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GTO is an orbit which has its main axis in the plane of Earth's equator. Moon orbit has a different plane. This means until GTO main axis isn't on the intersection of equator and Moon orbit, you can't get to the vicinity of the Moon by just raising apogee. Raising apogee won't change the direction of the main axis of the orbit, and that axis doesn't in general lay in the Moon orbit plane.
So how Beresheet is going to get to the Moon starting from GTO and having modest delta-V capabilities?
The burns are not necessarily in plane burns, and therefore maneuvers do not only increase the apogee. In fact, a 2013 video of the work in progress on the trajectory shows quite a significant out-of-plane burn to intersect the Moon's trajectory.
That said, the latest trajectory diagram I could find of their mission shows that all orbits are co-planar, or near co-planar. The press release says that the entire journey should last about 2.5 months. This should give plenty of time for the SpaceIL operators to correct target the maneuvers such that they do not need anything else than co-planar or near co-planer burns.
Official site offers static and dynamic view of the full path to the Moon:
Raw data (undocumented, possibly x, y, z ,vx, vy, vz geocentric):
http://live.spaceil.com/data/data_s1.txt (22/2 - 12/4)
http://live.spaceil.com/data/data_s2.txt (4/4 - 12/4)
http://live.spaceil.com/data/data_m.txt (22/2 - 16/12)
