Number Assignment

As shown in the figure above, a grid of $5 \times 5$ has 25 cells in total. For convenience, the top-left cell has coordinates $(1,1)$ and the bottom-right cell has coordinates $(5,5)$. Initially, $(1,1)$, $(4,5)$, $(5,4)$, and $(5,5)$ are filled with $0$. Every cell in row 1 and every cell in column 1 is already filled with an integer. These are $A$ through $H$. You must fill the remaining empty cells with distinct numbers from $1$ to $13$. In other words, $a$ through $m$ are distinct numbers from $1$ to $13$. The following rules must hold among these numbers.

• $A = a + e + i + l$
• $B = b + f + j + m$
• $C = c + g + k$
• $D = d + h$
• $E = a + b + c + d$
• $F = e + f + g + h$
• $G = i + j + k$
• $H = l + m$

In other words, except for $(1,1)$, each number in the first row equals the sum of the numbers in its column, and each number in the first column equals the sum of the numbers in its row.

For example, suppose $A$ through $H$ are given as in the figure on the right. If $a$ through $m$ are assigned as shown in the figure, the rules above hold. However, if $A$ through $H$ are $3, 22, 27, 9, 16, 34, 27, 14$, there is no way to assign $a$ through $m$ that satisfies the rules.

The input is given through standard input. The integers $A, B, C, D, E, F, G, H$ are given in order on one line. Each integer is between $3$ and $46$, inclusive.

Print to standard output the number of ways to assign distinct integers from $1$ to $13$ to $a$ through $m$ so that the rules are satisfied, on one line.