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 PyPy3 (7.3.0) (judge verdict: Accepted)

import sys, re
from math import ceil, floor, sqrt, pi, factorial, gcd
from copy import deepcopy
from collections import Counter, deque
from heapq import heapify, heappop, heappush
from itertools import accumulate, product, combinations, combinations_with_replacement
from bisect import bisect, bisect_left, bisect_right
from functools import reduce
from decimal import Decimal, getcontext
# input = sys.stdin.readline 
def i_input(): return int(input())
def i_map(): return map(int, input().split())
def i_list(): return list(i_map())
def i_row(N): return [i_input() for _ in range(N)]
def i_row_list(N): return [i_list() for _ in range(N)]
def s_input(): return input()
def s_map(): return input().split()
def s_list(): return list(s_map())
def s_row(N): return [s_input for _ in range(N)]
def s_row_str(N): return [s_list() for _ in range(N)]
def s_row_list(N): return [list(s_input()) for _ in range(N)]
def lcm(a, b): return a * b // gcd(a, b)
sys.setrecursionlimit(10 ** 6)
INF = float('inf')
# MOD = 10 ** 9 + 7
MOD = 998244353
num_list = []
str_list = []

def main():
    n, k = i_map()
    num_list = i_row_list(k)

    dp = [0] * (n+1)
    dp_sum = [0] * (n+1)
    dp[1] = 1
    dp_sum[1] = 1

    for i in range(2, n+1):
        for j in range(k):
            li = i - num_list[j][1]
            ri = i - num_list[j][0]
            if ri < 0:
                continue
            li = max(li, 0)
            dp[i] += dp_sum[ri] - dp_sum[li-1]
            dp[i] %= MOD
        dp_sum[i] = dp_sum[i-1] + dp[i]
    
    print(dp[n]%MOD)

if __name__ == '__main__':
    main()