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Ex - Max Limited Sequence

Score : $600$ points

Problem Statement

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 $1 \leq j \leq Q$.

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.

Input

Input is given from Standard Input in the following format:

$N$ $M$ $Q$

$L_1$ $R_1$ $X_1$

$\vdots$

$L_Q$ $R_Q$ $X_Q$

Output

Print the answer.

3 3 2
1 2 2
2 3 3

5

$A = (0, 2, 3), (1, 2, 3), (2, 0, 3), (2, 1, 3), (2, 2, 3)$ satisfy the conditions.

1 1 1
1 1 1

1

6 40000000 3
1 4 30000000
2 6 20000000
3 5 10000000

135282163