Inverted Triangles

On an infinite two-dimensional coordinate plane, there are $N$ points. The points are numbered from $1$ to $N$, and the coordinates of point $i$ are $(x_i, y_i)$. All point coordinates are distinct.

Choose three distinct numbers $a$, $b$, and $c$ between $1$ and $N$ inclusive. Connecting the points for these numbers with line segments forms a triangle. A triangle that satisfies all of the following conditions is called an inverted triangle.

• $x_a < x_b < x_c$.
• $y_b < y_a$.
• $y_b < y_c$.

Let $S$ be the sum of the areas of all inverted triangles that can be formed by choosing three points. Print the remainder when $2S$ is divided by $1,000,000,007$ (= $10^9 + 7$). It can be proven that $2S$ is always an integer.

The first line contains $N$. ($3 \le N \le 300\,000$)

Each of the next $N$ lines contains two integers $x_i$ and $y_i$, separated by a space. ($-10^9 \le x_i, y_i \le 10^9$)

Print the remainder when $2S$ is divided by $1,000,000,007$. This number is prime.