Okay.. I have recently picked up my psp again so I will give you some help because I've had my phat black 1001 for 6 years.
This tutorial will show you how to install 3.40 O-EA CFW to your psp, it's also the same one I used when I first installed CFW on mine.
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And heres the tutorial for the Pandora battery. NOTE: Regardless of what people tell you to do this with, I would not reccomend using a sharp object to open your battery. I used a butterknife then a razorblade to pop up the tab on chip ICO4. And make sure your battery isnt fully charged before doing this but if you see a flash it WON'T short your battery (I've done it atleast 12 times)
[ame=
https://www.youtube.com/watch?v=htgf1IqZGgo]YouTube - how to create a pandora battery with psp fat [REAL EASY][/ame]
Also if you can't solder the tab back on for some reason, here is a tutorial to write an MS boot code which allows you to boot your psp with a pandorized battery. NOTE: I have two memory sticks an 8G and a 128mb which is my magic memory stick that I use. If you don't have two memory sticks only one I'd highly recommend that you make a backup of your files on whatever memory stick you're using (just in case you wanna flash to a different firmware)
heres my DL link:
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It works in windows 7 and the readme is included in the file folder.
Heres a clip of me running tekken 3 on 3.40 O-EA + pm me if you have questions and rep if I helped

[ame=
https://www.youtube.com/watch?v=riOBshnKdQo]YouTube - ps1 games on psp[/ame]