Copy a ROM to your SD card, install a custom recovery (I prefer CWM), boot into recovery, wipe cache, Dalvik, data, and system, install the custom ROM, reboot the phone.
Every 4.4.2 ROM I have tried (Valudus, Cyanfox, CM11) will boot with a stock 4.1.2 bootloader or newer so you should be fine.
Do you have any warranty left on your phone? Does your phone have Knox installed?
Here’s the Steps I went through:
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I have a Bell S3 with 4.3 JB from OTA
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1.******Downloaded Odin to my PC
2.******Downloaded the image from* 66 CF Root….
3.******Loaded it to my phone (Using Odin)
4.******Made sure the phone is rooted after and it is
5.******Downloaded CWM Manager from Playstore
6.******Changed the Recovery image using the Manager to Samsung Galaxy S3 for AT&T (d2att)
7.******Downloaded Omni Rom then moved it to the SD card using My File app
8.******Rebooted to Recovery
9.******Backed up my phone
10.***Did a Factory reset
11.***Did a Wipe Cache
12.***Did a Wipe Davlik
13.***Flashed the Rom with no error
14.***Rebooted to System
15.***Samsung Logo showed up for a sec then the phone is dead
16.***Tried to turn it on by pressing the power button but not working, I thought the battery was drained when I rebooted, I plugged it in for half an hour but still not turning on. Was able to press the power_
17.***Vol down and home and it booted to Download mode
18.***Used Odin to load the image frok step 2 above and tried the same steps using different rom (Beanstalk-4.4) and the result is the same