Note: The only difference between this chapter and chapter 2 is that here, coordinates are only up to (\mathbf{3{,}000}).
Boss Rob just planted (N) happy little trees in his yard, which can be represented on a Cartesian plane. The (i)th tree is located at coordinates (t_i = (A_i, B_i)). Now, he's looking for the best spot to build a well in order to provide water to them. He considers the inconvenience of a potential well location (p) to be the sum of the squared Euclidean distances to every tree:
[\sum_{i=1}^{N} \Vert ,p - t_i \Vert ^ 2 ]
Rob wants to pick a location for his well, well... well. Help him determine the inconvenience for (Q) different potential well locations, ((X_1, Y_1), ..., (X_Q, Y_Q)). To reduce output size, please print the sum of inconveniences for all potential well locations, modulo (1{,}000{,}000{,}007).
Constraints
(1 \le T \le 55) (1 \le N, Q \le 500{,}000) (0 \le A_i, B_i, X_i, Y_i \le \mathbf{3{,}000}) All ((A_i, B_i)) are distinct within a given test case. All ((X_i, Y_i)) are distinct within a given test case.
The total sum of (N) and (Q) across all test cases is at most (3{,}000{,}000).
Input Format
Input begins with a single integer (T), the number of test cases. For each case, there is first a line containing a single integer (N). Then, (N) lines follow, the (i)th of which contains two space-separated integers (A_i) and (B_i). Then there is a line containing a single integer (Q). Then, (Q) lines follow, the (i)th of which contains two space-separated integers (X_i) and (Y_i).
Output Format
For the (i)th test case, print a line containing "Case #i: ", followed by a single integer, the sum of inconveniences for all (Q) well locations, modulo (1{,}000{,}000{,}007).
Sample Explanation
The first two sample cases are depicted below:
{{PHOTO_ID:3159344294326237|WIDTH:700}}
In the first case, the total inconvenience is (18 + 34 = 52):
- For the well at ((2, 5)), the inconvenience is the sum of the squared Euclidean distance to both trees, which is (3^2 + 3^2 = 18).
- For the well at ((6, 6)), the inconvenience is (32 + 2 = 34).
In the second case, the total inconvenience is (47 + 31 + 53 = 131):
- For the well at ((3, 1)), the inconvenience is (4 + 5 + 13 + 25 = 47).
- For the well at ((5, 2)), the inconvenience is (17 + 2 + 2 + 10 = 31).
- For the well at ((6, 5)), the inconvenience is (41 + 8 + 4 + 0 = 53).