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)

#[allow(unused_imports)]
use itertools::Itertools;
#[allow(unused_imports)]
use num::*;
use proconio::input;
#[allow(unused_imports)]
use proconio::marker::*;
#[allow(unused_imports)]
use std::collections::*;

#[derive(Copy, Clone, Debug, Eq, PartialEq)]
pub struct ModInt(u32);
impl ModInt {
    pub const MOD: u32 = 998244353;
    pub fn inv(self) -> Self {
        if self.0 == 0 {
            panic!();
        }
        self.pow(Self::MOD - 2)
    }
    pub fn pow<T: num::Unsigned + num::PrimInt>(self, e: T) -> Self {
        if e.is_zero() {
            return Self::new(1);
        }
        let mut res = self.pow(e >> 1);
        res *= res;
        if e & T::one() == T::one() {
            res *= self;
        }
        res
    }
    fn new(n: i64) -> Self {
        let m = Self::MOD as i64;
        let mut n = n % m;
        if n.is_negative() {
            n += m;
        }
        Self(n as u32)
    }
}
macro_rules! impl_from {
    ($ T : ty ) => {
        impl From<$T> for ModInt {
            fn from(n: $T) -> Self {
                use std::convert::TryFrom;
                Self::new(i64::try_from(n).unwrap())
            }
        }
    };
}
impl_from!(i32);
impl_from!(i64);
impl_from!(isize);
impl_from!(u32);
impl_from!(u64);
impl_from!(usize);
macro_rules! impl_into {
    ($ T : ty ) => {
        impl Into<$T> for ModInt {
            fn into(self) -> $T {
                self.0 as $T
            }
        }
    };
}
impl_into!(i32);
impl_into!(i64);
impl_into!(isize);
impl_into!(u32);
impl_into!(u64);
impl_into!(usize);
impl std::fmt::Display for ModInt {
    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
        write!(f, "{}", self.0)
    }
}
impl std::ops::Add for ModInt {
    type Output = Self;
    fn add(self, rhs: Self) -> Self {
        Self::new((self.0 + rhs.0) as i64)
    }
}
impl std::ops::AddAssign for ModInt {
    fn add_assign(&mut self, rhs: Self) {
        *self = *self + rhs;
    }
}
impl std::ops::Div for ModInt {
    type Output = Self;
    fn div(self, rhs: Self) -> Self {
        self * rhs.inv()
    }
}
impl std::ops::DivAssign for ModInt {
    fn div_assign(&mut self, rhs: Self) {
        *self = *self / rhs;
    }
}
impl std::ops::Mul for ModInt {
    type Output = Self;
    fn mul(self, rhs: Self) -> Self {
        Self::new((self.0 as i64) * (rhs.0 as i64))
    }
}
impl std::ops::MulAssign for ModInt {
    fn mul_assign(&mut self, rhs: Self) {
        *self = *self * rhs;
    }
}
impl std::ops::Sub for ModInt {
    type Output = Self;
    fn sub(self, rhs: Self) -> Self {
        Self::new((self.0 as i64) - (rhs.0 as i64))
    }
}
impl std::ops::SubAssign for ModInt {
    fn sub_assign(&mut self, rhs: Self) {
        *self = *self - rhs;
    }
}
impl num::Zero for ModInt {
    fn zero() -> Self {
        Self::new(0)
    }
    fn is_zero(&self) -> bool {
        *self == Self::zero()
    }
}
impl num::One for ModInt {
    fn one() -> Self {
        Self::new(1)
    }
    fn is_one(&self) -> bool {
        *self == Self::one()
    }
}

fn solve() {
    input! {
        n: usize, k: usize,
        lrv: [(usize, usize); k]
    };

    let mut imos = vec![ModInt::zero(); n + 2];
    imos[1] = ModInt::one();
    let mut cur = ModInt::zero();
    for i in 1..n {
        cur += imos[i];
        for (l, r) in &lrv {
            if i + l > n {
                continue;
            }
            imos[i + l] += cur;
            imos[(i + r + 1).min(n + 1)] -= cur;
        }
    }
    println!("{}", imos[n]);
}

fn main() {
    std::thread::Builder::new()
        .name("big stack size".into())
        .stack_size(256 * 1024 * 1024)
        .spawn(|| {
            solve();
        })
        .unwrap()
        .join()
        .unwrap();
}