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(clippy::needless_range_loop)]
#![allow(unused_macros)]
#![allow(dead_code)]
#![allow(unused_imports)]

use itertools::Itertools;
use proconio::input;
use proconio::marker::*;

pub const MOD: usize = 998244353;
pub fn mint(number: usize) -> ModInt {
    ModInt(number % MOD)
}
#[derive(Copy, Clone, Eq, PartialEq, std :: fmt :: Debug)]
pub struct ModInt(pub usize);
impl std::ops::Add for ModInt {
    type Output = Self;
    fn add(self, rhs: Self) -> Self {
        ModInt((self.0 + rhs.0) % MOD)
    }
}
#[allow(clippy::suspicious_op_assign_impl)]
impl std::ops::AddAssign for ModInt {
    fn add_assign(&mut self, rhs: Self) {
        self.0 += rhs.0;
        self.0 %= MOD;
    }
}
impl std::ops::Sub for ModInt {
    type Output = Self;
    fn sub(self, rhs: Self) -> Self {
        ModInt((self.0 + MOD - rhs.0) % MOD)
    }
}
#[allow(clippy::suspicious_op_assign_impl)]
impl std::ops::SubAssign for ModInt {
    fn sub_assign(&mut self, rhs: Self) {
        self.0 += MOD - rhs.0;
        self.0 %= MOD;
    }
}
impl std::ops::Mul for ModInt {
    type Output = Self;
    fn mul(self, rhs: Self) -> Self {
        ModInt((self.0 * rhs.0) % MOD)
    }
}
#[allow(clippy::suspicious_op_assign_impl)]
impl std::ops::MulAssign for ModInt {
    fn mul_assign(&mut self, rhs: Self) {
        self.0 *= rhs.0;
        self.0 %= MOD;
    }
}
impl std::ops::Div for ModInt {
    type Output = Self;
    fn div(self, rhs: Self) -> Self {
        ModInt((self.0 * rhs.pow(MOD - 2).0) % MOD)
    }
}
#[allow(clippy::suspicious_op_assign_impl)]
impl std::ops::DivAssign for ModInt {
    fn div_assign(&mut self, rhs: Self) {
        self.0 *= rhs.pow(MOD - 2).0;
        self.0 %= MOD;
    }
}
impl std::fmt::Display for ModInt {
    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
        f.write_str(&self.0.to_string())
    }
}
impl ModInt {
    pub fn pow(&self, power: usize) -> ModInt {
        let mut acc_base = *self;
        let mut acc_pow = power;
        let mut res = ModInt(1);
        while acc_pow > 0 {
            if acc_pow & 1 == 1 {
                res *= acc_base;
            }
            acc_base *= acc_base;
            acc_pow >>= 1;
        }
        res
    }
}
pub struct ModIntFact {
    memo: Vec<ModInt>,
    memo_inv: Vec<ModInt>,
    size: usize,
}
impl ModIntFact {
    pub fn new() -> Self {
        Self {
            memo: vec![ModInt(1)],
            memo_inv: vec![ModInt(1)],
            size: 0,
        }
    }
    pub fn extend(&mut self, size: usize) {
        if self.size >= size {
            return;
        }
        self.memo.resize(size + 1, ModInt(0));
        self.memo_inv.resize(size + 1, ModInt(0));
        for n in (self.size + 1)..=size {
            self.memo[n] = self.memo[n - 1] * ModInt(n);
            self.memo_inv[n] = self.memo[n].pow(MOD - 2);
        }
        self.size = size;
    }
    pub fn fact(&mut self, n: usize) -> ModInt {
        self.extend(n);
        self.memo[n]
    }
    pub fn fact_inv(&mut self, n: usize) -> ModInt {
        self.extend(n);
        self.memo_inv[n]
    }
    pub fn ncr(&mut self, n: usize, r: usize) -> ModInt {
        self.fact(n) * self.fact_inv(r) * self.fact_inv(n - r)
    }
    pub fn npr(&mut self, n: usize, r: usize) -> ModInt {
        self.fact(n) * self.fact_inv(n - r)
    }
    pub fn nhr(&mut self, n: usize, r: usize) -> ModInt {
        self.fact(n + r - 1) * self.fact_inv(r) * self.fact_inv(n - 1)
    }
}
impl std::iter::Sum for ModInt {
    fn sum<I: Iterator<Item = Self>>(iter: I) -> Self {
        iter.fold(ModInt(0), |x, y| x + y)
    }
}

use std::cmp;

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

    let mut ans = vec![mint(0); n + 1];
    let mut cum = vec![mint(0); n + 1];
    ans[1] = mint(1);
    cum[1] = mint(1);

    for i in 2..=n {
        let mut sum = mint(0);
        for &(l, r) in &r {
            sum += cum[i - cmp::min(l, i)] - cum[i - cmp::min(r + 1, i)];
        }
        ans[i] = sum;
        cum[i] = cum[i - 1] + sum;
    }

    println!("{}", ans[n]);
}