I (21M) have broken broken my girlfriends (22F) trust, how do I move forward?

For some background we live about 3 hours apart and across national borders, so seeing each other is difficult but I believe we have been doing a great job of making it work. We have been dating for 5 months and I really love this girl, I think she's the one and I know she feels the same about me. I seriously can not imagine being with anyone else and I think we have a truly special connection, she deserves the world.

Last night we were on a facetime call when she went to bed around 11, I stayed up for a little bit longer and went to bed at 11:30. I woke up an hour later and was very horny, I haven't seen her in a week so I guess I was pent up. For whatever reason I decided to go onto omegle and started sexting with this random girl. No pictures, videos or personal information was exchanged. There is no way for either of us to contact each other again, but we exchanged sexual messages. As soon as I was finished I felt awful and started crying. I know I would never do anything like this in person.

I have no idea why I did this, I've never done anything like this before. I can truly say that last night was the worst I've ever felt in my entire life, because I know I fucked up in a major way. I really love her and have imagined a future with her ever since we started dating, she's even moving to my city in a few months. So how could I do this? Why did I do this? This is very out of character for me and I know I will never do anything approaching this again.

I haven't told her yet and I'm not even sure if I should. On one hand she deserves to know the truth and it should be up to her if she wants to keep being with me after I've broken her trust. On the other hand I don't want to hurt her, the interaction meant absolutely nothing to me emotionally but I know I would be hurt if she had done the same. This guilt is eating me alive.

I know I will never do anything like this again and I cant even believe I did it this time, it doesn't feel real. She is truly amazing and doesn't deserve this but up until now our relationship has had no issues and we do a great job of communicating with each other., I'm not looking for pity. But I cant hurt her and I don't want to lose this girl. I know I fucked up bad.

TLDR: I've been in a long-distance relationship with a girl I love for about five months, and I genuinely see a future with her. Last night we fell asleep on facetime, I woke up feeling very horny and impulsively went into an anonymous chat room and sexted a stranger. No names, photos, videos, or personal information were exchanged, and there's no way for us to contact each other again. The moment it was over, I felt awful, started crying, and was overwhelmed with guilt because I know I violated my girlfriend's trust. This is completely out of character for me, I've never done anything like it before, and I know I never want to do it again. Now I'm struggling with whether to tell her, because I believe she deserves honesty, but I also don't want to hurt someone I love deeply over a meaningless mistake that I deeply regret.