Break-up or not (?)

I have no idea what to do since I found out that my boyfriend whom I dated for almost a year is a compulsive liar. Everything started when I accidentally discovered his Google drive backup that included his ex gfs nudes. Before I saw all those photos I was trusting his every word %100. We had an argument over those photos because it's both unethical for him to keep those photos and I don't think he should keep them while we're in an on going relationship because it made me insecure. Long story short he told me they were only back ups from an old phone and backed them up long time ago and he didn't even cherry picked the nudes from the others he simply just procrastinated to delete them. Even after we made up I felt like my trust for him was shattering so I went into his phone while he was sleeping (and I know it was a horrible act but the photos made me question his faithfulness) and found out he has been talking with his ex fbuddy for months and deleting all their messages and call history. He called her many times the day we got into the fight about the photos (no wonder why) . When I confronted him he told me that they only talked through phone and that was all and he also admitted that he was flirting with other girls to fix his low self esteem while proving himself that he "still has it". He insisted it might be a hypocritical act but it wasn't cheating because nothing physical happened. We broke up and I told him that I don't ever want to see him again. After I ended our relationship he admitted that he had been lying since the beginning of our relationship about lots of things. Such as his previous sexual intercourses, the places he has been, the books he read, his stories about past experiences. Even his so called dead dog. He voluntarily admitted all his lies by himself and said he didn't even realised he lied all those things until he lost me. Now he says he wants to come clean apologises repeatedly to save our relationship. I know he did it all out of his low self esteem but trust is my number one priority in a relationship and now I don't know what to do. I tought we were open and frank to each other since we started our relationship as two friends and to be honest it was an amazing relationship until the very end. We were able to communicate during our conflicts without disrespecting and upsetting each other (we still do), we helped each other in terms of our other psychological problems, we were basically trying to build a future together by supporting each other, we had great time together that I will never regret and now I can't decide whether the relationship deserves another shot or not cause I don't think I can handle the mistrust and being labelled as cheated gf. I can't bring myself to believe anything he says. He now started therapy and wants to build trust again but I don't know it will ever gonna happen. I suggested him them we should just hang out and stay low key without any labels or so and maybe see other people like we're in an open relationship because I think it's the only way to not end up as the stupid cheated one with lower rights. I don't wanna do this just with the "an eye for an eye" motive but I somehow want to protect myself. What do you suggest me or us to do?