Find $E(X_1|X_2\leq x_2, X_3\leq x_3)$ where $(X_1,X_2,X_3)$ is multivariate normal

Question

Let $(X_1\; X_2\; X_3)^{T} \sim N(\mu,\Sigma)$. We want to find $E(X_1|X_2\leq x_2, X_3\leq x_3)$. In my course we are taught conditional expectation in a rigorous way unlike I previously knew. So I understand the meaning of $E(X_1|X_2,X_3)$ but I don't understand the expression given above. I tried to solve this with brute force but I couldn't do it. I have found the conditional probability distribution to be $$F(X_1\le x|X_2\le x_2,X_3\le x_3)=\dfrac{\int_{-\infty}^{x_3}\int_{-\infty}^{x_2}\int_{-\infty}^{x}\exp\left(-\frac{1}{2}(\mathbf{y}-\mathbf{\mu})^{T}\Sigma^{-1/2}(\mathbf{y}-\mathbf{\mu})\right)\;\mathrm{d}y_1\mathrm{d}y_2\mathrm{d}y_3}{\int_{-\infty}^{x_3}\int_{-\infty}^{x_2}\int_{-\infty}^{\infty}\exp\left(-\frac{1}{2}(\mathbf{y}-\mathbf{\mu})^{T}\Sigma^{-1/2}(\mathbf{y}-\mathbf{\mu})\right)\;\mathrm{d}y_1\mathrm{d}y_2\mathrm{d}y_3}$$ Here we take $y=(y_1\; y_2 \; y_3)^{T}$. Can someone help me out?