AT_arc230_e.Minister

You are given positive integers $N$ and $M$.

A box contains one ball numbered $1$, one ball numbered $2$, $\dots$, one ball numbered $N$. There is also a blackboard with nothing written on it. Using these, Alice plays the following game.

First, Alice draws one ball from the box uniformly at random and keeps it at hand. Then, Alice performs the following sequence of operations $N-1$ times.

• Draw one ball from the box uniformly at random, independently of the previous operations, and keep it at hand.

• Write the sum of the numbers of the two balls at hand on the blackboard.

• Freely choose exactly one of the two balls at hand to discard, and keep the other at hand.

Alice's score in this game is defined as follows: it is $0$ if the maximum value of the integers written on the blackboard is $M$ or greater, and otherwise it is the number of the ball finally kept at hand.

Find the expected value, modulo $998244353$, of Alice's score when she acts to maximize the expected value of her score.

Definition of expected value modulo $998244353$

It can be proved that the sought expected value is always a rational number. Also, under the constraints of this problem, it can be proved that when that value is represented as an irreducible fraction $\frac{A}{B}$, we have $B {{}\not\equiv{}} 0 \pmod{998244353}$. Therefore, there is a unique integer $C$ such that $C \times B \equiv A \pmod{998244353}, 0 \leq C < 998244353$. Report this $C$.

给定正整数 $N$ 和 $M$。

一个盒子中包含编号为 $1$ 的球一个、编号为 $2$ 的球一个、……、编号为 $N$ 的球一个。另有一块黑板，初始为空。Alice 使用这些物品进行如下游戏：

首先，Alice 从盒子中均匀随机抽取一个球，并将其持于手中。随后，Alice 重复执行以下操作共 $N-1$ 次：

• 从盒子中独立地、均匀随机抽取一个球（与之前所有操作无关），并将其持于手中；
• 将手中两个球的编号之和写在黑板上；
• 自由选择手中两个球中的恰好一个丢弃，另一个继续持于手中。

定义 Alice 在本局游戏中的得分为：若黑板上所写整数的最大值 大于等于 $M$，则得分为 $0$；否则，得分为最后留在手中的那个球的编号。

当 Alice 以最大化其得分期望值的方式行动时，求其得分的期望值对 $998244353$ 取模的结果。

关于“对 $998244353$ 取模的期望值”的定义：

可以证明，所求期望值恒为有理数。此外，在本题约束下还可证明：若将该值表示为既约分数 $\frac{A}{B}$，则必有 $B {{}\not\equiv{}} 0 \pmod{998244353}$。因此，存在唯一整数 $C$ 满足

$$C \times B \equiv A \pmod{998244353}, \quad 0 \leq C < 998244353.$$

请输出该 $C$。

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

$N$ $M$

输入从标准输入中以如下格式给出：

$N$ $M$

Output the answer.

输出答案。

Sample 1 Explanation:
The following is one possible scenario of the game.

1. One ball is drawn from the box, and its number is $3$. This ball is kept at hand.

2. One ball is drawn from the box, and its number is $1$. This ball is kept at hand. The numbers of the two balls at hand are now $1,3$.

3. Write $1+3=4$ on the blackboard.

4. Discard the ball numbered $3$, keeping the ball numbered $1$ at hand.

5. One ball is drawn from the box, and its number is $2$. This ball is kept at hand. The numbers of the two balls at hand are now $1,2$.

6. Write $1+2=3$ on the blackboard.

7. Discard the ball numbered $1$, keeping the ball numbered $2$ at hand.

8. Since the maximum value of the integers written on the blackboard is not $5$ or greater, Alice obtains a score of $2$, the number of the ball finally kept at hand.

This scenario is not necessarily the result of Alice acting to maximize the expected value of her score.

The sought expected value is $\dfrac{5}{3} \bmod{998244353}=665496237$.

Constraints

• $2 \le N \le 3000$
• $1 \le M \le 2N$
• All input values are integers.

样例 1 解释：
以下是游戏的一种可能情形。

1. 从盒子中随机取出一个球，其编号为 $3$。该球被保留在手中。
2. 从盒子中随机取出一个球，其编号为 $1$。该球被保留在手中。此时手中两个球的编号为 $1,3$。
3. 在黑板上写下 $1+3=4$。
4. 丢弃编号为 $3$ 的球，仅保留编号为 $1$ 的球在手中。
5. 从盒子中随机取出一个球，其编号为 $2$。该球被保留在手中。此时手中两个球的编号为 $1,2$。
6. 在黑板上写下 $1+2=3$。
7. 丢弃编号为 $1$ 的球，仅保留编号为 $2$ 的球在手中。
8. 由于黑板上所写整数的最大值未达到 $5$ 或更大，Alice 的得分为 $2$，即最终保留在手中的球的编号。

该情形未必是 Alice 为最大化其得分期望值而采取的最优策略所导致的结果。

所求的期望值为 $\dfrac{5}{3} \bmod{998244353}=665496237$。

约束条件

• $2 \le N \le 3000$
• $1 \le M \le 2N$
• 所有输入值均为整数。