Should I let go? How? :(

My ex girlfriend and I met 13 years ago, when we were still in school. We became best friends and shared everything with each other. When we were about 19 year old we discovered we were in love - both of us had dated boys before so it was all new. We decided to give it a try, and it worked perfectly. We were together 6 year and half, and of course during those years we had ups and downs but we loved each other dearly. We were very happy , we made plans for our future and though we never actually lived in together we were inseparable. We were best friends and lovers.
Back in June, out of nowhere she said she needed break, and though I asked her multiple times if she was seeing someone else, she denied it. A couple of weeks later I found out that in fact, there was another girl. We grew apart and I decided to take the opportunity to move to another country to study (thought it would help me overcome the unbearable depression after our breakup). Just before I moved in mid September, she said she had "realized how much she loved me and that she was about to lose me", so we kinda got back together. We agreed to wait a couple months, and she would try to join me here next semester. Things were great the first two weeks. She was back to being the girl I fell in love with. And then, all of a sudden, she changed. I asked her and she confessed that she had met another girl (like 3 weeks ago) and that she is madly in love with her (they have only seen each other ONCE, but they "talk a lot and fell in love" - just to clarify this is a different girl than the one she was seeing in June). She wants us to stay friends because "I'll always be her best friend, but we can't be together because she loves someone else".

So here I am, in a foreign country where I don't no anybody, going through our breakup all over again
I just can't understand how can someone fall in and out of love so fast?
I really really love her, but I'm very hurt. I'm not sure how to handle the pain, because I'm not only losing my girlfriend but also my best friend. Should I try to stay friends with her? Should I move on?