Well based on your issue I don't think it is the power since LED 1 and 2 seem to be working fine. LED 3 also seems to be functioning as usual, but if it goes off could be an indication of bad graphics card. What you could do is try to connect it to an external monitor. I believe if LED 4 is off it is an indicator for a bad inverter or screen (Should be an easy fix).
+
I wasn't sure if you were saying the monitor was coming on an staying on or going off after trying to boot.
Have you tried to get into recovery mode? It seems as if the system powers on and gets stuck looping trying to boot it would be either a hardware or software issue and if you can run an AHT or get to the recovery menu you might be able to identify what is happening.
Have you tried to get into recovery mode? It seems as if the system powers on and gets stuck looping trying to boot it would be either a hardware or software issue and if you can run an AHT or get to the recovery menu you might be able to identify what is happening.