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C - False Hope /

Time Limit: 2 sec / Memory Limit: 1024 MB

問題文

3\times3 のマス目に 1 から 9 までの数字が書き込まれており、上から i 行目、左から j 列目 (1\leq i\leq3,1\leq j\leq3) に書き込まれている数字は c _ {i,j} です。

• どの 1\leq i\leq3 についても、c _ {i,1}=c _ {i,2}=c _ {i,3} ではない
• どの 1\leq j\leq3 についても、c _ {1,j}=c _ {2,j}=c _ {3,j} ではない
• c _ {1,1}=c _ {2,2}=c _ {3,3} ではない
• c _ {3,1}=c _ {2,2}=c _ {1,3} ではない

• はじめに知ったほうの 2 マスに書かれた数字が同じであり、最後に知ったマスに書かれた数字がそれと異なる。

制約

• c _ {i,j}\in\lbrace1,2,3,4,5,6,7,8,9\rbrace\ (1\leq i\leq3,1\leq j\leq3)
• c _ {i,1}=c _ {i,2}=c _ {i,3} ではない (1\leq i\leq3)
• c _ {1,j}=c _ {2,j}=c _ {3,j} ではない (1\leq j\leq3)
• c _ {1,1}=c _ {2,2}=c _ {3,3} ではない
• c _ {1,3}=c _ {2,2}=c _ {3,1} ではない

入力

c _ {1,1} c _ {1,2} c _ {1,3}
c _ {2,1} c _ {2,2} c _ {2,3}
c _ {3,1} c _ {3,2} c _ {3,3}


入力例 1

3 1 9
2 5 6
2 7 1


出力例 1

0.666666666666666666666666666667


入力例 2

7 7 6
8 6 8
7 7 6


出力例 2

0.004982363315696649029982363316


入力例 3

3 6 7
1 9 7
5 7 5


出力例 3

0.4


Score : 300 points

Problem Statement

There is a 3\times3 grid with numbers between 1 and 9, inclusive, written in each square. The square at the i-th row from the top and j-th column from the left (1\leq i\leq3,1\leq j\leq3) contains the number c _ {i,j}.

The same number may be written in different squares, but not in three consecutive cells vertically, horizontally, or diagonally. More precisely, it is guaranteed that c _ {i,j} satisfies all of the following conditions.

• c _ {i,1}=c _ {i,2}=c _ {i,3} does not hold for any 1\leq i\leq3.
• c _ {1,j}=c _ {2,j}=c _ {3,j} does not hold for any 1\leq j\leq3.
• c _ {1,1}=c _ {2,2}=c _ {3,3} does not hold.
• c _ {3,1}=c _ {2,2}=c _ {1,3} does not hold.

Takahashi will see the numbers written in each cell in random order. He will get disappointed when there is a line (vertical, horizontal, or diagonal) that satisfies the following condition.

• The first two squares he sees contain the same number, but the last square contains a different number.

Find the probability that Takahashi sees the numbers in all the squares without getting disappointed.

Constraints

• c _ {i,j}\in\lbrace1,2,3,4,5,6,7,8,9\rbrace\ (1\leq i\leq3,1\leq j\leq3)
• c _ {i,1}=c _ {i,2}=c _ {i,3} does not hold for any 1\leq i\leq3.
• c _ {1,j}=c _ {2,j}=c _ {3,j} does not hold for any 1\leq j\leq3.
• c _ {1,1}=c _ {2,2}=c _ {3,3} does not hold.
• c _ {3,1}=c _ {2,2}=c _ {1,3} does not hold.

Input

The input is given from Standard Input in the following format:

c _ {1,1} c _ {1,2} c _ {1,3}
c _ {2,1} c _ {2,2} c _ {2,3}
c _ {3,1} c _ {3,2} c _ {3,3}


Output

Print one line containing the probability that Takahashi sees the numbers in all the squares without getting disappointed. Your answer will be considered correct if the absolute error from the true value is at most 10 ^ {-8}.

Sample Input 1

3 1 9
2 5 6
2 7 1


Sample Output 1

0.666666666666666666666666666667


For example, if Takahashi sees c _ {3,1}=2,c _ {2,1}=2,c _ {1,1}=3 in this order, he will get disappointed.

On the other hand, if Takahashi sees c _ {1,1},c _ {1,2},c _ {1,3},c _ {2,1},c _ {2,2},c _ {2,3},c _ {3,1},c _ {3,2},c _ {3,3} in this order, he will see all numbers without getting disappointed.

The probability that Takahashi sees all the numbers without getting disappointed is \dfrac 23. Your answer will be considered correct if the absolute error from the true value is at most 10 ^ {-8}, so outputs such as 0.666666657 and 0.666666676 would also be accepted.

Sample Input 2

7 7 6
8 6 8
7 7 6


Sample Output 2

0.004982363315696649029982363316


Sample Input 3

3 6 7
1 9 7
5 7 5


Sample Output 3

0.4