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S Sig Perf 414, VIN 814
I've had the 3G stop working several times; from talking with Tesla service, you can force a hard reset by pulling the touchscreen fuse, waiting a few minutes and then reseating the fuse. That has worked fine for me; when the touchscreen comes back online, the 3G signal is found.
Picture of the correct fuse to yank is shown in another thread: Model S Technical / Mechanical Issues - Page 35
Picture of the correct fuse to yank is shown in another thread: Model S Technical / Mechanical Issues - Page 35