Inaho XI

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Points: 5 (partial)
Time limit: 2.0s
Memory limit: 64M

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Problem type

Given \(M\) points in \(N\) dimensional space, find the minimum "surface area" of a hyperrectangle required to contain all \(M\) points modulo \(10^9+7\).

As an example, the "surface area" of a \(2\)-dimensional hyperrectangle (rectangle) is the sum of its \(4\) side lengths. The "surface area" of a \(3\)-dimensional hyperrectangle (rectangular prism) is the sum of the areas of the \(6\) sides of the hyperrectangle.

Input Specification

The first line will contain two space-separated integers, \(N, M\ (2 \le N \le 10, 1 \le M \le 10^5)\), the number of dimensions and the number of points respectively.

The next \(M\) lines will each contain \(N\) integers, \(a_{i_1}, a_{i_2}, \ldots, a_{i_N}\ (-10^9 \le a_{i_j} \le 10^9)\).

Output Specification

Output the minimum "surface area" of a hyperrectangle required to contain all \(M\) points, modulo \(10^9+7\).

Subtasks

Subtask 1 [10%]

\(N = 2, M \le 100\)

Subtask 2 [20%]

\(M \le 100\)

Subtask 3 [70%]

No further constraints.

Sample Input 1

2 4
1 1
3 3
-1 2
0 0

Sample Output 1

14

Sample Input 2

5 3
1 4 2 3 4
0 -129 6 9 0
0 0 -10 9 5

Sample Output 2

183436

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