Box intersection volume

There are $N$ boxes in three dimensional space. Each box is given by two opposite corners, and every face is parallel to a coordinate plane.

Write a program that computes the volume of the region covered by all $N$ boxes at once, that is, the volume of their intersection.

The first line contains the number of boxes $N$ ($1 \le N \le 1000$).

Each of the next $N$ lines contains six integers $x_1$, $y_1$, $z_1$, $x_2$, $y_2$, $z_2$ separated by spaces. The first three are the coordinates of one corner and the last three are the coordinates of the opposite corner. All coordinates are between $1$ and $1000$, and $x_1 < x_2$, $y_1 < y_2$, $z_1 < z_2$.

Print the volume of the intersection of the $N$ boxes on the first line. If the intersection is empty or has no thickness, print $0$.