Time-resolved magneto-optical imaging reveals that the dynamics of the helicity-dependent all-optical switching (HD-AOS) of Co/Pt ferromagnetic multilayers occurs on the time scales from nanoseconds to seconds. We find HD-AOS proceeds by two stages. First, for an optimized laser fluence, the ultrashort laser pulse demagnetizes the film to 25% of the initial magnetization. Subsequent laser pulses aids nucleation of small reversed domains. The observed nucleation is stochastic and independent of the helicity of laser light. At the second stage circularly polarized light breaks the degeneracy between the magnetic domains promoting a preferred direction of domain wall motion. One circular polarization results in a collapse of the reversed magnetic domains. The other polarization causes the growth of reversed magnetic domain from the nucleation sites, via deterministic displacement of the domain wall resulting in magnetization reversal. This mechanism is supported by further imaging studies of deterministic laser-induced displacement of the domain walls when excited by circularly polarized optical pulses.