Score : $600$ points

问题描述

有 $N$ 棵树。在第 $0$ 天，每棵树上都没有果实。

从第 $1$ 天早晨开始及随后的每一天，第 $i$ 棵树将结出 $i$ 个新的果实，对于每个 $i = 1, 2, \ldots, N$。

高桥将进行 $Q$ 次采集。对于每个 $i = 1, 2, \ldots, Q$，第 $i$ 次采集发生在第 $D_i$ 天的晚上，收集此时第 $L_i$ 到第 $R_i$ 棵树上所有剩余的果实。

对于这 $Q$ 次采集中的每一次，输出高桥将收集到的果实总数，结果对 $998244353$ 取模。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

There are $N$ trees. On Day $0$, each tree bears no fruits.
On the morning of Day $1$ and subsequent days, the $i$-th tree bears $i$ new fruits for each $i = 1, 2, \ldots, N$.

Takahashi will perform $Q$ harvesting. For each $i = 1, 2, \ldots, Q$, the $i$-th harvesting takes place on the night of Day $D_i$, collecting all fruits remaining on the $L_i$-th through $R_i$-th trees at that point.

For each of the $Q$ harvesting, print the number of fruits Takahashi will collect, modulo $998244353$.

Constraints

• $1 \leq N \leq 10^{18}$
• $1 \leq Q \leq 2 \times 10^5$
• $1 \leq D_1 \lt D_2 \lt \cdots \lt D_Q \leq 10^{18}$
• $1 \leq L_i \leq R_i \leq N$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $Q$

$D_1$ $L_1$ $R_1$

$D_2$ $L_2$ $R_2$

$\vdots$

$D_Q$ $L_Q$ $R_Q$

Output

Print $Q$ lines. For each $i = 1, 2, \ldots, Q$, the $i$-th line should contain the number of fruits Takahashi will collect in the $i$-th harvesting, modulo $998244353$.

Sample Input 1

5 3
2 2 3
3 3 4
5 1 5

Sample Output 1

10
15
50

For each $i = 1, 2, 3, 4, 5$, let $A_i$ be the number of fruits remaining on the $i$-th tree. Now, we use the sequence $A = (A_1, A_2, A_3, A_4, A_5)$ to represent the numbers of fruits remaining in the trees.

• On Day $0$, we have $A = (0, 0, 0, 0, 0)$.
• On the morning of Day $1$, each tree bears new fruits, and we have $A = (1, 2, 3, 4, 5)$.
• On the morning of Day $2$, each tree bears new fruits, and we have $A = (2, 4, 6, 8, 10)$.
• On the night of Day $2$, Takahashi performs the $1$-st harvesting. $4 + 6 = 10$ fruits are collected, and we have $A = (2, 0, 0, 8, 10)$.
• On the morning of Day $3$, each tree bears new fruits, and we have $A = (3, 2, 3, 12, 15)$.
• On the night of Day $3$, Takahashi performs the $2$-nd harvesting. $3 + 12 = 15$ fruits are collected, and we have $A = (3, 2, 0, 0, 15)$.
• On the morning of Day $4$, each tree bears new fruits, and we have $A = (4, 4, 3, 4, 20)$.
• On the morning of Day $5$, each tree bears new fruits, and we have $A = (5, 6, 6, 8, 25)$.
• On the night of Day $5$, Takahashi performs the $3$-rd harvesting. $5 + 6 + 6 + 8 + 25 = 50$ fruits are collected, and we have $A = (0, 0, 0, 0, 0)$.

Sample Input 2

711741968710511029 1
82803157126515475 516874290286751784 588060532191410838

Sample Output 2

603657470

Be sure to print the numbers modulo $998244353$.

update @ 2024/3/10 10:52:40