Restoring Road Network

In Takahashi Kingdom, which once existed, there are N cities, and some pairs of cities are connected bidirectionally by roads. The following are known about the road network:

• People traveled between cities only through roads. It was possible to reach any city from any other city, via intermediate cities if necessary.
• Different roads may have had different lengths, but all the lengths were positive integers.

Snuke the archeologist found a table with N rows and N columns, A, in the ruin of Takahashi Kingdom. He thought that it represented the shortest distances between the cities along the roads in the kingdom.

Determine whether there exists a road network such that for each u and v, the integer A_{u, v} at the u-th row and v-th column of A is equal to the length of the shortest path from City u to City v. If such a network exist, find the shortest possible total length of the roads.

Constraints

• 1 \leq N \leq 300
• If i ≠ j, 1 \leq A_{i, j} = A_{j, i} \leq 10^9.
• A_{i, i} = 0

Inputs

Input is given from Standard Input in the following format:

N A_{1, 1} A_{1, 2} ... A_{1, N} A_{2, 1} A_{2, 2} ... A_{2, N} ... A_{N, 1} A_{N, 2} ... A_{N, N}

Outputs

If there exists no network that satisfies the condition, print -1. If it exists, print the shortest possible total length of the roads.

Examples

Input

3 0 1 3 1 0 2 3 2 0

Output

3

Input

3 0 1 3 1 0 1 3 1 0

Output

-1

Input

5 0 21 18 11 28 21 0 13 10 26 18 13 0 23 13 11 10 23 0 17 28 26 13 17 0

Output

82

Input

3 0 1000000000 1000000000 1000000000 0 1000000000 1000000000 1000000000 0

Output

3000000000

inputFormat

Inputs

Input is given from Standard Input in the following format:

N A_{1, 1} A_{1, 2} ... A_{1, N} A_{2, 1} A_{2, 2} ... A_{2, N} ... A_{N, 1} A_{N, 2} ... A_{N, N}

outputFormat

Outputs

If there exists no network that satisfies the condition, print -1. If it exists, print the shortest possible total length of the roads.

Examples

Input

3 0 1 3 1 0 2 3 2 0

Output

3

Input

3 0 1 3 1 0 1 3 1 0

Output

-1

Input

5 0 21 18 11 28 21 0 13 10 26 18 13 0 23 13 11 10 23 0 17 28 26 13 17 0

Output

82

Input

3 0 1000000000 1000000000 1000000000 0 1000000000 1000000000 1000000000 0

Output

3000000000

样例

3
0 1000000000 1000000000
1000000000 0 1000000000
1000000000 1000000000 0

3000000000