CF1173B.Nauuo and Chess

Nauuo is a girl who loves playing chess.

One day she invented a game by herself which needs $n$ chess pieces to play on a $m\times m$ chessboard. The rows and columns are numbered from $1$ to $m$. We denote a cell on the intersection of the $r$-th row and $c$-th column as $(r,c)$.

The game's goal is to place $n$ chess pieces numbered from $1$ to $n$ on the chessboard, the $i$-th piece lies on $(r_i,\,c_i)$, while the following rule is satisfied: for all pairs of pieces $i$ and $j$, $|r_i-r_j|+|c_i-c_j|\ge|i-j|$. Here $|x|$ means the absolute value of $x$.

However, Nauuo discovered that sometimes she couldn't find a solution because the chessboard was too small.

She wants to find the smallest chessboard on which she can put $n$ pieces according to the rules.

She also wonders how to place the pieces on such a chessboard. Can you help her?

Nauuo 是一位热爱下棋的女孩。

一天，她自己发明了一个游戏，需要 $n$ 枚棋子在 $m \times m$ 的棋盘上进行。棋盘的行与列均从 $1$ 编号至 $m$。我们将第 $r$ 行与第 $c$ 列相交的格子记为 $(r,c)$。

该游戏的目标是将编号为 $1$ 至 $n$ 的 $n$ 枚棋子放置在棋盘上，其中第 $i$ 枚棋子位于 $(r_i,\,c_i)$，且满足如下规则：对任意两枚棋子 $i$ 和 $j$，均有 $|r_i-r_j|+|c_i-c_j|\ge|i-j|$。此处 $|x|$ 表示 $x$ 的绝对值。

然而，Nauuo 发现有时她无法找到满足条件的摆放方案，因为棋盘太小了。

她希望找出能够按上述规则放置 $n$ 枚棋子的最小尺寸棋盘（即最小的 $m$）。

她还想知道：在这样大小的棋盘上，应如何具体摆放这些棋子？你能帮帮她吗？

The only line contains a single integer $n$ ($1\le n\le 1000$) — the number of chess pieces for the game.

唯一的一行包含一个整数 $n$（$1\le n\le 1000$）——游戏所用的棋子数量。

The first line contains a single integer — the minimum value of $m$, where $m$ is the length of sides of the suitable chessboard.

The $i$-th of the next $n$ lines contains two integers $r_i$ and $c_i$ ($1\le r_i,c_i\le m$) — the coordinates of the $i$-th chess piece.

If there are multiple answers, print any.

第一行包含一个整数——满足条件的棋盘边长 $m$ 的最小值。

接下来的 $n$ 行中，第 $i$ 行包含两个整数 $r_i$ 和 $c_i$（$1\le r_i,c_i\le m$）——表示第 $i$ 个棋子的坐标。

若存在多个答案，输出任意一个即可。

In the first example, you can't place the two pieces on a $1\times1$ chessboard without breaking the rule. But you can place two pieces on a $2\times2$ chessboard like this:

In the second example, you can't place four pieces on a $2\times2$ chessboard without breaking the rule. For example, if you place the pieces like this:

then $|r_1-r_3|+|c_1-c_3|=|1-2|+|1-1|=1$, $|1-3|=2$, $1 \lt 2$; and $|r_1-r_4|+|c_1-c_4|=|1-2|+|1-2|=2$, $|1-4|=3$, $2 \lt 3$. It doesn't satisfy the rule.

However, on a $3\times3$ chessboard, you can place four pieces like this:

在第一个例子中，你无法在 $1\times1$ 的棋盘上放置两个棋子而不违反规则。但你可以在 $2\times2$ 的棋盘上这样放置两个棋子：

在第二个例子中，你无法在 $2\times2$ 的棋盘上放置四个棋子而不违反规则。例如，若你按如下方式放置棋子：

则有 $|r_1-r_3|+|c_1-c_3|=|1-2|+|1-1|=1$，而 $|1-3|=2$，满足 $1 \lt 2$；且 $|r_1-r_4|+|c_1-c_4|=|1-2|+|1-2|=2$，而 $|1-4|=3$，满足 $2 \lt 3$。这不满足规则。

然而，在 $3\times3$ 的棋盘上，你可以这样放置四个棋子：