Regent air, from LA, was $1,200 PP plus $100 PP deviation fee (which we would need). Total $1,300. Am I understanding you correctly - Fly into NY or FL and then use RSSC's air for CROSS?
BA offers some very good fares from LA to London. We may opt to fly out a few days earlier and stop in London, then take an early AM flight from London to Nice. Fares from London to Nice, OW, are very reasonable.
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