Problem 212

Combined Volume of Cuboids

An axis-aligned cuboid, specified by parameters { (x_{0},y_{0},z_{0}), (dx,dy,dz) }, consists of all points (X,Y,Z) such that x_{0} X x_{0}+dx, y_{0} Y y_{0}+dy and z_{0} Z z_{0}+dz. The volume of the cuboid is the product, dx dy dz. 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},...,C_{50000} be a collection of 50000 axis-aligned cuboids such that C_{n} has parameters

x_{0} = S_{6n-5} modulo 10000

y_{0} = S_{6n-4} modulo 10000

z_{0} = S_{6n-3} modulo 10000

dx = 1 + (S_{6n-2} modulo 399)

dy = 1 + (S_{6n-1} modulo 399)

dz = 1 + (S_{6n} modulo 399)

where S_{1},...,S_{300000} come from the "Lagged Fibonacci Generator":

For 1 `k` 55, S_{k} = [100003 - 200003`k` + 300007`k`^{3}] (modulo 1000000)

For 56 `k`, S_{k} = [S_{k-24} + S_{k-55}] (modulo 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},...,C_{100}, is 723581599.

What is the combined volume of all 50000 cuboids, C_{1},...,C_{50000} ?

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