A15379 | 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:

$$$$|(((S_1\ \mathbin{op}_1\ S_2)\ \mathbin{op}_2\ S_3)\ \mathbin{op}_3\ S_4)\ \dots\ \mathbin{op}_{n-1}\ S_n| $$ </p><p>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 $998244353$ .