E. Manhattan Triangle
time limit per test
3 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

The Manhattan distance between two points $$$(x_1, y_1)$$$ and $$$(x_2, y_2)$$$ is defined as: $$$$$$|x_1 - x_2| + |y_1 - y_2|.$$$$$$

We call a Manhattan triangle three points on the plane, the Manhattan distances between each pair of which are equal.

You are given a set of pairwise distinct points and an even integer $$$d$$$. Your task is to find any Manhattan triangle, composed of three distinct points from the given set, where the Manhattan distance between any pair of vertices is equal to $$$d$$$.

Input

Each test consists of multiple test cases. The first line contains one integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases. The description of the test cases follows.

The first line of each test case contains two integers $$$n$$$ and $$$d$$$ ($$$3 \le n \le 2 \cdot 10^5$$$, $$$2 \le d \le 4 \cdot 10^5$$$, $$$d$$$ is even) — the number of points and the required Manhattan distance between the vertices of the triangle.

The $$$(i + 1)$$$-th line of each test case contains two integers $$$x_i$$$ and $$$y_i$$$ ($$$-10^5 \le x_i, y_i \le 10^5$$$) — the coordinates of the $$$i$$$-th point. It is guaranteed that all points are pairwise distinct.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.

Output

For each test case, output three distinct integers $$$i$$$, $$$j$$$, and $$$k$$$ ($$$1 \le i,j,k \le n$$$) — the indices of the points forming the Manhattan triangle. If there is no solution, output "$$$0\ 0\ 0$$$" (without quotes).

If there are multiple solutions, output any of them.

Example
Input
6
6 4
3 1
0 0
0 -2
5 -3
3 -5
2 -2
5 4
0 0
0 -2
5 -3
3 -5
2 -2
6 6
3 1
0 0
0 -2
5 -3
3 -5
2 -2
4 4
3 0
0 3
-3 0
0 -3
10 8
2 1
-5 -1
-4 -1
-5 -3
0 1
-2 5
-4 4
-4 2
0 0
-4 1
4 400000
100000 100000
-100000 100000
100000 -100000
-100000 -100000
Output
2 6 1
4 3 5
3 5 1
0 0 0
6 1 3
0 0 0
Note

In the first test case:

Points $$$A$$$, $$$B$$$, and $$$F$$$ form a Manhattan triangle, the Manhattan distance between each pair of vertices is $$$4$$$. Points $$$D$$$, $$$E$$$, and $$$F$$$ can also be the answer.

In the third test case:

Points $$$A$$$, $$$C$$$, and $$$E$$$ form a Manhattan triangle, the Manhattan distance between each pair of vertices is $$$6$$$.

In the fourth test case, there are no two points with a Manhattan distance of $$$4$$$, and therefore there is no suitable Manhattan triangle.