## Problem F. Four-dimensional polytope ≡

 Author: A. Baranov Time limit: 1 sec Input file: Standard input Memory limit: 256 Mb Output file: Standard output

### Statement

Consider some convex four-dimensional polytope that presented by a mesh composed of such elements as
vertices, edges, faces and three-dimensional bodies.

Each vertex is represented by four coordinates (x, y, z, w).
Each edge is a straight line segment and is represented by a pair of vertices connected by it.
Each face is a convex polygon and is represented by a set of its edges.
Each body is a convex polyhedron and is represented by a set of its faces.

The polytope itself is specified by set of three-dimensional bodies bounding it.

You program must calculate volume of a cross-section of the polytope by 3D sub-space with w = 0.

### Input format

Input data contains sequence of the mesh elements.

First there is the integer V, followed by exactly 4 ⋅ V real numbers that are coordinates of the vertices.

Next, integer E, followed by exactly 2 ⋅ E numbers of the vertices, defining the edges pairwise.

Next is the integer F, followed by exactly F faces represented in the following format.

First integer number of edges N, followed by N indices of edges of this face.

The bodies specified by set of their faces are written in the same way.

Indices of the all elements start from zero.

### Output format

Output data must contain the volume with an accuracy of at least 5 digits after decimal point.

### Constraints

It is guaranteed that all mesh elements are non-degenerate.

Vertex coordinates are in the range from  − 10 to 10.

Count of elements of each kind does not exceed 1000.

### Sample tests

No. Standard input Standard output
1
5
-1.00000 -1.00000 -1.00000 -0.50000
0.00000  1.00000 -1.00000 -0.50000
1.00000  0.00000 -1.00000 -0.50000
0.00000  0.00000  1.00000 -0.50000
0.00000  0.00000  0.00000  0.50000

10
0 1
0 2
0 3
1 2
1 3
2 3
0 4
1 4
2 4
3 4

10
3 0 1 3
3 1 2 5
3 0 2 4
3 3 4 5
3 0 6 7
3 1 6 8
3 2 6 9
3 3 7 8
3 4 7 9
3 5 8 9

5
4 0 1 2 3
4 0 4 5 7
4 1 5 6 9
4 2 4 6 8
4 3 7 8 9

0.12500

0.776s 0.152s 15