Just playing around with more examples... Say you want to go to Paris next spring...
A simple ORD-CDG-ORD (AA) for next May is $810+ for 8305 miles.
But, for another $38, you can do ORD-CDG(o)-HEL-DEL(o)-HEL-FCO-ORD, using AA non-stops across the pond and AY for the other segments giving you 18043 miles!! And full EQM and EQPs. And, you can VIP AA42 and AA111.
This, friends, is a steal.