Your hardware has gone bad. From my own testing, it appears to be an overheating issue, most likely related to bad capacitors or somthing along those lines. I'm pretty sure Nintendo knows about it, but they won't confirm anything.
Anyway, call the support center, and play dumb. Let them go though troubleshooting, and then they'll tell you to send it in for repair. In most cases, they'll extend your warranty for you so it's covered, but that probably depends on how old yours is. (Mine was about 2 months out of warranty when it died, and they extended my warranty out so I wouldn't have to pay out-of-pocket).
It's also handy if you live within 20 miles of the US headquarters, so you can just drive over there and have them fix it (or replace it, as they did in my case), while you wait,.....