Score : $650$ points

问题描述

我们有一个 $N$ 行和 $M$ 列的网格。我们将左起第 $j$ 列、上起第 $i$ 行的单元格记为 $(i,j)$。

已知整数序列 $A=(A_1,A_2,\dots,A_K)$ 和 $B=(B_1,B_2,\dots,B_L)$，其长度分别为 $K$ 和 $L$。

对于所有满足 $1 \le i \le K$ 和 $1 \le j \le L$ 的整数对 $(i,j)$，求以下问题答案之和并对 $998244353$ 取模。

• 一个棋子最初放置在点 $(1,A_i)$。有多少种路径可以通过重复以下移动 $(N-1)$ 次将棋子移动到点 $(N,B_j)$？
  • 假设棋子当前位于单元格 $(p,q)$，将其移动至 $(p+1,q-1)$、$(p+1,q)$ 或 $(p+1,q+1)$，且在整个过程中不能移出网格范围。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

We have a grid with $N$ rows and $M$ columns. We denote by $(i,j)$ the cell in the $i$-th row from the top and $j$-th column from the left.

You are given integer sequences $A=(A_1,A_2,\dots,A_K)$ and $B=(B_1,B_2,\dots,B_L)$ of lengths $K$ and $L$, respectively.

Find the sum, modulo $998244353$, of the answers to the following question over all integer pairs $(i,j)$ such that $1 \le i \le K$ and $1 \le j \le L$.

• A piece is initially placed at $(1,A_i)$. How many paths are there to take it to $(N,B_j)$ by repeating the following move $(N-1)$ times?
  • Let $(p,q)$ be the piece's current cell. Move it to $(p+1,q-1),(p+1,q)$, or $(p+1,q+1)$, without moving it outside the grid.

Constraints

• $1 \le N \le 10^9$
• $1 \le M,K,L \le 10^5$
• $1 \le A_i,B_j \le M$

Input

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

$N$ $M$ $K$ $L$

$A_1$ $A_2$ $\dots$ $A_K$

$B_1$ $B_2$ $\dots$ $B_L$

Output

Print the answer.

Sample Input 1

3 4 1 2
1
1 2

Sample Output 1

4

For $(i,j)=(1,1)$, the following two paths are possible:

• $(1,1) \rightarrow (2,1) \rightarrow (3,1)$
• $(1,1) \rightarrow (2,2) \rightarrow (3,1)$

For $(i,j)=(1,2)$, the following two paths are possible:

• $(1,1) \rightarrow (2,1) \rightarrow (3,2)$
• $(1,1) \rightarrow (2,2) \rightarrow (3,2)$

Thus, the answer is $2 + 2 =4$.

Sample Input 2

5 8 4 5
3 1 4 1
2 7 1 8 2

Sample Output 2

137

Sample Input 3

883671387 87719 10 12
86879 64174 47274 41688 17713 50897 53989 7210 30894 5714
60358 28835 48036 48450 67149 36558 35929 69025 77539 19195 60762 60721

Sample Output 3

941873621

update @ 2024/3/10 08:47:15