Chris and Magic Square

Description:

Input:

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.

Output:

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.

Sample Input:

3
4 0 2
3 5 7
8 1 6

Sample Output:

9

Sample Input:

4
1 1 1 1
1 1 0 1
1 1 1 1
1 1 1 1

Sample Output:

1

Sample Input:

4
1 1 1 1
1 1 0 1
1 1 2 1
1 1 1 1

Sample Output:

-1

Note:

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.