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

use std::str::FromStr;
use std::io::*;
use std::collections::*;
use std::cmp::*;

struct Scanner<I: Iterator<Item = char>> {
    iter: std::iter::Peekable<I>,
}

macro_rules! exit {
    () => {{
        exit!(0)
    }};
    ($code:expr) => {{
        if cfg!(local) {
            writeln!(std::io::stderr(), "===== Terminated =====")
                .expect("failed printing to stderr");
        }
        std::process::exit($code);
    }}
}

impl<I: Iterator<Item = char>> Scanner<I> {
    pub fn new(iter: I) -> Scanner<I> {
        Scanner {
            iter: iter.peekable(),
        }
    }

    pub fn safe_get_token(&mut self) -> Option<String> {
        let token = self.iter
            .by_ref()
            .skip_while(|c| c.is_whitespace())
            .take_while(|c| !c.is_whitespace())
            .collect::<String>();
        if token.is_empty() {
            None
        } else {
            Some(token)
        }
    }

    pub fn token(&mut self) -> String {
        self.safe_get_token().unwrap_or_else(|| exit!())
    }

    pub fn get<T: FromStr>(&mut self) -> T {
        self.token().parse::<T>().unwrap_or_else(|_| exit!())
    }

    pub fn vec<T: FromStr>(&mut self, len: usize) -> Vec<T> {
        (0..len).map(|_| self.get()).collect()
    }

    pub fn mat<T: FromStr>(&mut self, row: usize, col: usize) -> Vec<Vec<T>> {
        (0..row).map(|_| self.vec(col)).collect()
    }

    pub fn char(&mut self) -> char {
        self.iter.next().unwrap_or_else(|| exit!())
    }

    pub fn chars(&mut self) -> Vec<char> {
        self.get::<String>().chars().collect()
    }

    pub fn mat_chars(&mut self, row: usize) -> Vec<Vec<char>> {
        (0..row).map(|_| self.chars()).collect()
    }

    pub fn line(&mut self) -> String {
        if self.peek().is_some() {
            self.iter
                .by_ref()
                .take_while(|&c| !(c == '\n' || c == '\r'))
                .collect::<String>()
        } else {
            exit!();
        }
    }

    pub fn peek(&mut self) -> Option<&char> {
        self.iter.peek()
    }
}

use std::ops::*;
use std::fmt;

#[derive(Copy, Clone, PartialEq, Debug)]
struct ModInt(usize);

const MOD: usize = 998244353;

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

    fn inv(self) -> Self {
        self.pow(MOD - 2)
    }
}

impl Add for ModInt {
    type Output = Self;

    fn add(self, rhs: Self) -> Self {
        let mut tmp = self;
        tmp.0 += rhs.0;
        if tmp.0 >= MOD {
            tmp.0 -= MOD;
        }
        tmp
    }
}

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

impl Sub for ModInt {
    type Output = Self;

    fn sub(self, rhs: Self) -> Self {
        let mut ret = self;
        if ret.0 < rhs.0 {
            ret.0 += MOD;
        }
        ret.0 -= rhs.0;
        ret
    }
}

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

impl Mul for ModInt {
    type Output = Self;

    fn mul(self, rhs: ModInt) -> Self {
        let mut ret = self;
        ret.0 *= rhs.0;
        ret.0 %= MOD;
        ret
    }
}

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

impl Div for ModInt {
    type Output = ModInt;

    fn div(self, rhs: ModInt) -> Self {
        let mut ret = self;
        ret.0 *= rhs.inv().0;
        ret.0 %= MOD;
        ret
    }
}

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

impl Neg for ModInt {
    type Output = ModInt;
 
    fn neg(self) -> Self::Output {
        ModInt(MOD) - self
    }
}

impl fmt::Display for ModInt {
    fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
        write!(f, "{}", self.0)
    }
}

fn main() {
    let cin = stdin();
    let cin = cin.lock();
    let mut sc = Scanner::new(cin.bytes().map(|c| c.unwrap() as char));
    let N: usize = sc.get();
    let K: usize = sc.get();
    let mut L = vec![0; N];
    let mut R = vec![0; N];
    for i in 0..K {
        L[i] = sc.get();
        R[i] = sc.get();
    }
    let mut dp = vec![ModInt(0); N+1];
    dp[1] = ModInt(1);
    dp[2] = -ModInt(1);
    for i in 1..=N {
        let p = dp[i-1];
        dp[i] += p;
        for j in 0..K {
            if i + L[j] > N {
                continue;
            }
            let p = dp[i];
            dp[i+L[j]] += p;
            if i+R[j]+1 <= N {
                dp[i+R[j]+1] += -p;
            }
        }
    }
    println!("{}", dp[N]);
}