Help Yourself#10124

Help Yourself

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Bessie has been given $n$ segments on a 1D number line. The $i$ -th segment contains all reals $x$ such that $l_{i} \leq x \leq r_{i}$ l_i \le x \le r_i. Define the union of a set of segments to be the set of all $x$ that are contained within at least one segment. Define the complexity of a set of segments to be the number of connected regions represented in its union.

Bessie wants to compute the sum of the complexities over all $2^{n}$ 2^n subsets of the given set of $n$ segments, modulo $1 0^{9} + 7$ 10^9 + 7.

Normally, your job is to help Bessie. But this time, you are Bessie, and there's no one to help you. Help yourself!

Eingabe

The first line contains $n (1 \leq n \leq 1 0^{5})$ n~(1 \le n \le 10^5). Each of the next $n$ lines contains two integers $l_{i}$ l_i and $r_{i}$ r_i. It is guaranteed that $l_{i} < r_{i}$ l_i < r_i and all $l_{i}, r_{i}$ l_i, r_i are distinct integers in the range $1...2 n$ .

Ausgabe

Output the answer, modulo $1 0^{9} + 7$ 10^9 + 7.

Beispiele

The complexity of each nonempty subset is written below.

{[1, 6]} ⟹ 1, {[2, 3]} ⟹ 1, {[4, 5]} ⟹ 1
{[1, 6], [2, 3]} ⟹ 1, {[1, 6], [4, 5]} ⟹ 1, {[2, 3], [4, 5]} ⟹ 2
{[1, 6], [2, 3], [4, 5]} ⟹ 1

The answer is $1 + 1 + 1 + 1 + 1 + 2 + 1 = 8$ 1 + 1 + 1 + 1 + 1 + 2 + 1 = 8.

Beispiel 1