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Collect Resources


Problem Description

There is a rectangular field with two types of resources viz, resource A and resource B. At each cell in the field, both of the resources can be present. There are two types of factories which can collect resource A and B. If resource A is collected in a factory then resource B is lost and vice versa. Now, these two factories are located perpendicular to each other i.e the factory collecting resource A is at the top ( north) of the rectangular field and the factory collecting resource B is to the left ( west) of the rectangular field. The resources have to be brought to the factory on a straight path without taking any turns. This implies that if there is any south-north conveyor belt in the cell, but the cell north of it contains an east-west conveyor belt, then all resources transported on the south-north conveyor belt will be lost.

You have to design a conveyor belt system that maximizes the total amount of minerals mined, i.e the sum of the amount of resource A transported to the top and the sum of the amount of resource B transported to the left. You are given a 2-D matrix A and B of the same size denoting the amount of resource of the corresponding type in that cell.

Since the answer can be large, return it modulo 109 + 7.



Problem Constraints

1 <= |A| = |B| <= 103

1 <= |A[0]| = |B[0]| <=103

0 <= A[i][j], B[i][j] <= 109



Input Format

The first line contains a matrix that specifies the availability of the resource A in each of the cells. 

The second line contains a matrix that specifies the availability of the resource B in each of the cells.



Output Format

Return a single integer which specifies the maximum amount of resources that can be processed by the factories.



Example Input

Input 1:

   A = [
         [7, 8, 9]
         [4, 5, 6]
         [1, 2, 3]
       ]

B = [ [1, 2, 3] [4, 5, 6] [7, 8, 9] ]

Input 2:

   A = [
         [1, 0, 1]
       ]

B = [ [2, 1, 0] ]



Example Output

Output 1:

  63

Output 2:

  4



Example Explanation

Explanation 1:

  The first row can be sent to the top to collect resource A. The rest two rows will be sent to the left to collect resource B. 

Explanation 2:

  The first two columns can be sent to the left. The third column will be sent to the top.Total resource collected - 2 + 1 + 1 = 4 



NOTE: You only need to implement the given function. Do not read input, instead use the arguments to the function. Do not print the output, instead return values as specified. Still have a question? Checkout Sample Codes for more details.
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