing the relative phase between the two frequency components of the bichromatic laser field, an ultrarelativistic electron bunch colliding head-on with the laser pulse can be deflected in a controlled way, with the deflection angle being independent of the initial electron energy. The effect is predicted to be observable with laser powers and intensities close to those of current state-of-the-art petawatt laser systems. [ABSTRACT FROM AUTHOR]