Staircases
Description
You are given a matrix, consisting of $n$ rows and $m$ columns. The rows are numbered top to bottom, the columns are numbered left to right.
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:
- 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;
- 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.
Input
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.
Output
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.
Samples
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