103366: [Atcoder]ABC336 G - 16 Integers

Problem Statement

You are given $16$ non-negative integers $X_{i, j, k, l}$ $(i, j, k, l \in \lbrace 0, 1 \rbrace)$ in ascending order of $(i, j, k, l)$.
Let $N = \displaystyle \sum_{i=0}^1 \sum_{j=0}^1 \sum_{k=0}^1 \sum_{l=0}^1 X_{i,j,k,l}$.
Find the number, modulo $998244353$, of sequences $(A_1, A_2, ..., A_{N+3})$ of length $N + 3$ consisting of $0$s and $1$s that satisfy the following condition.

• For every quadruple of integers $(i, j, k, l)$ $(i, j, k, l \in \lbrace 0, 1 \rbrace)$, there are exactly $X_{i,j,k,l}$ integers $s$ between $1$ and $N$, inclusive, that satisfy:
  • $A_s = i$, $A_{s + 1} = j$, $A_{s + 2} = k$, and $A_{s + 3} = l$.

Constraints

• $X_{i, j, k, l}$ are all non-negative integers.
• $1 \leq \displaystyle \sum_{i=0}^1 \sum_{j=0}^1 \sum_{k=0}^1 \sum_{l=0}^1 X_{i,j,k,l} \leq 10^6$

Input

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

$X_{0,0,0,0}$ $X_{0,0,0,1}$ $X_{0,0,1,0}$ $X_{0,0,1,1}$ $X_{0,1,0,0}$ $X_{0,1,0,1}$ $X_{0,1,1,0}$ $X_{0,1,1,1}$ $X_{1,0,0,0}$ $X_{1,0,0,1}$ $X_{1,0,1,0}$ $X_{1,0,1,1}$ $X_{1,1,0,0}$ $X_{1,1,0,1}$ $X_{1,1,1,0}$ $X_{1,1,1,1}$ 

Output

Print the number, modulo $998244353$, of sequences that satisfy the condition in the problem statement.

Sample Input 1

0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0

Sample Output 1

1

This input corresponds to the case where $X_{1, 0, 1, 0}$ and $X_{1, 1, 0, 1}$ are $1$ and all others are $0$.
In this case, only one sequence satisfies the condition, which is $(1, 1, 0, 1, 0)$.

Sample Input 2

1 1 2 0 1 2 1 1 1 1 1 2 1 0 1 0

Sample Output 2

16

Sample Input 3

21 3 3 0 3 0 0 0 4 0 0 0 0 0 0 0

Sample Output 3

2024

Sample Input 4

62 67 59 58 58 69 57 66 67 50 68 65 59 64 67 61

Sample Output 4

741536606

问题描述

给你16个非负整数$X_{i, j, k, l}$ $(i, j, k, l \in \lbrace 0, 1 \rbrace)$，按照$(i, j, k, l)$的升序排列。

设$N = \displaystyle \sum_{i=0}^1 \sum_{j=0}^1 \sum_{k=0}^1 \sum_{l=0}^1 X_{i,j,k,l}$。

求满足以下条件的0和1组成的长度为$N + 3$的序列$(A_1, A_2, ..., A_{N+3})$的数量，对998244353取模。

• 对于每一个四元组$(i, j, k, l)$ $(i, j, k, l \in \lbrace 0, 1 \rbrace)$，有恰好$X_{i,j,k,l}$个1到$N$（包括1和N）之间的整数$s$满足以下条件：
  • $A_s = i$，$A_{s + 1} = j$，$A_{s + 2} = k$，$A_{s + 3} = l$。

限制条件

• $X_{i, j, k, l}$都是非负整数。
• $1 \leq \displaystyle \sum_{i=0}^1 \sum_{j=0}^1 \sum_{k=0}^1 \sum_{l=0}^1 X_{i,j,k,l} \leq 10^6$

输入

输入从标准输入按照以下格式给出：

$X_{0,0,0,0}$ $X_{0,0,0,1}$ $X_{0,0,1,0}$ $X_{0,0,1,1}$ $X_{0,1,0,0}$ $X_{0,1,0,1}$ $X_{0,1,1,0}$ $X_{0,1,1,1}$ $X_{1,0,0,0}$ $X_{1,0,0,1}$ $X_{1,0,1,0}$ $X_{1,0,1,1}$ $X_{1,1,0,0}$ $X_{1,1,0,1}$ $X_{1,1,1,0}$ $X_{1,1,1,1}$ 

输出

输出满足问题描述中条件的序列的数量，对998244353取模。

样例输入1

0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0

样例输出1

1

这个输入对应于$X_{1, 0, 1, 0}$和$X_{1, 1, 0, 1}$为1，其余为0的情况。

在这种情况下，只有一个序列满足条件，即$(1, 1, 0, 1, 0)$。

样例输入2

1 1 2 0 1 2 1 1 1 1 1 2 1 0 1 0

样例输出2

16

样例输入3

21 3 3 0 3 0 0 0 4 0 0 0 0 0 0 0

样例输出3

2024

样例输入4

62 67 59 58 58 69 57 66 67 50 68 65 59 64 67 61

样例输出4

741536606