Mac mini stuck in boot cycle

Today my mac mini suddenly froze up while I was working. The screen split vertically and about 1/4 or less of the left side of the screen then showed up on the right. I just had the spinning rainbow wheel and could not force quit applications. I ended up powering off with the power button, then when it tried to reboot, it just kept cycling in the boot cycle and never reach the login screen. I let it go for about 45 minutes like this, hoping that it would eventually "fix itself", but to no avail. I have never had a problem with overheating, but on the off chance that that was it, I let it sit idle for awhile to cool completely, then tried powering up again. While it was off and unplugged, I even tried holding the power button in for 10+ seconds (as I had read elsewhere), to see if that would help when it was plugged in again, but again, when I plugged it back in a powered up, it just cycled and never reached the login screen for me to be able to do anything.

Any suggestions on how to proceed would be greatly appreciated.