LED 3 on these units typically represents that the logic board and GPU are communicating properly. This LED will be on when the CPU is communicating properly with the GPU. The fact that yours is flashing for a brief second is a bit interesting, as most of the times with a video card failure it doesn't come on at all and you will just watch LEDs 1 and 2 stay on while the fans slowly ramp up. My initial thoughts are it could potentially be a defect with the graphics card, which on these models is unfortunately an integrated parts that requires logic board replacement and/or advanced microsoldering techniques to rectify. However, since it flashes and you hear a click, it is also possible that the PSU is defective in some way, or there is a loose connection somewhere. It may not be able to supply enough power to the GPU during the boot up process or to the system as a whole and shuts off as soon as more power is used to bring CPU and GPU online. This generation has a very reliable GPU, so if you wanted to start out with the cheapest component to try replacing (after verifying everything else is seated correctly), I would try replacing the PSU.
Please ensure you take all stated precautions when handling high voltage power supplies.
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