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: Time Limit Exceeded)

#!/usr/bin/env python3

import sys


def main():
    mod = 998244353
    sys.setrecursionlimit(10**6)    # 1000 -> 1000000
    def input():  return sys.stdin.readline().rstrip()
    def mi():     return map(int, input().split())    

    n, k = mi()
    intervals = [tuple(mi()) for _ in range(k)]
    N = n
    
    # N: 処理する区間の長さ
    LV = (N-1).bit_length()
    N0 = 2**LV
    data = [0]*(2*N0)
    lazy = [0]*(2*N0)

    def gindex(l, r):
        L = (l + N0) >> 1; R = (r + N0) >> 1
        lc = 0 if l & 1 else (L & -L).bit_length()
        rc = 0 if r & 1 else (R & -R).bit_length()
        for i in range(LV):
            if rc <= i:
                yield R
            if L < R and lc <= i:
                yield L
            L >>= 1; R >>= 1

    # 遅延伝搬処理
    def propagates(*ids):
        for i in reversed(ids):
            v = lazy[i-1]
            if not v:
                continue
            if lazy[2*i-1] + v >= mod:
                lazy[2*i-1] = (lazy[2*i-1] + v) % mod
            else:
                lazy[2*i-1] += v
            # lazy[2*i-1] = (lazy[2*i-1] + v) % mod
            if lazy[2*i] + v >= mod:
                lazy[2*i] = (lazy[2*i] + v) % mod
            else:
                lazy[2*i] += v
            # lazy[2*i] = (lazy[2*i] + v) % mod 
            if data[2*i-1] + v >= mod:
                data[2*i-1] = (data[2*i-1] + v) % mod
            else:
                data[2*i-1] += v           
            # data[2*i-1] = (data[2*i-1] + v) % mod
            if data[2*i] + v >= mod:
                data[2*i] = (data[2*i] + v) % mod
            else:
                data[2*i] += v                  
            # data[2*i] = (data[2*i] + v) % mod
            lazy[i-1] = 0

    # 区間[l, r)にxを加算
    def update(l, r, x):
        *ids, = gindex(l, r)
        propagates(*ids)
        L = N0 + l; R = N0 + r
        while L < R:
            if R & 1:
                R -= 1
                lazy[R-1] = (lazy[R-1] + x) % mod
                data[R-1] = (data[R-1] + x) % mod
            if L & 1:
                lazy[L-1] = (lazy[L-1] + x) % mod
                data[L-1] = (data[L-1] + x) % mod
                L += 1
            L >>= 1; R >>= 1
    
    # [l] を求める
    def get(l):
        propagates(*gindex(l, l+1))
        return data[N0+l-1]



    update(0, 1, 1)
    for i in range(n):
        p = get(i)
        # print(p)
        for l, r in intervals:
            if i + l < n:
                update(i + l, min(i + r + 1, n), p)
    
    # print(query(n-1, n))
    print(get(n-1))


if __name__ == "__main__":
    main()