Algorithm title: 10401

chess is a two-player board the game believed to have been India as early as the sixth century. However, in this problem we are not discuss about chess, rather we'll talk about a modified form of the classic N-queen Problem. I know you are familiar with plotting n-queens on a chess board with the help of a classic backtracking. If you write this algorithm now you'll find this there are ways of plotting 8 Queens In a 8x8 board provided no Queens attack each of the other.

In this problem we'll talk about injured queens who can move only like a king in horizontal and diagonal direction from Current position but can reach any row from the position like a normal chess queen. You'll have to find the number of possible arrangements with such injured Queens in a particular (n x N) board (with some additional constraints), such that no two queens all other.

Fig: Injured Queen at A6 can reach the adjacent grey squares. Queen atE4 can reach adjacent grey squares. The injured Queen positions are black and reachable places, are grey.

Input

Input file contains several lines of input. Each line expresses a certain board status. The length of these are status string is the board dimension N (0<n<=15) . The ' the ' of the ' denotes ' of the ' the ' the ' the ' of the ' the ' of the ' the ' of the ' the second character S of the second column and so on. So if the character of the status string is 2 , it means that we are looking for arrangements (no TW O injured Queen attack each other) which has injured queen in column A, row 2 . The possible numbers for rows are 1, 2, 3 ... D, E, F which indicates row 1, 2, 3 ... A . If any column contains ' ? ' It means the injured queen can is in any row. So a status string 1?4?? 3 means, are asked to find out total number of possible arrangements in a (6x6) Chessboa Rd which has three of its six injureD Queens at A1 , C4 and F3 . Also Note that there would be no invalid inputs. For example 1?51 are a invalid input because a (4x4) chessboard does not have a fifth R ow.

Output

For each line of the input produce one line of output. This line should contain a integer which indicates the total number of possible arrangements of the corresponding input s Tatus string.

Sample Input
??????
???????????????
??? 8?????
?????

Sample Output
2642
22696209911206174
2098208
0