Everlasting L

Matt loves letter L.

A point set P is (a, b)-L if and only if there exists x, y satisfying:

P = {(x, y), (x + 1, y), . . . , (x + a, y), (x, y + 1), . . . , (x, y + b)}(a, b ≥ 1)

A point set Q is good if and only if Q is an (a, b)-L set and gcd(a, b) = 1.

Matt is given a point set S. Please help him find the number of ordered pairs of sets (A, B) such that:

The first line contains only one integer T , which indicates the number of test cases.

For each test case, the first line contains an integer N (0 ≤ N ≤ 40000), indicating the size of the point set S.

Each of the following N lines contains two integers x_i, y_i, indicating the i-th point in S (1 ≤ x_i, y_i ≤ 200). It’s guaranteed that all (x_i, y_i) would be distinct.

For each test case, output a single line “Case #x: y”, where x is the case number (starting from 1) and y is the number of pairs.

2
6
1 1
1 2
2 1
3 3
3 4
4 3
9
1 1
1 2
1 3
2 1
2 2
2 3
3 1
3 2
3 3

Case #1: 2
Case #2: 6

n the second sample, the ordered pairs of sets Matt can choose are:
A = {(1, 1), (1, 2), (1, 3), (2, 1)} and B = {(2, 2), (2, 3), (3, 2)}
A = {(2, 2), (2, 3), (3, 2)} and B = {(1, 1), (1, 2), (1, 3), (2, 1)}
A = {(1, 1), (1, 2), (2, 1), (3, 1)} and B = {(2, 2), (2, 3), (3, 2)}
A = {(2, 2), (2, 3), (3, 2)} and B = {(1, 1), (1, 2), (2, 1), (3, 1)}
A = {(1, 1), (1, 2), (2, 1)} and B = {(2, 2), (2, 3), (3, 2)}
A = {(2, 2), (2, 3), (3, 2)} and B = {(1, 1), (1, 2), (2, 1)}
Hence, the answer is 6.