CF1295F.Good Contest

An online contest will soon be held on ForceCoders, a large competitive programming platform. The authors have prepared $n$ problems; and since the platform is very popular, $998244351$ coder from all over the world is going to solve them.

For each problem, the authors estimated the number of people who would solve it: for the $i$-th problem, the number of accepted solutions will be between $l_i$ and $r_i$, inclusive.

The creator of ForceCoders uses different criteria to determine if the contest is good or bad. One of these criteria is the number of inversions in the problem order. An inversion is a pair of problems $(x, y)$ such that $x$ is located earlier in the contest ($x \lt y$), but the number of accepted solutions for $y$ is strictly greater.

Obviously, both the creator of ForceCoders and the authors of the contest want the contest to be good. Now they want to calculate the probability that there will be no inversions in the problem order, assuming that for each problem $i$, any integral number of accepted solutions for it (between $l_i$ and $r_i$) is equally probable, and all these numbers are independent.

一场在线编程竞赛即将在 ForceCoders 这一大型编程竞赛平台上举行。出题人共准备了 $n$ 道题目；由于该平台非常受欢迎，全球将有 $998244351$ 名程序员参与解题。

对于每道题目，出题人均预估了其通过人数：第 $i$ 道题的通过人数将在 $l_i$ 到 $r_i$ 之间（含端点）。

ForceCoders 平台的创始人采用多种标准来评判一场竞赛是否优秀，其中一条标准便是题目顺序中的“逆序对”数量。一个逆序对是指一对题目 $(x, y)$，满足 $x$ 在比赛中排在 $y$ 之前（即 $x < y$），但 $y$ 的通过人数严格大于 $x$ 的通过人数。

显然，ForceCoders 平台创始人与竞赛出题人都希望这场竞赛是优秀的。现在，他们希望计算题目顺序中不出现任何逆序对的概率。假设对于每道题 $i$，其通过人数在区间 $[l_i, r_i]$ 内的任意整数值均等可能，且各题的通过人数相互独立。

The first line contains one integer $n$ ($2 \le n \le 50$) — the number of problems in the contest.

Then $n$ lines follow, the $i$-th line contains two integers $l_i$ and $r_i$ ($0 \le l_i \le r_i \le 998244351$) — the minimum and maximum number of accepted solutions for the $i$-th problem, respectively.

第一行包含一个整数 $n$（$2 \le n \le 50$）—— 表示比赛中的题目数量。

接下来有 $n$ 行，其中第 $i$ 行包含两个整数 $l_i$ 和 $r_i$（$0 \le l_i \le r_i \le 998244351$）—— 分别表示第 $i$ 题的通过人数下限与上限。

The probability that there will be no inversions in the contest can be expressed as an irreducible fraction $\frac{x}{y}$, where $y$ is coprime with $998244353$. Print one integer — the value of $xy^{-1}$, taken modulo $998244353$, where $y^{-1}$ is an integer such that $yy^{-1} \equiv 1$ $(mod$ $998244353)$.

比赛中不出现逆序对的概率可以表示为最简分数 $\frac{x}{y}$，其中 $y$ 与 $998244353$ 互质。请输出一个整数 —— $xy^{-1} \bmod 998244353$ 的值，这里 $y^{-1}$ 是满足 $yy^{-1} \equiv 1 \pmod{998244353}$ 的整数。

The real answer in the first test is $\frac{1}{2}$.

第一个测试用例的正确答案是 $\frac{1}{2}$。