Expectation of product of jointly normal variables conditional on inequality

Question

X, Y and Z are jointly normal with mean zero and a positive definite covariance matrix $\Sigma$. I know if I wanted to compute $E[X|Y+Z-X<A]$ (where A is a constant), I could do a change of variables and use a truncated normal (i.e. let $V\equiv Y+Z-X$ and $\sigma_V$ be the variance of $V$. Then, $E[X|Y+Z-X<A]=\frac{Cov\left(X,\frac{V}{\sigma_V}\right)}{Var\left(\frac{V}{\sigma_V}\right)}\mathbb{E}\left[\frac{V}{\sigma_V}\middle|\frac{V}{\sigma_V}<\frac{A}{\sigma_V}\right]$, which can be easily computed using a truncated standard normal). How would I approach computing $E[X^2|V<A]$ and $E[XY|V<A]$ in a similar way?