410393: GYM104015 K Staircases

You are given a matrix, consisting of $$$n$$$ rows and $$$m$$$ columns.

Each cell of the matrix can be either free or locked.

Let's call a path in the matrix a staircase if it:

• starts and ends in the free cell;
• visits only free cells;
• has one of the two following structures:
  1. the second cell is $$$1$$$ to the right from the first one, the third cell is $$$1$$$ to the bottom from the second one, the fourth cell is $$$1$$$ to the right from the third one, and so on;
  2. the second cell is $$$1$$$ to the bottom from the first one, the third cell is $$$1$$$ to the right from the second one, the fourth cell is $$$1$$$ to the bottom from the third one, and so on.

In particular, a path, consisting of a single cell, is considered to be a staircase.

Here are some examples of staircases:

Initially all the cells of the matrix are free.

You have to process $$$q$$$ queries, each of them flips the state of a single cell. So, if a cell is currently free, it makes it locked, and if a cell is currently locked, it makes it free.

Print the number of different staircases after each query. Two staircases are considered different if there exists such a cell that appears in one path and doesn't appear in the other path.

The first line contains three integers $$$n$$$, $$$m$$$ and $$$q$$$ ($$$1 \le n, m \le 1000$$$; $$$1 \le q \le 10^4$$$) — the sizes of the matrix and the number of queries.

Each of the next $$$q$$$ lines contains two integers $$$x$$$ and $$$y$$$ ($$$1 \le x \le n$$$; $$$1 \le y \le m$$$) — the description of each query.

Print $$$q$$$ integers — the $$$i$$$-th value should be equal to the number of different staircases after $$$i$$$ queries. Two staircases are considered different if there exists such a cell that appears in one path and doesn't appear in the other path.

2 2 8
1 1
1 1
1 1
2 2
1 1
1 2
2 1
1 1

5
10
5
2
5
3
1
0

3 4 10
1 4
1 2
2 3
1 2
2 3
3 2
1 3
3 4
1 3
3 1

49
35
24
29
49
39
31
23
29
27

1000 1000 2
239 634
239 634

1332632508
1333333000