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(non_snake_case)]
#![allow(unused_imports)]
#![allow(dead_code)]

use proconio::{input, fastout};
use proconio::marker::*;
use whiteread::parse_line;
use std::collections::*;
use num::*;
use num_traits::*;
use superslice::*;
use std::ops::*;

use itertools::Itertools;
use itertools_num::ItertoolsNum;


#[derive(Debug)]
pub struct ModInt<T> {
    v: T,
    m: T,
}
impl<T> ModInt<T>
where
    T: Copy,
{
    pub fn value(&self) -> T {
        self.v
    }
    pub fn modulo(&self) -> T {
        self.m
    }
}
impl<T> ModInt<T> {
    fn new_unchecked(v: T, modulo: T) -> Self {
        Self { v, m: modulo }
    }
}
impl<T> ModInt<T>
where
    T: Copy + RemAssign + PartialOrd,
{
    pub fn new(mut v: T, modulo: T) -> Self {
        if v >= modulo {
            v %= modulo;
        }
        Self::new_unchecked(v, modulo)
    }
}
impl<T> ModInt<T>
where
    T: Copy
        + Sub<Output = T>
        + ShrAssign
        + BitAnd<Output = T>
        + PartialEq
        + PartialOrd
        + Div<Output = T>
        + RemAssign,
    ModInt<T>: MulAssign,
{
    pub fn pow(self, e: T) -> Self {
        let zero = self.modulo() - self.modulo();
        let one = self.modulo() / self.modulo();
        let mut e = e;
        let mut result = Self::new_unchecked(one, self.modulo());
        let mut cur = self;
        while e > zero {
            if e & one == one {
                result *= cur;
            }
            e >>= one;
            cur *= cur;
        }
        result
    }
}
impl<T> Copy for ModInt<T> where T: Copy {}
impl<T> Clone for ModInt<T>
where
    T: Copy,
{
    fn clone(&self) -> Self {
        Self::new_unchecked(self.value(), self.modulo())
    }
}
impl<T> Add<T> for ModInt<T>
where
    T: AddAssign + SubAssign + RemAssign + Copy + PartialOrd,
{
    type Output = Self;
    fn add(self, mut rhs: T) -> Self::Output {
        if rhs >= self.modulo() {
            rhs %= self.modulo();
        }
        rhs += self.value();
        if rhs >= self.modulo() {
            rhs -= self.modulo();
        }
        Self::new_unchecked(rhs, self.modulo())
    }
}
impl<T> Sub<T> for ModInt<T>
where
    T: AddAssign + SubAssign + RemAssign + Copy + PartialOrd,
{
    type Output = Self;
    fn sub(self, mut rhs: T) -> Self::Output {
        if rhs >= self.modulo() {
            rhs %= self.modulo();
        }
        let mut result = self.value();
        result += self.modulo();
        result -= rhs;
        if result >= self.modulo() {
            result -= self.modulo();
        }
        Self::new_unchecked(result, self.modulo())
    }
}
impl<T> Mul<T> for ModInt<T>
where
    T: MulAssign + RemAssign + Copy + PartialOrd,
{
    type Output = Self;
    fn mul(self, mut rhs: T) -> Self::Output {
        if rhs >= self.modulo() {
            rhs %= self.modulo();
        }
        rhs *= self.value();
        rhs %= self.modulo();
        Self::new_unchecked(rhs, self.modulo())
    }
}
impl<T> Add<ModInt<T>> for ModInt<T>
where
    T: Copy,
    ModInt<T>: Add<T, Output = ModInt<T>>,
{
    type Output = Self;
    fn add(self, rhs: ModInt<T>) -> Self::Output {
        self + rhs.value()
    }
}
impl<T> Sub<ModInt<T>> for ModInt<T>
where
    T: Copy,
    ModInt<T>: Sub<T, Output = ModInt<T>>,
{
    type Output = Self;
    fn sub(self, rhs: ModInt<T>) -> Self::Output {
        self - rhs.value()
    }
}
impl<T> Mul<ModInt<T>> for ModInt<T>
where
    T: Copy,
    ModInt<T>: Mul<T, Output = ModInt<T>>,
{
    type Output = Self;
    fn mul(self, rhs: ModInt<T>) -> Self::Output {
        self * rhs.value()
    }
}
impl<T> Div<ModInt<T>> for ModInt<T>
where
    T: Copy,
    ModInt<T>: Div<T, Output = ModInt<T>>,
{
    type Output = Self;
    fn div(self, rhs: ModInt<T>) -> Self::Output {
        self / rhs.value()
    }
}
impl<T> AddAssign<T> for ModInt<T>
where
    T: Copy,
    ModInt<T>: Add<T, Output = ModInt<T>>,
{
    fn add_assign(&mut self, other: T) {
        *self = *self + other;
    }
}
impl<T> AddAssign<ModInt<T>> for ModInt<T>
where
    T: Copy,
    ModInt<T>: Add<ModInt<T>, Output = ModInt<T>>,
{
    fn add_assign(&mut self, other: ModInt<T>) {
        *self = *self + other;
    }
}
impl<T> SubAssign<T> for ModInt<T>
where
    T: Copy,
    ModInt<T>: Sub<T, Output = ModInt<T>>,
{
    fn sub_assign(&mut self, other: T) {
        *self = *self - other;
    }
}
impl<T> SubAssign<ModInt<T>> for ModInt<T>
where
    T: Copy,
    ModInt<T>: Sub<ModInt<T>, Output = ModInt<T>>,
{
    fn sub_assign(&mut self, other: ModInt<T>) {
        *self = *self - other;
    }
}
impl<T> DivAssign<T> for ModInt<T>
where
    T: Copy,
    ModInt<T>: Div<T, Output = ModInt<T>>,
{
    fn div_assign(&mut self, rhs: T) {
        *self = *self / rhs
    }
}
impl<T> DivAssign<ModInt<T>> for ModInt<T>
where
    T: Copy,
    ModInt<T>: Div<ModInt<T>, Output = ModInt<T>>,
{
    fn div_assign(&mut self, rhs: ModInt<T>) {
        *self = *self / rhs
    }
}
impl<T> MulAssign<T> for ModInt<T>
where
    T: Copy,
    ModInt<T>: Mul<T, Output = ModInt<T>>,
{
    fn mul_assign(&mut self, rhs: T) {
        *self = *self * rhs;
    }
}
impl<T> MulAssign<ModInt<T>> for ModInt<T>
where
    T: Copy,
    ModInt<T>: Mul<ModInt<T>, Output = ModInt<T>>,
{
    fn mul_assign(&mut self, rhs: ModInt<T>) {
        *self = *self * rhs;
    }
}
impl<T> Div<T> for ModInt<T>
where
    T: Copy
        + Add<Output = T>
        + Sub<Output = T>
        + Div<Output = T>
        + BitAnd<Output = T>
        + PartialEq
        + PartialOrd
        + SubAssign
        + ShrAssign
        + RemAssign
        + MulAssign,
{
    type Output = Self;
    fn div(self, mut rhs: T) -> Self::Output {
        if rhs >= self.modulo() {
            rhs %= self.modulo();
        }
        let one = self.modulo() / self.modulo();
        let two = one + one;
        self * Self::new_unchecked(rhs, self.modulo()).pow(self.modulo() - two)
    }
}

#[fastout]
fn solve() {
    const MOD: usize = 998244353;
    const INF: usize = std::usize::MAX;

    input!{
        n: usize,
        k: usize,
        lr: [(usize, usize); k]
    }

    let mut dp = vec![ModInt::new(0, MOD); n+10];
    let mut dp_sum = vec![ModInt::new(0, MOD); n+10];
    dp[0] = ModInt::new(1, MOD);
    dp_sum[0] = ModInt::new(1, MOD);

    for i in 1..n {
        let mut val = ModInt::new(0, MOD);
        for &(l, r) in lr.iter() {
            if l <= i {
                val += dp_sum[i - l];
                if r < i {
                    val -= dp_sum[i - r - 1]
                }
            }
        }
        dp[i] = val;
        dp_sum[i] = dp_sum[i-1] + dp[i];
    }
    // dbg!(&dp_sum);
    println!("{}", dp[n-1].value());
}

fn main() {
    solve()
}

#[cfg(test)]
mod test {
    use super::solve;
}