Why does my HTC One X+ reboot (and turns ON!) when not in active use?

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AC Question

HTC One X+ reboots (and turns ON!) when not in active use

HTC One X+, just over a year old. Yesterday, it started rebooting itself over and over, every 30 seconds to 2 minutes. Weirdly, it DOESN'T reboot when I'm actively using an app on it - just when it's sitting idle. So, it's not a boot loop per se - it boots fine, and I can use it - but as soon as I put it down, or put it in my pocket, it goes back into the re-re-start cycle.

At first I figured it might be a battery thing, so I let it drain all the way then charge. As soon as I plug it in to charge, it turns itself ON, and keeps rebooting. How it can do that even with fast boot turned off, I have no idea.

I tried booting in safe mode, and it still keeps restarting. Tried factory reset (3-4 times), and that doesn't help either.

When I try to boot to recovery, I get this horrible red triangle with exclamation mark. I tried wiping the cache from there, and when I do, it goes to a green circle screen with a progress bar. But the progress bar doesn't seem to start filling before it resets again.

I don't have root, and haven't really messed with it besides installing apps from play store - which I haven't done in more than a week. So I can't imagine anything I did caused the problem. But I also can't imagine that it's a hardware problem (e.g. fault in the power button) because it doesn't happen when it's getting active input.

There's no one who repairs HTC anything within 1000 miles or so - so taking it to a shop for repairs is not really an option. What else can I do?