Apps closing without warning

On Saturday night my phone started regularly killing off apps without any warning or error messages. Apps just stop.

At first I thought it was the game I was playing (We Rule) but then I noticed it happening with other apps: Including the web browser, and I think even the phone.

I reinstalled some of the apps I saw this problem affecting, and I even reset the phone.

What's the best way for me to find out what's causing this issue?

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