CF1809G.Prediction

Consider a tournament with $n$ participants. The rating of the $i$-th participant is $a_i$.

The tournament will be organized as follows. First of all, organizers will assign each participant an index from $1$ to $n$. All indices will be unique. Let $p_i$ be the participant who gets the index $i$.

Then, $n-1$ games will be held. In the first game, participants $p_1$ and $p_2$ will play. In the second game, the winner of the first game will play against $p_3$. In the third game, the winner of the second game will play against $p_4$, and so on — in the last game, the winner of the $(n-2)$-th game will play against $p_n$.

Monocarp wants to predict the results of all $n-1$ games (of course, he will do the prediction only after the indices of the participants are assigned). He knows for sure that, when two participants with ratings $x$ and $y$ play, and $|x - y| \gt k$, the participant with the higher rating wins. But if $|x - y| \le k$, any of the two participants may win.

Among all $n!$ ways to assign the indices to participants, calculate the number of ways to do this so that Monocarp can predict the results of all $n-1$ games. Since the answer can be large, print it modulo $998244353$.

考虑一场有 $n$ 名参赛者的锦标赛。第 $i$ 名参赛者的评分为 $a_i$。

锦标赛将按如下方式组织：首先，主办方将为每名参赛者分配一个从 $1$ 到 $n$ 的唯一索引。设 $p_i$ 表示获得索引 $i$ 的参赛者。

随后，将举行 $n-1$ 场比赛。第一场比赛由参赛者 $p_1$ 和 $p_2$ 进行；第二场比赛由第一场的胜者对阵 $p_3$；第三场比赛由第二场的胜者对阵 $p_4$；以此类推——最后一场（即第 $n-1$ 场）比赛由第 $(n-2)$ 场的胜者对阵 $p_n$。

Monocarp 希望预测全部 $n-1$ 场比赛的结果（当然，他仅在参赛者索引分配完成后才进行预测）。他确信：当两名评分为 $x$ 和 $y$ 的参赛者对战时，若 $|x - y| > k$，则评分更高者获胜；但若 $|x - y| \le k$，则两人中任意一人皆可获胜。

在全部 $n!$ 种为参赛者分配索引的方式中，计算使得 Monocarp 能够预测全部 $n-1$ 场比赛结果的分配方式数目。由于答案可能很大，请对 $998244353$ 取模后输出。

The first line contains two integers $n$ and $k$ ($2 \le n \le 10^6$; $0 \le k \le 10^9$).

The second line contains $n$ integers $a_1, a_2, \dots, a_n$ ($0 \le a_1 \le a_2 \le \dots \le a_n \le 10^9$).

第一行包含两个整数 $n$ 和 $k$（$2 \le n \le 10^6$；$0 \le k \le 10^9$）。

第二行包含 $n$ 个整数 $a_1, a_2, \dots, a_n$（$0 \le a_1 \le a_2 \le \dots \le a_n \le 10^9$）。

Print one integer — the number of ways to assign the indices to the participants so that Monocarp can predict the results of all $n-1$ games.

输出一个整数——表示为参赛者分配编号的方式数量，使得 Monocarp 能够预测全部 $n-1$ 场比赛的结果。

In the first example, a match with any pair of players can be predicted by Monocarp, so all $24$ ways to assign indices should be counted.

In the second example, suitable ways are $[1, 3, 2]$, $[2, 3, 1]$, $[3, 1, 2$] and $[3, 2, 1]$.

在第一个例子中，Monocarp 可以预测任意一对选手之间的比赛，因此所有 $24$ 种编号分配方式均应被计入。

在第二个例子中，符合条件的编号分配方式为 $[1, 3, 2]$、$[2, 3, 1]$、$[3, 1, 2]$ 和 $[3, 2, 1]$。