AmazingCaddy likes Sudoku very much. One day, he got some Sudoku problems and he wondered whether these problems are well designed. He needs your help now.
Sudoku is a logic-based, combinatorial number-placement puzzle. The objective is to fill a 9 × 9 grid with digits so that each column, each row, and each of the nine 3 × 3 regions contain all of the digits from 1 to 9.
-- Wikipedia
In this problem, the grid has three different sizes, with the formal (N^2) × (N^2), and N will be one in 2, 3 and 4. The rule is the same. The objective is to fill the (N^2) × (N^2) grid with characters so that each column, each row, and each of the N^2 regions (each size is N × N) contains all of the characters from 1 to 4(N = 2), 1 to 9(N = 3) or 1 to G (N = 4).
You task is that when you got a grid, you should tell the grid whether a puzzle or not. If it`s a puzzle, you should tell whether it is a minimal puzzle; else you should tell it has no solution or has multiple solutions.
A puzzle is a partially completed grid (the size is (N^2) × (N^2), N = 2, 3, 4), and has one and only one solution. If remove any character from a puzzle, it will lead to multiple solutions, then it is called a minimal puzzle.
The input contains several cases. For each case, the first line of the input is the N (2<= N <=4). The next N^2 lines will contain the grid. Each line will have N^2 characters, represent the grid. The empty cell will be represented as ’.’. All input data is legal, and you can get more information from the sample input.
For each case:
If the grid is not a puzzle, and has no solution, print “No Solution”, or has multiple solutions print “Multiple Solutions”.
If the grid is a puzzle, but not a minimal, print “Not Minimal”, or print N lines represent the answer in the same format in the input, and ‘.’ should be replaced with the right characters. You can get more information from the sample output.