Try another power supply. What model is it? And can you provide any more details about the condition of the console. Like dusty, bugs, were fans loud, did you ever observe any artifacting, YLOD in games, has it been previously repaired, is it sealed with warranty sticker...things like that build a picture that can hint at what may be going on.
1004 is an AC/DC power failure usually from flipping the power rocker off at the back when testing the console or when forcing it off after it becomes unresponsive. In your case it appears to be occuring on it's own when attempting to power on, which could very well be a bad power supply. So if you have one on hand, try it. If not, they're not too expensive to replace. And easily done for pretty much anyone. No soldering required.
If that's isn't it however, the chances this is something simple that won't require soldering is remote. Like maybe a ribbon cable is pinched. But that's unlikeky. Be sure to try another outlet and power cord. Just to rule that out.
