Problem:
Score : 400 points

Problem Statement

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.

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.

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.

Sample Input 1

5 2
1 1
3 4

Sample Output 1

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 4 possible ways to get to Cell 5:

1 \to 2 \to 3 \to 4 \to 5,

1 \to 2 \to 5,

1 \to 4 \to 5 and

1 \to 5.

Sample Input 2

5 2
3 3
5 5

Sample Output 2

0

Because S = \{ 3, 5 \} holds, you cannot reach to Cell 5.
Print 0.

Sample Input 3

5 1
1 2

Sample Output 3

5

Sample Input 4

60 3
5 8
1 3
10 15

Sample Output 4

221823067

Note that you have to print the answer modulo 998244353.

Solution in Python (3.8.2) (judge verdict: Accepted)


def solve():
    MOD = 998244353
    N, K = map(int, input().split())
    LR_L = [[int(x) for x in input().split()] for y in range(K)]
    # S = set()
    # for i in range(K):
    #     _l, _r = map(int, input().split())
    #     _s = set(list(range(_l, _r+1)))
    #     S |= _s

    # dp[i] : iに到達する経路数
    dp = [0] * (N+1)
    dp[1] = 1
    dp[2] = -1
    for i in range(1, N+1):
        dp[i] += dp[i-1]
        dp[i] %= MOD
        # for d in S:
        #     if i-d > 0:
        #         dp[i] = (dp[i] + dp[i-d]) % MOD
        for li, ri in LR_L:
            if i + li < N+1:
                dp[i+li] += dp[i]
                dp[i+li] %= MOD
            if i + ri + 1 < N+1:
                dp[i+ri+1] -= dp[i]
                dp[i+ri+1] %= MOD
    print(dp[N])


solve()