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Contest: Task: Related: TaskD TaskF

Score : $1600$ points

Problem Statement

Snuke found a random number generator. It generates an integer between $0$ and $N-1$ (inclusive). An integer sequence $A_0, A_1, \cdots, A_{N-1}$ represents the probability that each of these integers is generated. The integer $i$ ($0 \leq i \leq N-1$) is generated with probability $A_i / S$, where $S = \sum_{i=0}^{N-1} A_i$. The process of generating an integer is done independently each time the generator is executed.

Now, Snuke will repeatedly generate an integer with this generator until the following condition is satisfied:

  • For every $i$ ($0 \leq i \leq N-1$), the integer $i$ has been generated at least $B_i$ times so far.

Find the expected number of times Snuke will generate an integer, and print it modulo $998244353$. More formally, represent the expected number of generations as an irreducible fraction $P/Q$. Then, there exists a unique integer $R$ such that $R \times Q \equiv P \pmod{998244353},\ 0 \leq R < 998244353$, so print this $R$.

From the constraints of this problem, we can prove that the expected number of generations is a finite rational number, and its integer representation modulo $998244353$ can be defined.

Constraints

  • $1 \leq N \leq 400$
  • $1 \leq A_i$
  • $\sum_{i=0}^{N-1} A_i \leq 400$
  • $1 \leq B_i$
  • $\sum_{i=0}^{N-1} B_i \leq 400$
  • All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$
$A_0$ $B_0$
$A_1$ $B_1$
$\vdots$
$A_{N-1}$ $B_{N-1}$

Output

Print the expected number of times Snuke will generate an integer, modulo $998244353$.

Sample Input 1

2
1 1
1 1

Sample Output 1

3

The expected number of times Snuke will generate an integer is $3$.

Sample Input 2

3
1 3
2 2
3 1

Sample Output 2

971485877

The expected number of times Snuke will generate an integer is $132929/7200$.

Sample Input 3

15
29 3
78 69
19 15
82 14
9 120
14 51
3 7
6 14
28 4
13 12
1 5
32 30
49 24
35 23
2 9

Sample Output 3

371626143