Codeforces 520D. Cubes 状态模拟+贪心

Codeforces 520D

Description

Once Vasya and Petya assembled a figure of m cubes, each of them is associated with a number between 0 and m - 1 (inclusive, each number appeared exactly once). Let's consider a coordinate system such that the OX is the ground, and the OY is directed upwards. Each cube is associated with the coordinates of its lower left corner, these coordinates are integers for each cube.

The figure turned out to be stable. This means that for any cube that is not on the ground, there is at least one cube under it such that those two cubes touch by a side or a corner. More formally, this means that for the cube with coordinates (x, y) either y = 0, or there is a cube with coordinates (x - 1, y - 1), (x, y - 1) or (x + 1, y - 1).

Now the boys want to disassemble the figure and put all the cubes in a row. In one step the cube is removed from the figure and being put to the right of the blocks that have already been laid. The guys remove the cubes in such order that the figure remains stable. To make the process more interesting, the guys decided to play the following game. The guys take out the cubes from the figure in turns. It is easy to see that after the figure is disassembled, the integers written on the cubes form a number, written in the m-ary positional numerical system (possibly, with a leading zero). Vasya wants the resulting number to be maximum possible, and Petya, on the contrary, tries to make it as small as possible. Vasya starts the game.

Your task is to determine what number is formed after the figure is disassembled, if the boys play optimally. Determine the remainder of the answer modulo 10⁹ + 9.

Input

The first line contains number m (2 ≤ m ≤ 10⁵).

The following m lines contain the coordinates of the cubes x_i, y_i ( - 10⁹ ≤ x_i ≤ 10⁹, 0 ≤ y_i ≤ 10⁹) in ascending order of numbers written on them. It is guaranteed that the original figure is stable.

No two cubes occupy the same place.

Output

In the only line print the answer to the problem.

Sample Input

Input

3
2 1
1 0
0 1

Output

19

Input

5
0 0
0 1
0 2
0 3
0 4

Output

2930

先取编号大的 再取编号小的  
每次取完都要更新状态  使图保持稳定
（(x,y)的下面有(x-1,y-1),(x,y-1)或(x+1,y-1)或y=0 都算稳定状态）
每次取前都要判断上方三个的支撑是不是只有当前箱子  
若其中一个是 则不能取 取走当前箱子上方三个都稳定 才能取
取的数放在前面 取得的结果为m进制的数
最后转化为十进制输出

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