(EX) boyfriend

alright so me and my ex boyfriend dated for what would have been ten months today and we broke up two weeks ago because i slept with another guy a couple times because he neglected me a lot and (not trying to justify) but i was lonely. he was told at a party (not by me) and the next day ended things. we had something pretty serious going on and we said we were in love. the only chance i got to talk was over text and he told me that we were dont and to go to hell along with calling me trash and saying he didnt love me- i dont believe thats true- hes just hurting. i wrote this long letter which isnt necessarily asking for him back but more so apologizing. we havent spoken in a while. i tried to give him the letter and when i said hi he shook his head and walked away.. how do i get this guy to even think about working through things? like where do i even start ++ is it even possible?