Magic Rectangle

You are given a rectangle with $3$ rows and $N$ columns. You want to write each of the integers from $1$ to $3N$ exactly once into its cells so that the numbers increase along every row from left to right and along every column from top to bottom. In how many ways can this be done?

Some cells may already be filled in. Because the number of ways can be very large, output it modulo $1000007$.

The first line contains a natural number $N$ ($1 \le N \le 200$).

The next three lines describe the rectangle row by row from top to bottom, and within each row from left to right. Each of these lines contains $N$ integers $a_{i,j}$ ($0 \le a_{i,j} \le 3N$). A value $a_{i,j} = 0$ means the corresponding cell is not yet fixed; otherwise the cell already holds the value $a_{i,j}$.

Print the number of valid ways to fill the rectangle, modulo $1000007$, on a single line.