Sumdoku

Sumdoku is a variant of the game Sudoku. As in Sudoku, the aim is to fill in a $9$-by-$9$ grid with the digits $1$ through $9$ so that each digit $1$ through $9$ occurs exactly once in each row, exactly once in each column and exactly once in each of the $9$ $3$-by-$3$ sub-squares subject to constraints on the choices. In Sudoku, the constraints are that certain squares must contain fixed values. In Sumdoku, the constraints are on the sum of adjacent squares within each $3$-by-$3$ sub-square. In the illustration below, the symbols $<$, $=$ and $>$ indicate that the sum of the values on either side (or above and below) the symbol must have sum less than $10$, equal to $10$ or greater than $10$, respectively.

\includegraphics[width=0.4\textwidth ]{sumdoku.png}

Write a program to solve Sumdoku problems.

Input

The input consists of a single data set containing $15$ lines of input, each line consisting of the characters <, = or >. Rows $1$, $3$, $5$, $6$, $8$, $10$, $11$, $13$ and $15$ contain $6$ characters corresponding to constraints on the sum of values to the left and right of the symbol. Rows $2$, $4$, $7$, $9$, $12$ and $14$ contain $9$ characters corresponding to constraints on the sum of values above and below the symbol. Note: Solutions of some problems may not be unique. The judging program will just check whether your solution satisfies the constraints of the problem (row, column, $3$-by-$3$ box and inequality constraints).

Output

Output $9$ lines consisting of $9$ decimal digits each, separated by a single space. The value in the $j^\text {th}$ position in the $i^{\text {th}}$ line of the $9$ output lines is the solution value in column $j$ of row $i$.

Sample Input 1 Sample Output 1
<==<=>
><>><<><=
>=<==<
><<><>><<
<<>==<
=<=>=>
>><<>=<><
><>>><
=====<<>>
<><<>>
<><><<
<>><>><>=
=>><=>
<><<><><>
<>=>=<
5 3 7 8 2 1 6 4 9
9 6 4 5 3 7 8 2 1
8 1 2 9 4 6 7 3 5
6 4 5 3 7 8 1 9 2
7 8 1 4 9 2 5 6 3
3 2 9 6 1 5 4 7 8
2 7 8 1 6 9 3 5 4
1 9 3 7 5 4 2 8 6
4 5 6 2 8 3 9 1 7