Conditional Probability, Independence & Bayes · Challenge problem

In a game show there are three doors; a car is behind one of them, chosen uniformly at random, and goats are behind the other two. You pick door $1$. The host, who knows where the car is, then opens one of doors $2$ and $3$ that hides a goat. When both of those doors hide goats, this host opens door $3$ with probability $\tfrac14$ and door $2$ with probability $\tfrac34$. The host opens door $3$. What is the probability that you win the car if you switch to door $2$? Express your answer as a common fraction.