102695: [AtCoder]ABC269 F - Numbered Checker

Problem Statement

We have a grid with $N$ rows and $M$ columns. The square $(i,j)$ at the $i$-th row from the top and $j$-th column from the left has an integer $(i-1) \times M + j$ written on it.
Let us perform the following operation on this grid.

• For every square $(i,j)$ such that $i+j$ is odd, replace the integer on that square with $0$.

Answer $Q$ questions on the grid after the operation.
The $i$-th question is as follows:

• Find the sum of the integers written on all squares $(p,q)$ that satisfy all of the following conditions, modulo $998244353$.
  • $A_i \le p \le B_i$.
  • $C_i \le q \le D_i$.

Constraints

• All values in the input are integers.
• $1 \le N,M \le 10^9$
• $1 \le Q \le 2 \times 10^5$
• $1 \le A_i \le B_i \le N$
• $1 \le C_i \le D_i \le M$

Input

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

$N$ $M$
$Q$
$A_1$ $B_1$ $C_1$ $D_1$
$A_2$ $B_2$ $C_2$ $D_2$
$\vdots$
$A_Q$ $B_Q$ $C_Q$ $D_Q$

Output

Print $Q$ lines.
The $i$-th line should contain the answer to the $i$-th question as an integer.

Sample Input 1

5 4
6
1 3 2 4
1 5 1 1
5 5 1 4
4 4 2 2
5 5 4 4
1 5 1 4

Sample Output 1

28
27
36
14
0
104

The grid in this input is shown below.

This input contains six questions.

• The answer to the first question is $0+3+0+6+0+8+0+11+0=28$.
• The answer to the second question is $1+0+9+0+17=27$.
• The answer to the third question is $17+0+19+0=36$.
• The answer to the fourth question is $14$.
• The answer to the fifth question is $0$.
• The answer to the sixth question is $104$.

Sample Input 2

1000000000 1000000000
3
1000000000 1000000000 1000000000 1000000000
165997482 306594988 719483261 992306147
1 1000000000 1 1000000000

Sample Output 2

716070898
240994972
536839100

For the first question, note that although the integer written on the square $(10^9,10^9)$ is $10^{18}$, you are to find it modulo $998244353$.

Sample Input 3

999999999 999999999
3
999999999 999999999 999999999 999999999
216499784 840031647 84657913 415448790
1 999999999 1 999999999

Sample Output 3

712559605
648737448
540261130

问题描述

我们有一个包含$N$行和$M$列的网格。左上角的第$(i-1)$行和第$j$列的正方形上写着$(i-1) \times M + j$。

• 对于每个满足$i+j$为奇数的正方形$(i,j)$，将其上的整数替换为$0$。

回答操作后网格上的$Q$个问题。

• 找到满足以下所有条件的正方形$(p,q)$上写的整数的和（对$998244353$取模）。

• $A_i \le p \le B_i$。
• $C_i \le q \le D_i$。

限制条件

• 输入中的所有值都是整数。
• $1 \le N,M \le 10^9$
• $1 \le Q \le 2 \times 10^5$
• $1 \le A_i \le B_i \le N$
• $1 \le C_i \le D_i \le M$

输入

输入通过标准输入以以下格式给出：

$N$ $M$
$Q$
$A_1$ $B_1$ $C_1$ $D_1$
$A_2$ $B_2$ $C_2$ $D_2$
$\vdots$
$A_Q$ $B_Q$ $C_Q$ $D_Q$

输出

打印$Q$行。

第$i$行应包含第$i$个问题的答案（为整数）。

样例输入1

5 4
6
1 3 2 4
1 5 1 1
5 5 1 4
4 4 2 2
5 5 4 4
1 5 1 4

样例输出1

28
27
36
14
0
104

这个输入中的网格如下所示。

这个输入包含六个问题。

• 第一个问题的答案是$0+3+0+6+0+8+0+11+0=28$。
• 第二个问题的答案是$1+0+9+0+17=27$。
• 第三个问题的答案是$17+0+19+0=36$。
• 第四个问题的答案是$14$。
• 第五个问题的答案是$0$。
• 第六个问题的答案是$104$。

样例输入2

1000000000 1000000000
3
1000000000 1000000000 1000000000 1000000000
165997482 306594988 719483261 992306147
1 1000000000 1 1000000000

样例输出2

716070898
240994972
536839100

对于第一个问题，需要注意的是，虽然写在正方形$(10^9,10^9)$上的整数是$10^{18}$，但你需要对$998244353$取模。

样例输入3

999999999 999999999
3
999999999 999999999 999999999 999999999
216499784 840031647 84657913 415448790
1 999999999 1 999999999

样例输出3

712559605
648737448
540261130