Problem Summary: Minimum Cost to Make a Valid Path in a Grid

You are given a grid with m x n dimensions, where each cell contains an arrow pointing to one of four directions:

1. Right (1): Move to the cell on the right.
2. Left (2): Move to the cell on the left.
3. Down (3): Move to the cell below.
4. Up (4): Move to the cell above.

• You start at the top-left corner (0, 0) and aim to reach the bottom-right corner (m-1, n-1) by following the arrows.
• If necessary, you can modify the arrow in a cell at a cost of 1.
• The goal is to calculate the minimum cost required to create at least one valid path from the start to the end of the grid.

Examples:

1. Input: [[1,1,1,1],[2,2,2,2],[1,1,1,1],[2,2,2,2]]
   

   Output: 3

   Explanation: Modify arrows in 3 cells to create a valid path from (0, 0) to (3, 3).

2. Input: [[1,1,3],[3,2,2],[1,1,4]]

   Output: 0
   Explanation: A valid path already exists without any modifications.

3. Input: [[1,2],[4,3]]
   Output: 1
   Explanation: Modify one cell's arrow to create a valid path.

Constraints:

• 1 <= m, n <= 100
• Each cell contains a value from 1 to 4.