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[fflarestar]

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[svim]

OK, so starting with Odin you might be better off using an older version. There are some who find the most recent versions not being compatible with older phones.
http://odindownload.com/
Hopefully you won't still need to restore the PIT file but there's a link for one in this page:
http://forum.xda-developers.com/galaxy-s3/help/solved-pit-bricked-dead-problem-gt-t2535367

If the Recovery Mode on your S3 isn't working again when you re-flash the firmware, it sounds like it was inadvertently hosed while you were trying to restore things. You might want to think about installing a third-party Recovery like TWRP.
https://dl.twrp.me/i9300/
https://wiki.cyanogenmod.org/w/All_About_Recovery_Images
Whether you want to stay with the Samsung OEM ROM or try a different third-party ROM you'll still need to root your phone to install TWRP. This link will walk you through the process but note it's a couple of years old so adjust things accordingly:
http://galaxys3root.com/galaxy-s3-root/how-to-root-galaxy-s3-on-android-4-34-4-2-new/

 

[fflarestar]

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[svim]

Taking into consideration all the things you've tried, it does tend to indicate there's a hardware issue, I'm taking a guess that it's not a problem with the CPU chip but more likely the NAND flash chip is failing. But one or the other doesn't really matter, neither is user replaceable so you'll need to replace the motherboard anyway. If you're so inclined here's a step-by-step guide:
https://www.ifixit.com/Guide/Samsung+Galaxy+S+III+Motherboard+Replacement/12696
The S3 model is over four years old now (released in 2012) so maybe it's time to get a new phone perhaps? Or even a pre-owned, refurbished phone? What country do you live in?

 

[fflarestar]

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[svim]

Take a look into something like a Moto G4, it's apparently a little more than what you've budgeted but it's got some good specs for its price. This article is more U.S.-centric but it also has a lot of info on what to look for in a budget phone even if its suggested Moto G4 doesn't appeal to you:
http://thewirecutter.com/reviews/best-budget-android-phone/
Also, this site is a good reference since when you're shopping around for a replacement phone your selection is often limited by which carrier you use:
http://willmyphonework.net/

 

[fflarestar]

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