What are three ways that government might gain legitimacy

Well, experimentally, it's impossible, because of human nature. But theoretically, if extremely lawful people were appointed, that's one way. Another is if they had a serious reason to (like if they were given a legitimately extreme threat) and one more way is if there were one council or group to monitor them. the problem is, the council could become corrupt as well, and that system has certainly been tried.