Finding availability for such a ticket will be a major problem. You might find it easier to get a string of one way tickets and buy the shorter segments in between.
For F you need a lot of luck and the F version is more expensive, you would waste miles on the other flights where you cannot find availability.
A good starting point is the ANA website to search for more carriers than you can see on the LH/LX website.
I also believe you are trying to squeeze too much into 4 weeks.