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 C++ (GCC 9.2.1) (judge verdict: Accepted)

#include <iostream>
#include <vector>
#include <string>
#define input std::cin

using vector = std::vector<int>;
using lint = long long int;
using string = std::string;

struct range {
  int l, r;
};

int modulo = 998244353;

int main() {
  int n, k; input >> n >> k;
  std::vector<range> ranges; ranges.reserve(k);
  for (int i=0; i<k; i++) {
	int l, r; input >> l >> r;
	ranges.push_back({l,r});
  }
  vector cells (n+1); cells[n] = 0; cells[n-1] = 1;
  for (int i=n-2; i>=0; i--) {
	cells[i] = 0;
	for (int j=0; j<k; j++) {
	  range &rng = ranges[j];
	  int sum = (cells[std::min(i+rng.l, n)] -
				 (lint)cells[std::min(i+rng.r+1, n)] +
				 modulo)%modulo;
	  cells[i] = (cells[i] + (lint)sum) % modulo;
	}
	if (i == 0)
	  std::cout << cells[i] << std::endl;
	cells[i] = (cells[i] + (lint)cells[i+1]) % modulo;
  }
}