AT_abc464_f.Random Vault Heist

Kowasugi Bank has $N$ safes. Safe $i$ contains $A_i$ yen.

One day, a robber entered the bank. The robber repeats the following operation until the total amount stolen reaches at least $X$ yen:

• Choose one safe uniformly at random from the safes not yet opened, open it, and steal all the money inside.

Find the expected value, modulo $998244353$, of the total amount stolen by the robber.

Definition of expected value modulo $998244353$

It can be proved that the expected value sought is always a rational number. Moreover, under the constraints of this problem, it can also be proved that when this value is expressed as an irreducible fraction $\frac{P}{Q}$, it satisfies $Q {{}\not\equiv{}} 0 \pmod{998244353}$. Therefore, there exists a unique integer $R$ satisfying $R \times Q \equiv P \pmod{998244353}, 0 \leq R < 998244353$. Find this $R$.

小和杉银行有 $N$ 个保险箱。第 $i$ 个保险箱中存有 $A_i$ 日元。

某日，一名劫匪闯入了该银行。劫匪重复执行以下操作，直到所盗取的总金额至少达到 $X$ 日元：

• 在尚未打开的保险箱中均匀随机选择一个保险箱，将其打开，并盗走其中全部的钱款。

求劫匪所盗取的总金额的期望值（对 $998244353$ 取模）。

关于“对 $998244353$ 取模的期望值”的定义：

可以证明，本题所求的期望值恒为有理数。此外，在本题的约束条件下还可证明：若将该有理数表示为既约分数 $\frac{P}{Q}$，则必有 $Q {{}\not\equiv{}} 0 \pmod{998244353}$。因此，存在唯一的整数 $R$ 满足

$$R \times Q \equiv P \pmod{998244353},\quad 0 \leq R < 998244353.$$

请输出该 $R$。

The input is given from Standard Input in the following format:

$N$ $X$
$A_1$ $A_2$ $\ldots$ $A_N$

输入从标准输入中按以下格式给出：

$N$ $X$
$A_1$ $A_2$ $\ldots$ $A_N$

Output the answer.

输出答案。

Sample 1 Explanation:
There are two choices for the first safe to open.
If safe $1$ is opened first, the robber also opens safe $2$, so the total amount stolen is $13$ yen.
If safe $2$ is opened first, the total already reaches at least $X$ yen at that point, so the total amount stolen is $10$ yen.
Thus, the expected value is $\frac{13 + 10}{2} = \frac{23}{2}$.

Sample 2 Explanation:
Regardless of the order in which they are opened, the total amount stolen when two safes have been opened is $2$ yen. Thus, the expected value is $2$.

Constraints

• $1 \le N \le 40$
• $1 \le A_i \le 10^{16}$
• $1 \le X \le \sum_{i=1}^{N} A_i$
• All input values are integers.

样例 1 解释：
打开第一个保险箱有两种选择。
若首先打开保险箱 $1$，劫匪还会接着打开保险箱 $2$，因此总共窃取的金额为 $13$ 日元。
若首先打开保险箱 $2$，此时总金额已至少达到 $X$ 日元，因此总共窃取的金额为 $10$ 日元。
因此，期望值为 $\frac{13 + 10}{2} = \frac{23}{2}$。

样例 2 解释：
无论以何种顺序打开保险箱，当打开两个保险箱时，总共窃取的金额恒为 $2$ 日元。因此，期望值为 $2$。

限制条件

• $1 \le N \le 40$
• $1 \le A_i \le 10^{16}$
• $1 \le X \le \sum_{i=1}^{N} A_i$
• 所有输入值均为整数。