Notebook Boot and Lockup

I've got a 15" HP Spectre x360 laptop. It's now 16 months old, so past the warranty. At the 13 month mark, the computer shut down without warning and then would not restart. I plug in the power and see the LED light on. When I press the power button, the LED light for the power button comes on and I hear the fans start, but the screen remains black and all activity ceases after about 2 seconds. Initially when I had this problem, I took it to an authorized service center. They were going to try to save all my data and re-install windows, only when they tried the computer, about a week after the initial problem it booted up completely fine. The computer worked totally fine for about a week after that and then the same problem happened again. I took it back to the shop, this time after a month, due to scheduling constraints, and again they go to re-install and it works fine again. This pattern has continued until now.

The computer randomly shuts down. I try booting it once a day after the initial shut down and it usually takes a week before it will start up again, then it will work for a week or two and then shut down without warning again. This does not seem like normal behavior even out of warranty. What are my options for a fix?