Intervals

The set of all real numbers from $p$ to $q$, inclusive, is written $\[p,q]$. This is called an interval. $p > q$ is possible, in which case the set is empty.

The intersection of $k$ intervals $\[p_1, q_1]$, $\[p_2, q_2]$, $\cdots$, $\[p_k,q_k]$ is the set of values from $P = \max{(p_1, p_2, \cdots , p_k)}$ to $Q = \min{(q_1, q_2, \cdots , q_k)}$ inclusive, so it can be written as the interval $\[P,Q]$.

The length of an interval $\[p,q]$ is defined as $\max{(q-p, 0)}$.

You are given $N$ intervals $I_1$, $I_2$, $\cdots$, $I_N$, where $I_i = \[s_i,e_i]$. Write a program that, for each of the $2^N-1$ ways to choose one or more of the intervals, computes the sum of the lengths of the intersections of the chosen intervals and the number of intersections whose length is at least $1$.

The first line contains one integer $N$ ($1 \le N \le 10^5$), the number of intervals.

Each of the next $N$ lines contains two integers $s_i$ and $e_i$ ($0 \le s_i, e_i \le 10^9$) separated by a single space, describing $I_i = \[s_i, e_i]$.

Print on the first line two integers separated by a space: the sum of the intersection lengths over all $2^N-1$ ways to choose one or more intervals, and the number of intersections with length at least $1$. Since these numbers can be very large, print each one modulo $1\,000\,000\,007$.