Show that $$\operatorname{vol}(X(t)) = 2^{r-s} \pi^s \frac{t^n}{n!}$$ where $t > 0$, $n = r + 2s$, and $$X(t) := \left\{\mathbf{v} = (x_1,\ldots,x_r,y_1,z_1,\ldots,y_s,z_s) \in \Bbb R^{r+2s}: \sum_{i=1}^r |x_i| + 2\sum_{j=1}^s \sqrt{y_j^2 + z_j^2} < t\right\}$$
Essentially, we want to evaluate $$\int_{\Bbb R^{r+2s}} \mathbf{1}_{X(t)}(\mathbf{v}) \, \mathrm d \mathbf{v} = \int_{\Bbb R^{r+2s}} \mathbf{1}_{X(t)}(x_1,\ldots,x_r,y_1,z_1,\ldots,y_s,z_s)\, \mathrm{d}x_1\ldots \mathrm{d}x_n \mathrm{d}y_1\mathrm{d}z_1\ldots \mathrm{d}y_s \mathrm{d}z_s$$ but I have no clue how to proceed. Perhaps we could make a change of variables? I have been considering $y_j = \rho_j \cos\theta_j$ and $z_j = \rho_j \sin \theta_j$. This gives $\mathrm{d}y_j = \cos\theta_j\mathrm{d} \rho_j - \rho_j \sin\theta_j \mathrm{d}\theta_j$ and $\mathrm{d}z_j = \sin\theta_j \mathrm{d}\rho_j + \rho_j \cos\theta_j \mathrm{d}\theta_j$. How should I proceed? A solution with details would be greatly appreciated!
Context: This calculation appears in Algebraic Number Theory while developing Minkowski bounds in order to calculate class numbers of number fields. $n$ is the degree of the field extension, $r$ is the number of real embeddings, and $s$ is the number of pairs of complex embeddings. Clearly, $r + 2s = n$.