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, fastout};

#[fastout]
fn main() {
    input!{
        n: usize,
        k: usize,
        lr:[(i64, i64); k]
    }
    let mut dp = vec![ModInt::from_usize(0); n+1];
    let mut dpsum = vec![ModInt::from_usize(0); n + 1];

    dp[1] = ModInt::from_usize(1);
    dpsum[1] = ModInt::from_usize(1);
    for i in 2..=n {
        for j in 0..k {
            let (l, r) = lr[j];
            let li = (i as i64 - r).max(1);
            let ri = i as i64 - l;
            if ri < 0 {
                continue;
            }
            dp[i] += dpsum[ri as usize] - dpsum[(li-1) as usize];

        }
        dpsum[i] = dpsum[i-1] + dp[i];
    }

    let ans = dp[n].val();
    println!("{}", ans);
}
use std::ops::*;

const MOD: usize = 998244353;
#[derive(Copy, Clone)]
struct ModInt {
    n: usize,
}

impl ModInt {
    fn val(self)->usize {
        self.n
    }
    fn from_usize(n: usize) -> ModInt {
        ModInt { n: n % MOD }
    }
    fn pow(self, power: usize) -> ModInt {
        if power == 0 {
            return ModInt::from_usize(1);
        }
        if power % 2 == 0 {
            let t = self.pow(power/2);
            return t * t;
        }
        self * self.pow(power-1)
    }
    fn inv(self) -> ModInt {
        self.pow(MOD - 2)
    }
}

impl Add for ModInt {
    type Output = ModInt;

    fn add(self, rhs: ModInt) -> ModInt {
        let mut tmp = self;
        tmp.n += rhs.n;
        if tmp.n >= MOD {
            tmp.n -= MOD;
        }
        tmp
    }
}

impl AddAssign for ModInt {
    fn add_assign(&mut self, rhs: ModInt) {
        *self = *self + rhs;
    }
}

impl Sub for ModInt {
    type Output = ModInt;

    fn sub(self, rhs: ModInt) -> ModInt {
        let mut tmp = self;
        if tmp.n < rhs.n {
            tmp.n += MOD;
        }
        tmp.n -= rhs.n;
        tmp
    }
}

impl SubAssign for ModInt {
    fn sub_assign(&mut self, rhs: ModInt) {
        *self = *self - rhs;
    }
}

impl Mul for ModInt {
    type Output = ModInt;

    fn mul(self, rhs: ModInt) -> ModInt {
        let mut tmp = self;
        tmp.n *= rhs.n;
        tmp.n %= MOD;
        tmp
    }
}

impl MulAssign for ModInt {
    fn mul_assign(&mut self, rhs: ModInt) {
        *self = *self * rhs;
    }
}

impl Div for ModInt {
    type Output = ModInt;

    fn div(self, rhs: ModInt) -> ModInt {
        let mut tmp = self;
        tmp.n *= rhs.inv().n;
        tmp.n %= MOD;
        tmp
    }
}

impl DivAssign for ModInt {
    fn div_assign(&mut self, rhs: ModInt) {
        *self = *self / rhs;
    }
}