Marginalizing the product of two conditional distributions sharing a joint

Question

I have a joint normal distribution $$\mathcal{N}\left(\left.\begin{bmatrix}X\\ Y\\ Z\end{bmatrix}\right|\mu, \Sigma\right) $$ I want to find the following quantity $$\int \mathcal{N}(X|Y)\mathcal{N}(Z|X)dX$$ My approach so far has been to rewrite the problem as $$\int\mathcal{N}(X|Y)\dfrac{\mathcal{N}(X|Z)\mathcal{N}(Z)}{\mathcal{N}(X)}dX=\mathcal{N}(Z)\int\dfrac{\mathcal{N}(X|Y)\mathcal{N}(X|Z)}{\mathcal{N}(X)}dX$$ The goal of this approach is that then I could use the product of Gaussians (https://compbio.fmph.uniba.sk/vyuka/ml/old/2008/handouts/matrix-cookbook.pdf (Sec. 8.1.8)) to first rewrite the numerator and then use the formula for the quotient of Gaussians (https://davmre.github.io/blog/statistics/2015/03/27/gaussian_quotient) to address the result and the denominator. The issue I am having is that the math becomes incredibly tedious and I have some very nasty matrix equations that would need to be simplified. I am curious if there are any simple formulas or properties that could help address this problem.