#D9004. Nezzar and Chocolate Bars
Nezzar and Chocolate Bars
Nezzar and Chocolate Bars
Nezzar buys his favorite snack — n chocolate bars with lengths l_1,l_2,…,l_n. However, chocolate bars might be too long to store them properly!
In order to solve this problem, Nezzar designs an interesting process to divide them into small pieces. Firstly, Nezzar puts all his chocolate bars into a black box. Then, he will perform the following operation repeatedly until the maximum length over all chocolate bars does not exceed k.
- Nezzar picks a chocolate bar from the box with probability proportional to its length x.
- After step 1, Nezzar uniformly picks a real number r ∈ (0,x) and divides the chosen chocolate bar into two chocolate bars with lengths r and x-r.
- Lastly, he puts those two new chocolate bars into the black box.
Nezzar now wonders, what is the expected number of operations he will perform to divide his chocolate bars into small pieces.
It can be shown that the answer can be represented as P/Q, where P and Q are coprime integers and Q not ≡ 0 (mod 998 244 353). Print the value of P⋅ Q^{-1} mod 998 244 353.
Input
The first line contains two integers n and k (1 ≤ n ≤ 50, 1 ≤ k ≤ 2000).
The second line contains n integers l_1, l_2, …, l_n (1 ≤ l_i, ∑_{i=1}^{n} l_i ≤ 2000).
Output
Print a single integer — the expected number of operations Nezzar will perform to divide his chocolate bars into small pieces modulo 998 244 353.
Examples
Input
1 1 2
Output
4
Input
1 1 1
Output
0
Input
1 5 1234
Output
15630811
Input
2 1 2 3
Output
476014684
Input
10 33 10 20 30 40 50 60 70 80 90 100
Output
675105648
inputFormat
Input
The first line contains two integers n and k (1 ≤ n ≤ 50, 1 ≤ k ≤ 2000).
The second line contains n integers l_1, l_2, …, l_n (1 ≤ l_i, ∑_{i=1}^{n} l_i ≤ 2000).
outputFormat
Output
Print a single integer — the expected number of operations Nezzar will perform to divide his chocolate bars into small pieces modulo 998 244 353.
Examples
Input
1 1 2
Output
4
Input
1 1 1
Output
0
Input
1 5 1234
Output
15630811
Input
2 1 2 3
Output
476014684
Input
10 33 10 20 30 40 50 60 70 80 90 100
Output
675105648
样例
1 1
1
0