The Problem
Let $Y \sim N(0,B)$ and $Z \sim N(0,A)$. I would like to find $$\boxed{\mathbb{E}\left[|YZ|\right]}$$ This seems like a natural question that would arise in probability theory, but I have not been able to find any results online or in textbooks. (Note: Here $Y$,$Z$ are NOT necessarily independent. However, they are jointly normal with known pdf $f_{Y,Z}(y,z)$; nevertheless simply evaluating the double integral to directly find $\mathbb{E}[|YZ|]$ directly seems to not work.)
The Attempt
By the law of total expectation, we have that $\mathbb{E} \left[|YZ| \right] = \mathbb{E} \left[\mathbb{E}[(\mid YZ \mid )|Z] \right]$. So, first we will try to find $\mathbb{E}[(\mid YZ \mid )|Z]$. Well, $$\mathbb{E}[(\mid YZ \mid )|Z]=|Z| \mathbb{E}[(\mid Y\mid) |Z]$$
We note that $|Y|$ follows a half-normal distribution. This is where I am stuck. I'm not sure where to go from here, perhaps I need to use some results pertaining to the folded normal dist. Or maybe try a different approach entirely to find $\mathbb{E}[|YZ|]$.
Edit: We have that $Y|Z \sim N \left(0+\frac{\sqrt{B}}{\sqrt{A}}\rho(z-0),(1-\rho^2 )B \right)$ where $\rho$ is the correlation coefficient. Then $|Y|Z|$ follows a folded normal distribution, so we can get $\mathbb{E}[|Y|Z|]$ from that. The issue here is that we want $\mathbb{E}[(|Y|)|Z]$.
Any help with this problem is immensely appreciated. Thank you!
The answer is found in Corollary 3.1 in this - Guassian Integrals Involving Absolute Value Functions