It does sound like you're taking a risk to wait until.3 days before but it's safe or you can go weeks before and they can give you an appointment closer to the date you need to fly out. I waited until the 73 hours before my trip for 3 of us and it worked out. The person before me got an appointment 2 weeks later and it was on target for when she needed it.
If you don't want to cut it that close, get an "itinerary" that says your flying out sooner.
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