Shuts down how? It just goes completely black? Or does the screen go black but the lights are still on? That's probably a software problem already. The only way to fix it is either going to a store or contacting Nintendo (I'm not too sure reformatting might so the trick, but it might not). I have a New 3ds XL, no problems (but it has been dropped multiple times already)
I don't know what Nintendo is like where you live, but when my 3DS XL's camera was defect and I sent it to Nintendo for it to be repaired they were letting me choose whether I wanted my data to be reset or kept if possible. I still had all my data when I got my 3DS back. I'm from The Netherlands btw.