Kraniki

Radek, the boss of the firm Radek and Friends, wants to flood every document shelf at the rival company Mati and Partners. For a perfect sabotage, he asked his friend, the plumber Janusz, to install small water taps above each shelf.

The shelves are segments in the plane. Shelf $i$ is the segment between points $(l_i, h_i)$ and $(r_i, h_i)$. The tap above shelf $i$ is placed at $\left(\frac{l_i + r_i}{2}, h_i + 0.5\right)$. The floor is the $OX$ axis.

When the tap above shelf $i$ is opened, that shelf is flooded. Water then drips straight down from the ends of the flooded shelves. It may flood shelves below, or drip onto the floor, which has a natural drainage system.

Radek considers the taps in a fixed order. When he considers tap $i$, he opens it only if shelf $i$ is not flooded yet.

Radek has not chosen the order yet. He picks one of the $n!$ orders uniformly at random. Find the expected number of taps Radek has to open.

The first line contains one natural number $n$ ($1 \le n \le 500\,000$), the number of shelves at Mati and Partners.

Each of the next $n$ lines describes one shelf. Line $i$ contains two natural numbers $l_i$ and $r_i$ ($1 \le l_i < r_i \le 2 \cdot n$). Here $h_i = i$.

All numbers $l_i$ and $r_i$ are pairwise distinct. Together they form a permutation of $1$ to $2 \cdot n$.

Print one number: the expected number of taps Radek has to open, modulo $10^9 + 7$. Formally, let the answer be $p/q$ with $q \ne 0$ and $\gcd(p, q) = 1$. Print $p \cdot q^{-1} \pmod{10^9 + 7}$, where $q^{-1}$ is the unique integer from $1$ to $10^9 + 6$ such that $q \cdot q^{-1} \equiv 1 \pmod{10^9 + 7}$. The denominator of the reduced answer is proven not to be divisible by $10^9 + 7$.

In the first test, the average is $\frac{1}{6} \cdot (3 + 2 + 3 + 2 + 1 + 1) = 2$. For order 1, 3, 2, Radek opens taps 1 and 3. After tap 3 is opened, shelf 2 is also flooded, so he skips tap 2. For orders 3, 1, 2 and 3, 2, 1, opening tap 3 floods every shelf, so no other tap is opened.