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 Rust (1.42.0) (judge verdict: Time Limit Exceeded)

#[allow(unused_imports)] use proconio::{input, marker::{Bytes, Chars, Usize1, Isize1}};
#[allow(unused_imports)] use std::cmp::{min, max};
#[allow(unused_imports)] use superslice::Ext as _;

const INF: usize = 99999999999999;
const MOD: usize = 998244353;

fn proc_r(n: usize, pos: usize, ca: &mut [usize], set: &Vec<usize>, num: usize ) -> usize {
	if pos == n {
		return 1;
	}
	if ca[pos] != INF {
		return num * ca[pos] % MOD;
	}
	let mut sum = 0;
	for i in 0..set.len() {
		if pos+set[i] > n {
			break;
		}
		sum += proc_r(n, pos+set[i], ca, set, num);
	}
	ca[pos] = sum % MOD;
	return num * ca[pos] % MOD;
}

#[proconio::fastout]
fn main() {
	input! {
		n: usize,
		k: usize,
		lr: [(usize, usize); k],
	}

	let mut set: Vec<usize> = vec![];
	for i in 0..k {
		for j in lr[i].0..=lr[i].1 {
			set.push(j);
		}
	}
	set.sort();
	let mut ca = vec![INF; n];
	let ans = proc_r(n-1, 1-1, &mut ca, &set, 1);

	println!("{}", ans);
}