Iphone 5c keeps shutting itself off…and other issues?

I have a 32GB iphone 5c and about a week ago it started shutting itself off for no reason. It doesn't matter if I'm in the middle of something or if I've just unlocked it, it will shut itself off, display the black screen with the apple logo for about 10-15 seconds and then start working again by itself. I'm running the latest software updates and everything so I'm not sure what's going on. Also, when texting for about the past week as well it seems to be going in slow motion. I can have a whole sentence typed and the letters slowly appear onscreen. Kind of delays when I hit send on that particular text, and then it goes back to normal. I have over 9GB free of my memory so I have no clue why its all of a sudden having issues. Any advice would be greatly appreciated. Thanks!

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