Codeforces Round 698 (Div. 1) |
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Finished |

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combinatorics

fft

math

probabilities

*3500

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F. Nezzar and Chocolate Bars

time limit per test

5 secondsmemory limit per test

512 megabytesinput

standard inputoutput

standard outputNezzar buys his favorite snack — $$$n$$$ chocolate bars with lengths $$$l_1,l_2,\ldots,l_n$$$. 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 $$$r \in (0,x)$$$ 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 $$$\frac{P}{Q}$$$, where $$$P$$$ and $$$Q$$$ are coprime integers and $$$Q \not \equiv 0$$$ ($$$\bmod 998\,244\,353$$$). Print the value of $$$P\cdot Q^{-1} \mod 998\,244\,353$$$.

Input

The first line contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le n \le 50, 1 \le k \le 2000$$$).

The second line contains $$$n$$$ integers $$$l_1, l_2, \ldots, l_n$$$ ($$$1 \le l_i$$$, $$$\sum_{i=1}^{n} l_i \le 2000$$$).

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

Codeforces (c) Copyright 2010-2024 Mike Mirzayanov

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