Phase-space distribution of a selected region for Set-up I in the CR (left) and CT (right) case. We fix the impact parameter of the incoming single at the Hill radius and vary the inner BBH SMA and phase angle. This particular section of the phase space is between SMAs of 1 and 5 au. The plot is colour-coded in terms of final outcome as explained in Section 4.4: Escaper from Hill sphere 12 (yellow), 13 (blue), and 23 (pink). The black region corresponds to a three-body GW merger where two of the three objects merge inside the Hill sphere.