Problem9650--ABC179 —— D - Leaping Tak

9650: ABC179 —— D - Leaping Tak

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Time Limit : 1.000 sec  Memory Limit : 512 MiB

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 $998244353$.

Input

Input is given from Standard Input in the following format:

```
$N$ $K$
$L_1$ $R_1$
$L_2$ $R_2$
$:$
$L_K$ $R_K$
```

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.

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

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