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You simultaneously roll 5 dice, and if all 5 dice are the same value, you win....

You simultaneously roll 5 dice, and if all 5 dice are the same value, you win. If your dice are not all the same, you get to re-roll all 5 dice again. You get three tries in total. What is the probability of winning? That is, what is the probability that in your three tries, at least one of your rolls consists of five-of-a-kind?

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