Between Uniforms

Alice, Bob, and Chet play a game where each one selects a real number between $0$ and $1$. The winner is the person who selects a value between the other two values. Let $A,B,$ and $C$ be the numbers that Alice, Bob, and Chet select, respectively. It is known that $A \sim \unif{0}{1}$ and $B \sim \unif{1/2}{b}$, where $1/2 < b < 1$. Chet gets to select whatever number what wants. What value should Chet select to maximize his probability of winning when $b = 3/4$?