CF1619G.Unusual Minesweeper

Polycarp is very fond of playing the game Minesweeper. Recently he found a similar game and there are such rules.

There are mines on the field, for each the coordinates of its location are known ( $x_i, y_i$ ). Each mine has a lifetime in seconds, after which it will explode. After the explosion, the mine also detonates all mines vertically and horizontally at a distance of $k$ (two perpendicular lines). As a result, we get an explosion on the field in the form of a "plus" symbol ('+'). Thus, one explosion can cause new explosions, and so on.

Also, Polycarp can detonate anyone mine every second, starting from zero seconds. After that, a chain reaction of explosions also takes place. Mines explode instantly and also instantly detonate other mines according to the rules described above.

Polycarp wants to set a new record and asks you to help him calculate in what minimum number of seconds all mines can be detonated.

The first line of the input contains an integer $t$ ( $1 \le t \le 10^4$ ) — the number of test cases in the test.

An empty line is written in front of each test suite.

Next comes a line that contains integers $n$ and $k$ ( $1 \le n \le 2 \cdot 10^5$ , $0 \le k \le 10^9$ ) — the number of mines and the distance that hit by mines during the explosion, respectively.

Then $n$ lines follow, the $i$ -th of which describes the $x$ and $y$ coordinates of the $i$ -th mine and the time until its explosion ( $-10^9 \le x, y \le 10^9$ , $0 \le timer \le 10^9$ ). It is guaranteed that all mines have different coordinates.

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

Print $t$ lines, each of the lines must contain the answer to the corresponding set of input data — the minimum number of seconds it takes to explode all the mines.

Picture from examplesFirst example:

• $0$ second: we explode a mine at the cell $(2, 2)$ , it does not detonate any other mine since $k=0$ .
• $1$ second: we explode the mine at the cell $(0, 1)$ , and the mine at the cell $(0, 0)$ explodes itself.
• $2$ second: we explode the mine at the cell $(1, 1)$ , and the mine at the cell $(1, 0)$ explodes itself.

Second example:

• $0$ second: we explode a mine at the cell $(2, 2)$ we get:

- $1$ second: the mine at coordinate $(0, 0)$ explodes and since $k=2$ the explosion detonates mines at the cells $(0, 1)$ and $(1, 0)$ , and their explosions detonate the mine at the cell $(1, 1)$ and there are no mines left on the field.