CF1598E.Staircases

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:
  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.

你被给定一个由 $n$ 行和 $m$ 列组成的矩阵。行从上到下编号，列从左到右编号。

矩阵中的每个单元格要么是空闲的，要么是被锁定的。

我们称矩阵中的一条路径为阶梯路径（staircase），如果它满足以下条件：

• 起点和终点均为空闲单元格；
• 仅经过空闲单元格；
• 具有以下两种结构之一：
  1. 第二个单元格位于第一个单元格右侧一格，第三个单元格位于第二个单元格下方一格，第四个单元格位于第三个单元格右侧一格，依此类推；
  2. 第二个单元格位于第一个单元格下方一格，第三个单元格位于第二个单元格右侧一格，第四个单元格位于第三个单元格下方一格，依此类推。

特别地，仅包含单个单元格的路径也被视为一条阶梯路径。

以下是若干阶梯路径的示例：

初始时，矩阵中所有单元格均为空闲状态。

你需要处理 $q$ 个查询，每个查询将翻转（切换）某一个单元格的状态：若当前为空闲，则变为锁定；若当前为锁定，则变为空闲。

在每次查询后，请输出当前矩阵中不同的阶梯路径总数。若存在某个单元格，它出现在其中一条路径中但不出现在另一条路径中，则认为这两条阶梯路径不同。

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.

第一行包含三个整数 $n$、$m$ 和 $q$（$1 \le n, m \le 1000$；$1 \le q \le 10^4$）—— 分别表示矩阵的尺寸以及查询次数。

接下来的 $q$ 行，每行包含两个整数 $x$ 和 $y$（$1 \le x \le n$；$1 \le y \le m$）—— 描述每次查询。

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.

输出 $q$ 个整数——第 $i$ 个值应等于执行前 $i$ 次查询后不同楼梯路径的数量。若存在某个格子，它出现在其中一条路径中但不出现在另一条路径中，则认为这两条楼梯路径不同。