CF711B.Chris and Magic Square

ZS the Coder and Chris the Baboon arrived at the entrance of Udayland. There is a n × n magic grid on the entrance which is filled with integers. Chris noticed that exactly one of the cells in the grid is empty, and to enter Udayland, they need to fill a positive integer into the empty cell.

Chris tried filling in random numbers but it didn't work. ZS the Coder realizes that they need to fill in a positive integer such that the numbers in the grid form a magic square. This means that he has to fill in a positive integer so that the sum of the numbers in each row of the grid (), each column of the grid (), and the two long diagonals of the grid (the main diagonal — and the secondary diagonal — ) are equal.

Chris doesn't know what number to fill in. Can you help Chris find the correct positive integer to fill in or determine that it is impossible?

ZS 编码员和 Chris 猿猴抵达了乌代兰（Udayland）的入口。入口处有一个 $n \times n$ 的魔法方阵，其中填满了整数。Chris 发现方阵中恰好有一个格子是空的，而要进入乌代兰，他们需要在该空格中填入一个正整数。

Chris 尝试随机填入一些数字，但均未成功。ZS 编码员意识到，他们必须填入一个正整数，使得整个方阵成为一个幻方。也就是说，他需要填入一个正整数，使得方阵中每一行的数字之和（）、每一列的数字之和（），以及两条主对角线的数字之和（主对角线 — ，副对角线 — ）都相等。

Chris 不知道该填入什么数字。你能帮 Chris 找出应填入的正确正整数，或判断这是不可能的吗？

The first line of the input contains a single integer n (1 ≤ n ≤ 500) — the number of rows and columns of the magic grid.

n lines follow, each of them contains n integers. The j-th number in the i-th of them denotes a__i, j (1 ≤ a__i, j ≤ 109 or a__i, j = 0), the number in the i-th row and j-th column of the magic grid. If the corresponding cell is empty, a__i, j will be equal to 0. Otherwise, a__i, j is positive.

It is guaranteed that there is exactly one pair of integers i, j (1 ≤ i, j ≤ n) such that a__i, j = 0.

输入的第一行包含一个整数 $n$（$1 \leq n \leq 500$）—— 表示魔方阵的行数与列数。

接下来有 $n$ 行，每行包含 $n$ 个整数。其中第 $i$ 行的第 $j$ 个数表示 $a_{i,j}$（$1 \leq a_{i,j} \leq 10^9$ 或 $a_{i,j} = 0$），即魔方阵中第 $i$ 行、第 $j$ 列的数字。若对应格子为空，则 $a_{i,j} = 0$；否则 $a_{i,j}$ 为正整数。

保证恰好存在唯一一对整数 $i, j$（$1 \leq i, j \leq n$），使得 $a_{i,j} = 0$。

Output a single integer, the positive integer x (1 ≤ x ≤ 1018) that should be filled in the empty cell so that the whole grid becomes a magic square. If such positive integer x does not exist, output  - 1 instead.

If there are multiple solutions, you may print any of them.

输出一个整数，即应填入空格中的正整数 $x$（$1 \le x \le 10^{18}$），使得整个网格成为一个幻方。若不存在满足条件的正整数 $x$，则输出 $-1$。

若存在多个解，可输出其中任意一个。

In the first sample case, we can fill in 9 into the empty cell to make the resulting grid a magic square. Indeed,

The sum of numbers in each row is:

4 + 9 + 2 = 3 + 5 + 7 = 8 + 1 + 6 = 15.

The sum of numbers in each column is:

4 + 3 + 8 = 9 + 5 + 1 = 2 + 7 + 6 = 15.

The sum of numbers in the two diagonals is:

4 + 5 + 6 = 2 + 5 + 8 = 15.

In the third sample case, it is impossible to fill a number in the empty square such that the resulting grid is a magic square.

在第一个样例中，我们可以在空格中填入数字 9，使得最终的网格成为一个幻方。事实上：

每行数字之和为：
4 + 9 + 2 = 3 + 5 + 7 = 8 + 1 + 6 = 15。

每列数字之和为：
4 + 3 + 8 = 9 + 5 + 1 = 2 + 7 + 6 = 15。

两条对角线上的数字之和为：
4 + 5 + 6 = 2 + 5 + 8 = 15。

在第三个样例中，无法在空格中填入任意数字，使得最终的网格成为幻方。