What can I infer about a correlation based on another correlation that shares one variable?

I have three vectors, $$X$$, $$Y$$, and $$Z$$. Each element of $$X$$ is independently normally distributed $$X_isim N(0,sigma^2_X)$$.

The elements of $$Y$$ and $$Z$$ are jointly normally distributed:

$$left[ begin{array}{c}Y_i\ Z_iend{array}right]sim Nleft(left[begin{array}{c}0\ 0end{array}right],left[begin{array}{cc}sigma^2_Y & rho \ rho & sigma^2_Zend{array}right]right)$$

So, naturally, $$E(X’Y)=E(X’Z)=0$$.

However, in finite samples, $$X’Y$$ doesn’t have to be zero.

What can I say about $$E(X’Z)$$ conditional on my sample observation of $$X’Y$$? In other words, can I say anything about $$E(X’Z|X’Y)$$?

I feel like this is probably not a super difficult problem but I’ve come at it a few different ways and not really been able to make much headway (aside from a simulation, which gives me an answer but not why, or how conditional it is on the parameters I set), which makes me think I’m missing something obvious.

While in the middle of writing this up it occurred to me that I could probably calculate a distribution of $$X’Z$$ conditional on $$X’Y$$ from the Wishart distribution. But I, uh, hope it’s simpler than that.