Expected Beauty

Morgan the robot has an array $A$ of size $N$, indexed from $1$ to $N$. The value of each element is random: $A_i$ can be any integer from $L_i$ to $R_i$ (inclusive), each with equal probability.

Morgan defines the beauty of $A$ as follows. Morgan has a variable score that starts at $0$. An operation on an array $a$ is:

• Choose an index $i$ with $1 \le i < |a|$ and $a_i = a_{i+1}$. If no such $i$ exists, the operation cannot be performed.
• Add the value of $a_i$ to score and remove $a_i$ from the array.
• The remaining elements are joined together, keeping their order.

The beauty of $A$ is the maximum value of score$^2$ that Morgan can get after performing zero or more operations on $A$.

Because the array is random, Morgan wants the expected beauty of $A$. Morgan's algorithm is too slow, so Morgan asks for help computing the expected value.

The first line contains an integer $N$ ($1 \le N \le 200000$). Each of the next $N$ lines contains two integers $L_i$ and $R_i$ ($1 \le L_i \le R_i \le 10^8$).

Let $M = 998244353$. The expected value can be written as an irreducible fraction $p/q$, where $q \not\equiv 0 \pmod M$. Print one integer $x$ on a single line such that $0 \le x < M$ and $x \cdot q \equiv p \pmod M$.