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D. Rainbow Rectangles
time limit per test
4 seconds
memory limit per test
512 megabytes
input
standard input
output
standard output

Shrimpy Duc is a fat and greedy boy who is always hungry. After a while of searching for food to satisfy his never-ending hunger, Shrimpy Duc finds M&M candies lying unguarded on a $$$L \times L$$$ grid. There are $$$n$$$ M&M candies on the grid, the $$$i$$$-th M&M is currently located at $$$(x_i + 0.5, y_i + 0.5),$$$ and has color $$$c_i$$$ out of a total of $$$k$$$ colors (the size of M&Ms are insignificant).

Shrimpy Duc wants to steal a rectangle of M&Ms, specifically, he wants to select a rectangle with integer coordinates within the grid and steal all candies within the rectangle. Shrimpy Duc doesn't need to steal every single candy, however, he would like to steal at least one candy for each color.

In other words, he wants to select a rectangle whose sides are parallel to the coordinate axes and whose left-bottom vertex $$$(X_1, Y_1)$$$ and right-top vertex $$$(X_2, Y_2)$$$ are points with integer coordinates satisfying $$$0 \le X_1 < X_2 \le L$$$ and $$$0 \le Y_1 < Y_2 \le L$$$, so that for every color $$$1 \le c \le k$$$ there is at least one M&M with color $$$c$$$ that lies within that rectangle.

How many such rectangles are there? This number may be large, so you only need to find it modulo $$$10^9 + 7$$$.

Input

The first line contains three positive integers $$$n, k, L$$$ $$$(1 \leq k \leq n \leq 2 \cdot 10^3, 1 \leq L \leq 10^9 )$$$ — the number of M&Ms, the number of colors and the length of the grid respectively.

The following $$$n$$$ points describe the M&Ms. Each line contains three integers $$$x_i, y_i, c_i$$$ $$$(0 \leq x_i, y_i < L, 1 \le c_i \le k)$$$ — the coordinates and color of the $$$i$$$-th M&M respectively.

Different M&Ms have different coordinates ($$$x_i \ne x_j$$$ or $$$y_i \ne y_j$$$ for every $$$i \ne j$$$), and for every $$$1 \le c \le k$$$ there is at least one M&M with color $$$c$$$.

Output

Output a single integer — the number of rectangles satisfying Shrimpy Duc's conditions, modulo $$$10^9 + 7$$$.

Examples
Input
4 2 4
3 2 2
3 1 1
1 1 1
1 2 1
Output
20
Input
5 3 10
6 5 3
5 3 1
7 9 1
2 3 2
5 0 2
Output
300
Input
10 4 10
5 4 4
0 0 3
6 0 1
3 9 2
8 7 1
8 1 3
2 1 3
6 3 2
3 5 3
4 3 4
Output
226
Note

Grid for the first sample: