Description:
A compass points directly toward the morning star. It can only point in one of eight directions: the four cardinal directions (N, S, E, W) or some combination (NW, NE, SW, SE). Otherwise, it will break.
The directions the compass can point.
There are $$$n$$$ distinct points with integer coordinates on a plane. How many ways can you put a compass at one point and the morning star at another so that the compass does not break?
Input Format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^4$$$). The description of the test cases follows.
The first line of each test case contains a single integer $$$n$$$ ($$$2 \leq n \leq 2 \cdot 10^5$$$) — the number of points.
Then $$$n$$$ lines follow, each line containing two integers $$$x_i$$$, $$$y_i$$$ ($$$-10^9 \leq x_i, y_i \leq 10^9$$$) — the coordinates of each point, all points have distinct coordinates.
It is guaranteed that the sum of $$$n$$$ over all test cases doesn't exceed $$$2 \cdot 10^5$$$.
Output Format:
For each test case, output a single integer — the number of pairs of points that don't break the compass.
Note:
In the first test case, any pair of points won't break the compass:
- The compass is at $$$(0,0)$$$, the morning star is at $$$(-1,-1)$$$: the compass will point $$$\text{SW}$$$.
- The compass is at $$$(0,0)$$$, the morning star is at $$$(1,1)$$$: the compass will point $$$\text{NE}$$$.
- The compass is at $$$(-1,-1)$$$, the morning star is at $$$(0,0)$$$: the compass will point $$$\text{NE}$$$.
- The compass is at $$$(-1,-1)$$$, the morning star is at $$$(1,1)$$$: the compass will point $$$\text{NE}$$$.
- The compass is at $$$(1,1)$$$, the morning star is at $$$(0,0)$$$: the compass will point $$$\text{SW}$$$.
- The compass is at $$$(1,1)$$$, the morning star is at $$$(-1,-1)$$$: the compass will point $$$\text{SW}$$$.
In the second test case, only two pairs of points won't break the compass:
- The compass is at $$$(6,9)$$$, the morning star is at $$$(10,13)$$$: the compass will point $$$\text{NE}$$$.
- The compass is at $$$(10,13)$$$, the morning star is at $$$(6,9)$$$: the compass will point $$$\text{SW}$$$.