Note there is a repair manual available online-- not sure where I found mine, but search for "Fuji GX680 Repair Manual". It's a bit large, as it contains about 4 or 5 separate books. (
https://learncamerarepair.com)
If both lights are blinking alternately, and there's a 3 second beep from the film holder, that sounds like "A trouble has occurred on the camera and/or lens". Try reseating the lens (inspect contacts), using a different lens if you have one, or it's most likely, a problem with the mirror drive. (Page 57, Troubleshooting, "Fuji GX680 Electrical System)
If you power-cycle the camera, and flip the mirror up, does it go up?
If you then flip the mirror switch to down, does it drop all the way?
Remove the film back, and verify the operation.
If it's not closing all the way, something's jamming, or the motor has a problem. If it closes all the way, then it's the photocoupler that detects the mirror's position-- could be mechanical or electrical, will take further research.