Time-resolved imaging reveals that the helicity dependent all-optical switching (HD-AOS) of Co/Pt ferromagnetic multilayers proceeds by two stages. First one involves the helicity independent and stochastic nucleation of reversed magnetic domains. At the second stage circularly polarized light breaks the degeneracy between the magnetic domains and promotes the preferred direction of domain wall (DW) motion. The growth of the reversed domain from the nucleation cite, for a particular helicity, leads to full magnetic reversal. This study demonstrates a novel mechanism of HD-AOS mediated by the deterministic displacement of DWs.