9790: ABC262 —— Ex - Max Limited Sequence
[Creator : ]
Description
Find the number, modulo $998244353$, of integer sequences $A = (A_1, \dots, A_N)$ of length $N$ that satisfy all of the following conditions:
- $0 \leq A_i \leq M$ for all $i$ such that $1 \leq i \leq N$.
- The maximum value of $A_{L_j}, \dots, A_{R_j}$ is $X_j$ for all $j$ such that $1 \leq j \leq Q$.
- $0 \leq A_i \leq M$ for all $i$ such that $1 \leq i \leq N$.
- The maximum value of $A_{L_j}, \dots, A_{R_j}$ is $X_j$ for all $j$ such that $1 \leq j \leq Q$.
Input
Input is given from Standard Input in the following format:
```
$N$ $M$ $Q$
$L_1$ $R_1$ $X_1$
$\vdots$
$L_Q$ $R_Q$ $X_Q$
```
```
$N$ $M$ $Q$
$L_1$ $R_1$ $X_1$
$\vdots$
$L_Q$ $R_Q$ $X_Q$
```
Output
Print the answer.
Constraints
- $1 \leq N \leq 2 \times 10^5$
- $1 \leq M \lt 998244353$
- $1 \leq Q \leq 2 \times 10^5$
- $1 \leq L_i \leq R_i \leq N \, (1 \leq i \leq Q)$
- $1 \leq X_i \leq M \, (1 \leq i \leq Q)$
- All values in input are integers.
- $1 \leq M \lt 998244353$
- $1 \leq Q \leq 2 \times 10^5$
- $1 \leq L_i \leq R_i \leq N \, (1 \leq i \leq Q)$
- $1 \leq X_i \leq M \, (1 \leq i \leq Q)$
- All values in input are integers.
Sample 1 Input
3 3 2
1 2 2
2 3 3
Sample 1 Output
5
A=(0,2,3),(1,2,3),(2,0,3),(2,1,3),(2,2,3) satisfy the conditions.
Sample 2 Input
1 1 1
1 1 1
Sample 2 Output
1
Sample 3 Input
6 40000000 3
1 4 30000000
2 6 20000000
3 5 10000000
Sample 3 Output
135282163