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_parens)]
#![allow(unused_imports)]
#![allow(non_upper_case_globals)]
#![allow(non_snake_case)]
#![allow(unused_mut)]
#![allow(unused_variables)]
#![allow(dead_code)]

use itertools::Itertools;
use proconio::input;
use proconio::marker::{Chars, Usize1};

#[allow(unused_macros)]
#[cfg(debug_assertions)]
macro_rules! mydbg {
    //($arg:expr) => (dbg!($arg))
    //($arg:expr) => (println!("{:?}",$arg));
      ($($a:expr),*) => {
          eprintln!(concat!($(stringify!($a), " = {:?}, "),*), $($a),*);
      }
}
#[cfg(not(debug_assertions))]
macro_rules! mydbg {
    ($($arg:expr),*) => {};
}

macro_rules! echo {
    ($($a:expr),*) => {
          $(println!("{}",$a))*
      }
}

use std::cmp::*;
use std::collections::*;
use std::ops::{Add, Div, Mul, Sub};

#[allow(dead_code)]
static INF_I64: i64 = 92233720368547758;
#[allow(dead_code)]
static INF_I32: i32 = 21474836;
#[allow(dead_code)]
static INF_USIZE: usize = 18446744073709551;
#[allow(dead_code)]
static M_O_D: usize = 1000000007;
#[allow(dead_code)]
static PAI: f64 = 3.1415926535897932;

trait IteratorExt: Iterator {
    fn toVec(self) -> Vec<Self::Item>;
}

impl<T: Iterator> IteratorExt for T {
    fn toVec(self) -> Vec<Self::Item> {
        self.collect()
    }
}

trait CharExt {
    fn toNum(&self) -> usize;
    fn toAlphabetIndex(&self) -> usize;
    fn toNumIndex(&self) -> usize;
}
impl CharExt for char {
    fn toNum(&self) -> usize {
        return *self as usize;
    }
    fn toAlphabetIndex(&self) -> usize {
        return self.toNum() - 'a' as usize;
    }
    fn toNumIndex(&self) -> usize {
        return self.toNum() - '0' as usize;
    }
}

trait VectorExt {
    fn joinToString(&self, s: &str) -> String;
}
impl<T: ToString> VectorExt for Vec<T> {
    fn joinToString(&self, s: &str) -> String {
        return self
            .iter()
            .map(|x| x.to_string())
            .collect::<Vec<_>>()
            .join(s);
    }
}

trait StringExt {
    fn get_reverse(&self) -> String;
}
impl StringExt for String {
    fn get_reverse(&self) -> String {
        self.chars().rev().collect::<String>()
    }
}

trait UsizeExt {
    fn pow(&self, n: usize) -> usize;
}
impl UsizeExt for usize {
    fn pow(&self, n: usize) -> usize {
        return ((*self as u64).pow(n as u32)) as usize;
    }
}

pub trait Monoid {
    type T: Clone + PartialEq + std::fmt::Debug;
    fn id() -> Self::T;
    fn op(a: &Self::T, b: &Self::T) -> Self::T;
    fn op2(a: &Self::T, b: &Self::T) -> Self::T;
    fn op3(a: &Self::T, b: &Self::T) -> Self::T;
}
struct SUM;
impl Monoid for SUM {
    type T = u64;
    fn id() -> Self::T {
        0
    }
    fn op(a: &Self::T, b: &Self::T) -> Self::T {
        (*a + *b) % 998244353
    }
    fn op2(a: &Self::T, b: &Self::T) -> Self::T {
        (*a + *b) % 998244353
    }
    fn op3(a: &Self::T, b: &Self::T) -> Self::T {
        (*a + *b) % 998244353
    }
}

struct SqrtDecomposition<M: Monoid> {
    sqrt_N: usize,
    sub: Vec<M::T>,
    lazy: Vec<M::T>,
    a: Vec<M::T>,
}
impl<M: Monoid> SqrtDecomposition<M> {
    //fn new(N: usize) -> SqrtDecomposition<M> {
    fn new(a: &Vec<M::T>) -> SqrtDecomposition<M> {
        let sqrt_N = (a.len() + 511) / 512;
        let mut sub = vec![M::id(); sqrt_N];
        let mut lazy = vec![M::id(); sqrt_N];
        let mut b = vec![M::id(); sqrt_N * 512];
        for i in 0..a.len() {
            b[i] = a[i].clone();
            sub[i / 512] = M::op(&sub[i / 512], &b[i]);
        }
        SqrtDecomposition {
            sqrt_N: sqrt_N,
            sub: sub,
            a: b,
            lazy: lazy,
        }
    }
    fn update(&mut self, i: usize, v: M::T) {
        let sub_index = i / 512;
        self.a[i] = v;
        let s = sub_index * 512;
        let e = sub_index * 512 + 512;
        self.sub[sub_index] = M::id();
        for j in s..e {
            self.sub[sub_index] = M::op(&self.sub[sub_index], &self.a[j]);
        }
    }
    fn query(&mut self, l: usize, r: usize) -> M::T {
        let sub_index = l / 512;
        let mut ret = M::id();
        for i in sub_index..self.sqrt_N {
            let s = i * 512;
            let e = i * 512 + 512;
            if l >= s && e <= r {
                ret = M::op(&ret, &self.sub[i]);
            } else {
                for j in max(s, l)..min(e, r) {
                    self.a[j] = M::op(&self.a[j], &ret);
                }
            }
        }
        ret
    }
    fn update_range(&mut self, l: usize, r: usize, v: M::T) {
        let sub_l = l / 512;
        let sub_r = (r + 511) / 512;
        for i in sub_l..sub_r {
            let s = i * 512;
            let e = s + 512;
            if s >= l && e < r {
                self.lazy[i] = M::op2(&self.lazy[i], &v);
            } else {
                for j in max(s, l)..min(e, r) {
                    self.a[j] = M::op3(&self.a[j], &v);
                }
            }
        }
    }
    fn get(&mut self, l: usize) -> M::T {
        let sub_index = l / 512;
        if self.lazy[sub_index] != M::id() {
            let s = sub_index * 512;
            let e = s + 512;
            for j in s..e {
                self.a[j] = M::op3(&self.a[j], &self.lazy[sub_index]);
            }
            self.lazy[sub_index] = M::id();
        }
        self.a[l].clone()
    }
    fn query_custom<T>(
        &mut self,
        l: usize,
        r: usize,
        //check: fn(M::T) -> bool,
        check: T,
    ) -> Option<(M::T, usize)>
    where
        T: Fn(&M::T) -> bool,
    {
        let sub_index = l / 512;
        let mut ret = M::id();
        let mut index = 0;
        'outer: for i in sub_index..self.sqrt_N {
            let s = i * 512;
            let e = i * 512 + 512;

            if check(&self.sub[i]) {
                for j in s..e {
                    if check(&self.a[j]) {
                        ret = self.a[j].clone();
                        index = j;
                        break 'outer;
                    }
                }
            }
        }
        if ret == M::id() {
            return None;
        } else {
            return Some((ret, index));
        }
    }
}

fn main() {
    input! {
       N: usize,
       K:usize,
    }
    let L = (N + 511) / 512;
    let mut masu = vec![0; N + 11];
    let mut s: SqrtDecomposition<SUM> = SqrtDecomposition::new(&masu);
    let mut m = vec![];
    for _ in 0..K {
        input! {
            l:usize,
            r:usize,
        }
        m.push((l, r));
    }
    s.update(0, 1);
    for i in 0..N {
        if s.get(i) == 0 {
            continue;
        }
        for j in 0..K {
            let (l, mut r) = m[j];
            if i + l >= N {
                continue;
            }
            r += 1;
            let z = s.get(i);
            s.update_range(i + l, min(i + r, N), z);
        }
    }

    //mydbg!(s.sub, s.lazy, s.a);
    echo!(s.get(N - 1));
}