Xavier, Yolanda, and Zoe play a game that has $7$ rounds. In each round there is a single winner, and the outcomes of the rounds are independent. For each round the probability that Xavier wins is $\frac{1}{4}$ and Yolanda is twice as likely to win as Zoe. What is the probability that Xavier wins two rounds, Yolanda wins three rounds, and Zoe wins two rounds?