General :: Galaxy S3 - Phone Doesn't Boot?

I have a ~1.5 year old Samsung Galaxy S3 (North American) rooted, with Cyanogenmod 11 installed.

Ever since installing CM11, my phone has been restarting itself with some regularity.

Today, with over half the battery left, the phone seemingly turned itself off. I do not believe this is an issue with the cyanogenmod OS and I also don't believe the issue is related to the restarting I've been experiencing. These restarts, in the vast majority of occasions, started with a brief freeze, then screen turning off, before the phone automatically starting up again. However, this time, the phone did not freeze before the screen went black, and the phone did not restart itself. I repeatedly tried pressing the power button to unlock, and when that didn't work, I tried long-pressing to either turn the phone off or back on. When that also failed to work, I pulled the battery, and put it back in. The Samsung logo came up, and then the Galaxy S3 logo with the Cyanogenmod logo came up, then the screen went black again.

And this is where I'm at currently. After the screen goes black, I can't get it to come back on. Long pressing the power button does not turn it back on. Only when I pull the battery and put it back in, does the boot process start. Every time, during the first few seconds of the boot process, the phone turns itself off. Sometimes it only lasts until the Samsung logo, sometimes to the GS3/CM logo.

If I plug the phone in, with the battery inserted, I briefly see a battery on the screen, but again, the phone shuts down.

If I plug the phone in, with the battery out, the LED turns red, the phone vibrates, but nothing happens, and the LED goes out.

I've read about power buttons getting stuck, but this doesn't seem to be the issues since my phone does not go into a boot loop sequence.

I've tried leaving the battery out for an hour or so, but there is no change.

I've also tried pressing power+home+vol down and power+home+vol up, and they seem to work, until again, the phone shuts off. So one of these combinations leads me to warnings about changing OS before shutting down after about the same number of seconds, and the other leads me to the boot logo with the blue text on the top left corner before shutting down.

To me, this doesn't seem to be an issue with software, but with the phone staying powered..somehow.