410257: GYM103993 L Intersection and Union

You are given $$$n$$$ segments on the coordinate axis. The $$$i$$$-th segment is $$$[l_i, r_i]$$$. Let's denote the set of all integer points belonging to the $$$i$$$-th segment as $$$S_i$$$.

Let $$$A \cup B$$$ be the union of two sets $$$A$$$ and $$$B$$$, $$$A \cap B$$$ be the intersection of two sets $$$A$$$ and $$$B$$$, and $$$A \oplus B$$$ be the symmetric difference of $$$A$$$ and $$$B$$$ (a set which contains all elements of $$$A$$$ and all elements of $$$B$$$, except for the ones that belong to both sets).

Let $$$[\mathbin{op}_1, \mathbin{op}_2, \dots, \mathbin{op}_{n-1}]$$$ be an array where each element is either $$$\cup$$$, $$$\oplus$$$, or $$$\cap$$$. Over all $$$3^{n-1}$$$ ways to choose this array, calculate the sum of the following values:

$$$$$$|(((S_1\ \mathbin{op}_1\ S_2)\ \mathbin{op}_2\ S_3)\ \mathbin{op}_3\ S_4)\ \dots\ \mathbin{op}_{n-1}\ S_n|$$$$$$

In this expression, $$$|S|$$$ denotes the size of the set $$$S$$$.

The first line contains one integer $$$n$$$ ($$$2 \le n \le 3 \cdot 10^5$$$).

Then, $$$n$$$ lines follow. The $$$i$$$-th of them contains two integers $$$l_i$$$ and $$$r_i$$$ ($$$0 \le l_i \le r_i \le 3 \cdot 10^5$$$).

Print one integer — the sum of $$$|(((S_1\ \mathbin{op}_1\ S_2)\ \mathbin{op}_2\ S_3)\ \mathbin{op}_3\ S_4)\ \dots\ \mathbin{op}_{n-1}\ S_n|$$$ over all possible ways to choose $$$[\mathbin{op}_1, \mathbin{op}_2, \dots, \mathbin{op}_{n-1}]$$$. Since the answer can be huge, print it modulo $$$998244353$$$.

4
3 5
4 8
2 2
1 9

162

4
1 9
3 5
4 8
2 2

102