9650: ABC179 —— D - Leaping Tak
[Creator : ]
Description
There are $N$ cells arranged in a row, numbered $1, 2, \ldots, N$ from left to right.
Tak lives in these cells and is currently on Cell $1$. He is trying to reach Cell $N$ by using the procedure described below.
You are given an integer $K$ that is less than or equal to $10$, and $K$ non-intersecting segments $[L_1, R_1], [L_2, R_2], \ldots, [L_K, R_K]$. Let $S$ be the union of these $K$ segments. Here, the segment $[l, r]$ denotes the set consisting of all integers $i$ that satisfy $l \leq i \leq r$.
- When you are on Cell $i$, pick an integer $d$ from $S$ and move to Cell $i + d$. You cannot move out of the cells.
To help Tak, find the number of ways to go to Cell $N$, modulo $998244353$.
Tak lives in these cells and is currently on Cell $1$. He is trying to reach Cell $N$ by using the procedure described below.
You are given an integer $K$ that is less than or equal to $10$, and $K$ non-intersecting segments $[L_1, R_1], [L_2, R_2], \ldots, [L_K, R_K]$. Let $S$ be the union of these $K$ segments. Here, the segment $[l, r]$ denotes the set consisting of all integers $i$ that satisfy $l \leq i \leq r$.
- When you are on Cell $i$, pick an integer $d$ from $S$ and move to Cell $i + d$. You cannot move out of the cells.
To help Tak, find the number of ways to go to Cell $N$, modulo $998244353$.
Input
Input is given from Standard Input in the following format:
```
$N$ $K$
$L_1$ $R_1$
$L_2$ $R_2$
$:$
$L_K$ $R_K$
```
```
$N$ $K$
$L_1$ $R_1$
$L_2$ $R_2$
$:$
$L_K$ $R_K$
```
Output
Print the number of ways for Tak to go from Cell $1$ to Cell $N$, modulo $998244353$.
Constraints
- $2 \leq N \leq 2 \times 10^5$
- $1 \leq K \leq \min(N, 10)$
- $1 \leq L_i \leq R_i \leq N$
- $[L_i, R_i]$ and $[L_j, R_j]$ do not intersect ($i \neq j$)
- All values in input are integers.
- $1 \leq K \leq \min(N, 10)$
- $1 \leq L_i \leq R_i \leq N$
- $[L_i, R_i]$ and $[L_j, R_j]$ do not intersect ($i \neq j$)
- All values in input are integers.
Sample 1 Input
5 2
1 1
3 4
Sample 1 Output
4
The set S is the union of the segment [1,1] and the segment [3,4], therefore S={1,3,4} holds.
There are 44 possible ways to get to Cell 5:
- 1→2→3→4→5,
- 1→2→5,
- 1→4→5 and
- 1→5.
Sample 2 Input
5 2
3 3
5 5
Sample 2 Output
0
Because S={3,5} holds, you cannot reach to Cell 5. Print 0.
Sample 3 Input
5 1
1 2
Sample 3 Output
5
60 3
5 8
1 3
10 15
221823067