Fence

A pasture where factorials grow lies in the standard coordinate system, where the $x$ coordinate increases to the right and the $y$ coordinate increases upward. A cell is a square with side length 1 and sides parallel to the coordinate axes, such that both coordinates of its lower left corner $A = (x_A, y_A)$ are nonnegative integers. The value of such a cell is $x_A! \times y_A!$, where $a! = 1 \times 2 \times \cdots \times a$ for $a \ge 1$ and $0! = 1$.

A fence is a polygon whose sides are parallel to the coordinate axes. Given a fence, compute the sum of the values of all cells inside the fence, modulo $10^9 + 7$.

The first line contains a positive integer $n$ ($n \le 100$), the number of vertices of the fence.

The $k$-th of the next $n$ lines contains integers $x_k$ and $y_k$ ($0 \le x_k, y_k \le 10^9$), the coordinates of the $k$-th vertex of the fence. The vertices are listed in order along the polygon.

You may assume that the vertices form a polygon whose sides are parallel to the coordinate axes. In particular, the polygon does not intersect itself, and two adjacent sides are never parallel.

Print the requested sum modulo $10^9 + 7$ on the first line.

The figure shows the fence from the third example and the value of each cell. The values of the cells inside the fence add up to 119.