Combined Volume of Cuboids

An axis-aligned cuboid, specified by parameters ${(x_{0}, y_{0}, z_{0}), (d x, d y, d z)}$ , consists of all points $(X, Y, Z)$ such that $x_{0} \leq X \leq x_{0} + d x$ , $y_{0} \leq Y \leq y_{0} + d y$ and $z_{0} \leq Z \leq z_{0} + d z$ . The volume of the cuboid is the product, $d x \times d y \times d z$ . The combined volume of a collection of cuboids is the volume of their union and will be less than the sum of the individual volumes if any cuboids overlap.

Let $C_{1}, \dots, C_{50000}$ be a collection of $50000$ axis-aligned cuboids such that $C_{n}$ has parameters

\begin{aligned} x_{0} & = S_{6 n - 5} mod 10000 \\ y_{0} & = S_{6 n - 4} mod 10000 \\ z_{0} & = S_{6 n - 3} mod 10000 \\ d x & = 1 + (S_{6 n - 2} mod 399) \\ d y & = 1 + (S_{6 n - 1} mod 399) \\ d z & = 1 + (S_{6 n} mod 399) \end{aligned}

where $S_{1}, \dots, S_{300000}$ come from the "Lagged Fibonacci Generator":

For $1 \leq k \leq 55$ , $S_{k} = [100003 - 200003 k + 300007 k^{3}] (\mod 1000000)$ .
For $56 \leq k$ , $S_{k} = [S_{k - 24} + S_{k - 55}] (\mod 1000000)$ .

Thus, $C_{1}$ has parameters ${(7, 53, 183), (94, 369, 56)}$ , $C_{2}$ has parameters ${(2383, 3563, 5079), (42, 212, 344)}$ , and so on.

The combined volume of the first $100$ cuboids, $C_{1}, \dots, C_{100}$ , is $723581599$ .

What is the combined volume of all $50000$ cuboids, $C_{1}, \dots, C_{50000}$ ?

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.