Expectation of a function of three random variables

Question

I have three random variables $X, Y,$ and $Z$. $X=a$ whenever $Y \geq b$, and $X=0$ whenever $Y<b$, where $a$ and $b$ are constants. I have some function $g(X,Y,Z)$ of all three variables. I know the pdfs of $f_y$ and $f_z$. Also, $Y$ and $Z$ are independent. However, $X$ and $Y$ are dependent since $X$ is a function of $Y$. My question is how do I find $E(g(X,Y,Z))$ over the random variables I have?