Score : $800$ points

Problem Statement

You are given a sequence of length $N$, $A = (A_1, ..., A_N)$, and an integer $K$. How many permutations of $A$ are there such that no two adjacent elements sum to less than $K$? Find the count modulo $998244353$.

Constraints

• $2 \leq N \leq 2 \times 10^5$
• $0 \leq K \leq 10^9$
• $0 \leq A_i \leq 10^9$
• All values in the input are integers.

Input

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

$N$ $K$

$A_1$ $A_2$ $\dots$ $A_N$

Output

Print the answer.

4 5
1 2 3 4

4

The following four permutations satisfy the condition:

• $(1, 4, 2, 3)$
• $(1, 4, 3, 2)$
• $(2, 3, 4, 1)$
• $(3, 2, 4, 1)$

4 3
1 2 3 3

12

There are $12$ permutations of $A$, all of which satisfy the condition.

10 7
3 1 4 1 5 9 2 6 5 3

108