Given an m x n matrix, return a new matrix answer where answer[row][col] is the rank of matrix[row][col].

The rank is an integer that represents how large an element is compared to other elements. It is calculated using the following rules:

• The rank is an integer starting from 1.

• If two elements p and q are in the same row or column, then:

  • If p < q then rank(p) < rank(q).
  • If p == q then rank(p) == rank(q).
  • If p > q then rank(p) > rank(q).

• The rank should be as small as possible.

The test cases are generated so that answer is unique under the given rules.

Examples

Example 1

Input:  matrix = [[1,2],[3,4]]
Output: [[1,2],[2,3]]

The rank of matrix[0][0] is 1 because it is the smallest integer in its row and column. The rank of matrix[0][1] is 2 because matrix[0][1] > matrix[0][0] and matrix[0][0] is rank 1. The rank of matrix[1][0] is 2 because matrix[1][0] > matrix[0][0] and matrix[0][0] is rank 1. The rank of matrix[1][1] is 3 because matrix[1][1] > matrix[0][1], matrix[1][1] > matrix[1][0], and both matrix[0][1] and matrix[1][0] are rank 2.

Example 2

Input:  matrix = [[7,7],[7,7]]
Output: [[1,1],[1,1]]

Example 3

Input:  matrix = [[20,-21,14],[-19,4,19],[22,-47,24],[-19,4,19]]
Output: [[4,2,3],[1,3,4],[5,1,6],[1,3,4]]

Constraints

• m == matrix.length
• n == matrix[i].length
• 1 <= m, n <= 500
• -10⁹ <= matrix[row][col] <= 10⁹