Write a program to solve a Sudoku puzzle by filling the empty cells. A sudoku solution must satisfy all of the following rules:

1. Each of the digits 1-9 must occur exactly once in each row.
2. Each of the digits 1-9 must occur exactly once in each column.
3. Each of the digits 1-9 must occur exactly once in each of the 9 3x3 sub-boxes of the grid.

Problem Note:

• The given board contains only digits 1-9 and the character '.'.
• Empty cells are shown by the character ' . '
• You may assume that the given Sudoku puzzle will have a single unique solution.
• The board size is always 9x9.

Example

Input: A given sudoku puzzle:

Output: Solution numbers are marked in blue.