Now that's interesting. You can get a "free" LAX-JFK roundtrip ( approx 5,000 miles) by going eastbound first!

1 United Airlines 956 Apr 25 9:00 LAX 17:17 JFK
2 United Airlines 883 Apr 26 10:15 EWR 16:15 (Next day) NRT
United Airlines 897 Apr 27 17:55 NRT 23:45 SIN
3 United Airlines 890 May 1 7:20 SIN 15:05 NRT
United Airlines 852 May 1 15:55 NRT 19:51 JFK
4 United Airlines 897 May 2 7:00 JFK 9:55 LAX
Your Original Choice - 1169.76 USD per person

Also consider these less expensive alternatives:
Leg Flight info Date Depart Arrive
1 United Airlines 956 Apr 25 9:00 LAX 17:17 JFK
2 United Airlines 81 Apr 26 7:00 EWR 10:10 SFO
United Airlines 805 Apr 26 12:45 SFO 23:40 (Next day) SIN
3 United Airlines 806 May 1 6:45 SIN 12:26 LAX
Alternate Low Fare Option #1 - 1155.00 USD per person