Here's 3/4 of the solution, for 1/2 the budget:
UA:$208 LAX-MIA 7080PQM 2.93cpm
Mon 28 Sep
UA1489 LAX-EWR 2048-0509(+1 day) 739
Tue 29 Sep
UA1740 EWR-MIA 0716-1023 739
UA1082 MIA-EWR 1122-1420 739
UA1493 EWR-LAX 1505-1804 752
http://goo.gl/flights/cRvs
That leaves you <2000mi to go. Assuming you get 500mi min on the challenge, any four segments should do. (If this is incorrect, I'm sure someone will chime in.)
Good luck!