Jaeger-LeCoultre

My JLC Gran'Sport Reverso, about 4-5 years old, has stopped. I have never had to had this watch repaired before (and it's the only higher-end watch I've owned) and am unsure how to go about it. I live in Austin, TX and have come across this place: . I don... 

...both you and your watch will be so much happier. N

So do I just find the nearest authorized dealer? And then do they just take care of it shipping it to JLC for repair? Or do I work directly with a JLC authorized repair center? Thanks for the guidance... I've never had to deal with this before... FYI, the...