It would help if you laid out your trip in order. For starters, without using any more European segments (all 4 are spoken for) you have to get from Paris to South America some way, and then eventually get back to London to catch that last flight to Cairo.

You might want to download one of the OWE downloadable timetables. Cathay offers one, as does American and OW themselves. They are far from perfect, especially looking for specific flights more than a couple of months into the future, but are about the best planning tool available.