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: Accepted)

use proconio::input;
use std::cmp;

fn main() {
    input! {
        n: usize,
        k: usize,
        kukan: [(usize, usize); k],
    }

    let mo = 998244353;
    let mut dp = BIT::new(n);
    dp.add(0, 1);

    for i in 1..n {
        let mut tmp = 0;
        for j in 0..k {
            if i < kukan[j].0 {
                continue;
            }
            let start = i - kukan[j].0;
            let end = cmp::max(0, i as isize - kukan[j].1 as isize) as usize;

            tmp += dp.sum(end as isize, start as isize + 1);
            tmp %= mo;
        }
        dp.add(i, tmp);
    }

    let ans = dp.sum(n as isize - 1, n as isize);
    println!("{}", ans);
}

struct BIT {
    n: usize,
    d: Vec<usize>,
}

impl BIT {
    fn new(n: usize) -> Self {
        BIT {
            n: n,
            d: vec![0; n + 1],
        }
    }

    fn add(&mut self, mut i: usize, x: usize) {
        let mo = 998244353;
        i += 1;
        while i <= self.n {
            self.d[i] = self.d[i] + x;
            self.d[i] %= mo;
            i += i & i.wrapping_neg();
        }
    }

    fn sum_one(&self, mut i: isize) -> usize {
        let mo = 998244353;
        let mut x = 0;
        i += 1;
        let mut i = i as usize;
        while i != 0 {
            x = x + self.d[i];
            x %= mo;
            i -= i & i.wrapping_neg();
        }
        x
    }

    fn sum(&self, l: isize, r: isize) -> usize {
        let mo = 998244353;
        (self.sum_one(r - 1) + mo - self.sum_one(l - 1)) % mo
    }
}