Permanently Banned Wrongfully

Hey all, I've been playing league of legends since season one, took a break for a few years and got back into it about a month ago. I have been playing a lot over the last month with friends, and enjoying getting back into the game. I also want to note that there is a real value in this account to me, I've spent a lot of money and time over the years on this account and formed a lot of memories playing this game. I checked my emails when I got up this morning and saw an email saying I was permabanned, I assumed this would be just a phishing email but, when I tried to log in, I found out that I was permabanned. It says I've been permanently banned for "the use of prohibited third party software". I'm really at a loss for what to do. I submitted a ticket, figuring that was the best place to start. I've since logged on and checked my match history on a website to find that, hours after I went to bed, someone had played a game on my account. If you check yourself, they played as teemo (that wasn't me). Every other game in my match history I can remember playing, but that game I didn't play. I understand this sounds incredibly unbelievable, I've read dozens of posts like this on the forums and other sites over the years and have always seen them as probably someone's last ditch effort at getting unbanned when they did something wrong. But this is an account I really care about. I have never hacked, I think my rank, cs, score etc. are enough to attest to that. But please, please can someone tell me what I can do to get riot to believe me? Feeling so lost. EDIT: I've checked now and there were actually about 4 games played after I finished playing for the night.
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