SOLVE
LATER
Let $$S$$ be a set of intervals of integers and $$f(S)$$ be the number of integers $$x$$ for which there exist an interval $$T$$ with $$T \in S$$ and $$x, x+1 \in T$$. In particular we have that $$f(\{ [l,r] \}) = r - l$$.
Given an integer array $$A$$ of length $$2N$$. Consider a sequence of intervals $$[l_1, r_1], [l_2, r_2], \dots, [l_N, r_N]$$ valid if $$r_1 < r_2 < \dots < r_N$$, $$l_i < r_i$$ and $$(l_1, r_1, l_2, r_2, \dots, l_N, r_N)$$ is a permutation of $$A$$.
Find the sum $$f(\{ [l_1, r_1] \} \cup \{ [l_2, r_2] \} \cup \dots \cup \{ [l_N, r_N] \})$$ modulo $$10^9 + 7$$ over all possible sequences of intervals.
$$\textbf{Input}$$
The first line contains one integer - $$N (1 \le N \le 10^5)$$.
The next line contains $$2N$$ integers - $$A_1, A_2, \dots, A_{2N}$$ $$(1 \le A_1 < A_2 < \dots < A_{2N} \le 10^9)$$.
$$\textbf{Output}$$
Output the answer modulo $$10^9 + 7$$.
The valid intervals are $$\{ ([1,2], [3,4]), ([1,3], [2,4]), ([2,3], [1,4]) \}$$ contributing with $$2, 3, 3$$ respectively to the answer.