Polyhedron

Input

The first line contains the number of test cases $T$ (1 \leq T \leq 100). Each test case begins with a line containing the number of points $N$ (4 \leq N \leq 30). Each of the next $N$ lines contains three integers: $X$, $Y$, $Z$ (-1000 \leq X, Y, Z \leq 1000) — the coordinates of a point. Some points may coincide! It is guaranteed that any convex polyhedron containing all the given points has positive volume.

Output

For each test case, output a line in the format "Case #$A$:", where $A$ is the test number (starting from $1$), followed by $M$ more lines, where $M$ is the number of distinct face types (the type of a face is determined by the number of its vertices).

On each of the following $M$ lines, output two numbers: $k$ — the number of vertices in a face, and q_k — the number of faces with $k$ vertices in the polyhedron. After $k$, print a colon ":", a space, and then q_k. Output the results in increasing order of $k$.

The number of vertices of a face is determined geometrically: if a given point lies on the edge of a face, it is not counted as a vertex; if several points coincide at the same vertex, they are counted as one vertex.

Examples