I have had one Core pack it in (as they say), but it flashed white on reset indicating that the ARM could not talk to the CC3000. I believe the CC3000 was beyond repair on that device.
I had a core that was embedded in a project fail in the exact same fashion: breathing cyan forever, can’t put it in blue flashing mode. Last time it was online was Nov 14th 2017. I’ve been trying to revive it for hours, but I think it may be hopeless.
Then put the Core in DFU mode by holding down both the RST and MODE buttons, releasing RST and continuing to hold down MODE until it blinks yellow and issue the commands below, in order.
After completing some of the steps the color may change to something else; if so note what color/pattern it is, then use the button combination to get back to blinking yellow. The core should be blinking yellow before each of the commands listed below is entered.
Then put the Core in listening mode (blinking blue) if not already in that state by pressing and releasing RST. Then holding down MODE until it blinks dark blue. Then issue the command:
particle serial wifi
The normal sequence is white, blinking green, blinking cyan (light blue), fast blinking cyan, and finally breathing cyan. If you do not get to breathing cyan, please copy and paste the terminal output in your reply and let me know how far it did get.