My ex (24M) broke up with me (23F) and I’m thinking of writing him a letter?

My ex and I had a huge fight a little more than a month ago and he said he needed time to process things. However, I gnatted a lot and 2 weeks ago he broke up with me, saying he has felt tortured and can’t stand it anymore, he said he doesn’t know how to make himself care about me again. I stopped texting him and decided to go into NC for some time, but we saw each other a couple of times at the gym. I’ve apologised many times but I feel like he will never forgive me (I made a lot of mistakes and I deeply regret them). I told him I’m gonna change and eliminate my negative traits but he doesn’t seem to believe. I’ve decided to send him a letter maybe a month after the break up, saying how much I love him, how much I’m sorry, etc, etc (I’ve already written it and I think it’s very emotional and poetic and it will make him feel something) but I’m scared of what will happen. Do you guys think sending a letter is appropriate in my situation?

Tl dr - My ex (24M) broke up with me (23F) and I’m thinking of writing him a letter. Do you think that’s appropriate?
My ex (24M) broke up with me (23F) and I’m thinking of writing him a letter?
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