What does it take to prevent mass shootings?

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About 1000 students walked out of school and gathered at the Minnesota Capitol as part of nationwide protests against political inaction on preventing gun violence. April 2018 in St. Paul, Minnesota. Credit: Fibonacci Blue/CC BY 2.0.

President Trump today listed steps he believes will stop mass shootings. He suggested creating tools to identify early warning signs for shooters, reforming mental health laws, and preventing people with "red flags" to get guns. But what has the government been doing up to this point to stop attacks? What should they be doing?