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)

use input_mcr::*;
use std::cmp::*;

use modular::*;
type Mod = Modular<Mod_998_244_353>;

struct SegmentTree {
    n: usize,
    xs: Vec<Mod>,
}

impl SegmentTree {
    pub fn new(n: usize) -> SegmentTree {
        let n = n.next_power_of_two();
        let xs = vec![Mod::new(0); 2 * n];
        SegmentTree { n, xs }
    }
    pub fn add(&mut self, l: usize, r: usize, x: Mod) {
        let mut l = l + self.n;
        let mut r = r + self.n;
        while l < r {
            // eprintln!("l = {}, r = {}", l, r);
            if l & 1 != 0 {
                self.xs[l] += x;
                l += 1;
            }
            if r & 1 != 0 {
                r -= 1;
                self.xs[r] += x;
            }
            l >>= 1;
            r >>= 1;
        }
    }
    pub fn get(&self, i: usize) -> Mod {
        let mut i = self.n + i;
        let mut res = Mod::new(0);
        while i != 0 {
            res += self.xs[i];
            i /= 2;
        }
        res
    }
}

fn main() {
    input! {
        n: usize,
        m: usize,
        rs: [(usize,usize); m],
    }
    let mut seg = SegmentTree::new(3 * n + 10);
    seg.add(1, 2, Mod::new(1));
    for i in 1..n {
        for &(l, r) in &rs {
            let t = seg.get(i);
            seg.add(i + l, i + r + 1, t);
        }
    }
    let res = seg.get(n);
    println!("{}", res.as_u32());
}
pub mod input_mcr {
    // ref: tanakh <https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8>
    #[macro_export(local_inner_macros)]
    macro_rules! input {
        (source = $s:expr, $($r:tt)*) => {
            let mut parser = Parser::from_str($s);
            input_inner!{parser, $($r)*}
        };
        (parser = $parser:ident, $($r:tt)*) => {
            input_inner!{$parser, $($r)*}
        };
        (new_stdin_parser = $parser:ident, $($r:tt)*) => {
            let stdin = std::io::stdin();
            let reader = std::io::BufReader::new(stdin.lock());
            let mut $parser = Parser::new(reader);
            input_inner!{$parser, $($r)*}
        };
        ($($r:tt)*) => {
            input!{new_stdin_parser = parser, $($r)*}
        };
    }

    #[macro_export(local_inner_macros)]
    macro_rules! input_inner {
        ($parser:ident) => {};
        ($parser:ident, ) => {};
        ($parser:ident, $var:ident : $t:tt $($r:tt)*) => {
            let $var = read_value!($parser, $t);
            input_inner!{$parser $($r)*}
        };
    }

    #[macro_export(local_inner_macros)]
    macro_rules! read_value {
        ($parser:ident, ( $($t:tt),* )) => {
            ( $(read_value!($parser, $t)),* )
        };
        ($parser:ident, [ $t:tt ; $len:expr ]) => {
            (0..$len).map(|_| read_value!($parser, $t)).collect::<Vec<_>>()
        };
        ($parser:ident, chars) => {
            read_value!($parser, String).chars().collect::<Vec<char>>()
        };
        ($parser:ident, char_) => {
            read_value!($parser, String).chars().collect::<Vec<char>>()[0]
        };
        ($parser:ident, usize1) => {
            read_value!($parser, usize) - 1
        };
        ($parser:ident, line) => {
            $parser.next_line()
        };
        ($parser:ident, line_) => {
            $parser.next_line().chars().collect::<Vec<char>>()
        };
        ($parser:ident, $t:ty) => {
            $parser.next::<$t>().expect("Parse error")
        };
    }

    use std::collections::VecDeque;
    use std::io;
    use std::io::BufRead;
    use std::str;

    pub struct Parser<R> {
        pub reader: R,
        buf: VecDeque<u8>,
        parse_buf: Vec<u8>,
    }

    impl Parser<io::Empty> {
        pub fn from_str(s: &str) -> Parser<io::Empty> {
            Parser {
                reader: io::empty(),
                buf: VecDeque::from(s.as_bytes().to_vec()),
                parse_buf: vec![],
            }
        }
    }

    impl<R: BufRead> Parser<R> {
        pub fn new(reader: R) -> Parser<R> {
            Parser {
                reader: reader,
                buf: VecDeque::new(),
                parse_buf: vec![],
            }
        }
        pub fn update_buf(&mut self) {
            loop {
                let (len, complete) = {
                    let buf2 = self.reader.fill_buf().unwrap();
                    self.buf.extend(buf2.iter());
                    let len = buf2.len();
                    (len, buf2.last() < Some(&0x20))
                };
                self.reader.consume(len);
                if complete {
                    break;
                }
            }
        }
        pub fn next<T: str::FromStr>(&mut self) -> Result<T, T::Err> {
            loop {
                while let Some(c) = self.buf.pop_front() {
                    if c > 0x20 {
                        self.buf.push_front(c);
                        break;
                    }
                }
                self.parse_buf.clear();
                while let Some(c) = self.buf.pop_front() {
                    if c <= 0x20 {
                        self.buf.push_front(c);
                        break;
                    } else {
                        self.parse_buf.push(c);
                    }
                }
                if self.parse_buf.is_empty() {
                    self.update_buf();
                } else {
                    return unsafe { str::from_utf8_unchecked(&self.parse_buf) }.parse::<T>();
                }
            }
        }
        pub fn next_line(&mut self) -> String {
            loop {
                while let Some(c) = self.buf.pop_front() {
                    if c >= 0x20 {
                        self.buf.push_front(c);
                        break;
                    }
                }
                self.parse_buf.clear();
                while let Some(c) = self.buf.pop_front() {
                    if c < 0x20 {
                        self.buf.push_front(c);
                        break;
                    } else {
                        self.parse_buf.push(c);
                    }
                }
                if self.parse_buf.is_empty() {
                    self.update_buf();
                } else {
                    return unsafe { str::from_utf8_unchecked(&self.parse_buf) }.to_string();
                }
            }
        }
    }
}
pub mod modular {
    use std::marker::PhantomData;
    use std::ops::*;

    pub trait ModP {
        fn as_u64() -> u64;
    }

    #[allow(non_camel_case_types)]
    #[derive(Clone, Copy, Debug)]
    pub struct Mod_1_000_000_007();
    impl ModP for Mod_1_000_000_007 {
        fn as_u64() -> u64 {
            1_000_000_007
        }
    }

    #[allow(non_camel_case_types)]
    #[derive(Clone, Copy, Debug)]
    pub struct Mod_998_244_353();
    impl ModP for Mod_998_244_353 {
        fn as_u64() -> u64 {
            998_244_353
        }
    }

    #[derive(Clone, Copy, PartialEq, Debug)]
    pub struct Modular<P>(u32, PhantomData<P>);

    pub trait ToModular<P> {
        fn to_mod(self) -> Modular<P>;
    }

    impl<P: ModP + Copy> ToModular<P> for usize {
        fn to_mod(self) -> Modular<P> {
            let m = P::as_u64() as usize;
            Modular((self % m) as u32, PhantomData)
        }
    }

    impl<P: ModP + Copy> ToModular<P> for i64 {
        fn to_mod(self) -> Modular<P> {
            let m = P::as_u64() as i64;
            Modular(((self % m + m) % m) as u32, PhantomData)
        }
    }

    impl<P: ModP + Copy> ToModular<P> for i32 {
        fn to_mod(self) -> Modular<P> {
            let m = P::as_u64() as i32;
            Modular(((self % m + m) % m) as u32, PhantomData)
        }
    }

    impl<P: ModP + Copy> ToModular<P> for u64 {
        fn to_mod(self) -> Modular<P> {
            let m = P::as_u64() as u64;
            Modular((self % m) as u32, PhantomData)
        }
    }

    impl<P: ModP + Copy> ToModular<P> for u32 {
        fn to_mod(self) -> Modular<P> {
            let m = P::as_u64() as u32;
            Modular((self % m) as u32, PhantomData)
        }
    }

    impl<P: ModP + Copy> Modular<P> {
        pub fn new<T: ToModular<P>>(x: T) -> Modular<P> {
            x.to_mod()
        }
        pub fn pow(self, n: u64) -> Modular<P> {
            if n == 0 {
                return Modular(1, PhantomData);
            }
            let t = self.pow(n / 2);
            if n % 2 == 0 {
                t * t
            } else {
                t * t * self
            }
        }
        pub fn as_u32(self) -> u32 {
            self.0
        }
    }

    impl<P: ModP + Copy> Add for Modular<P> {
        type Output = Self;
        fn add(self, rhs: Self) -> Self::Output {
            let t = (self.0 + rhs.0) as u64;
            let u = if t < P::as_u64() { t } else { t - P::as_u64() };
            Modular(u as u32, PhantomData)
        }
    }

    impl<P: ModP + Copy> AddAssign for Modular<P> {
        fn add_assign(&mut self, rhs: Self) {
            *self = *self + rhs;
        }
    }

    impl<P: ModP + Copy> Sub for Modular<P> {
        type Output = Self;
        fn sub(self, rhs: Self) -> Self::Output {
            let t = if self.0 >= rhs.0 {
                self.0 as u64 - rhs.0 as u64
            } else {
                self.0 as u64 + P::as_u64() - rhs.0 as u64
            };
            assert!(t < P::as_u64());
            Modular(t as u32, PhantomData)
        }
    }

    impl<P: ModP + Copy> SubAssign for Modular<P> {
        fn sub_assign(&mut self, rhs: Self) {
            *self = *self - rhs;
        }
    }

    impl<P: ModP + Copy> Neg for Modular<P> {
        type Output = Modular<P>;
        fn neg(self) -> Modular<P> {
            Modular((P::as_u64() - self.0 as u64) as u32, PhantomData)
        }
    }

    impl<P: ModP + Copy> Mul for Modular<P> {
        type Output = Self;
        fn mul(self, rhs: Self) -> Self::Output {
            Modular(
                ((self.0 as u64 % P::as_u64()) * (rhs.0 as u64 % P::as_u64()) % P::as_u64()) as u32,
                PhantomData,
            )
        }
    }

    impl<P: ModP + Copy> MulAssign for Modular<P> {
        fn mul_assign(&mut self, rhs: Self) {
            *self = *self * rhs;
        }
    }

    impl<P: ModP + Copy> Div for Modular<P> {
        type Output = Self;
        fn div(self, rhs: Self) -> Self::Output {
            if rhs.0 == 0 {
                loop {}
            }
            assert!(rhs.0 != 0);
            self * rhs.pow(P::as_u64() - 2)
        }
    }

    impl<P: ModP + Copy> DivAssign for Modular<P> {
        fn div_assign(&mut self, rhs: Self) {
            *self = *self / rhs;
        }
    }

    pub fn gen_fact_table<P: ModP + Copy>(n: usize) -> Vec<Modular<P>> {
        let mut res = vec![Modular::<P>::new(1); n + 1];
        for i in 0..n {
            res[i + 1] = res[i] * Modular::<P>::new((i + 1) as u32);
        }
        res
    }

    pub fn gen_inv_table<P: ModP + Copy>(n: usize) -> Vec<Modular<P>> {
        let mut res = vec![Modular::new(0); n + 1];
        res[1] = Modular::new(1);
        for i in 2..n + 1 {
            res[i] =
                -res[P::as_u64() as usize % i] * Modular::<P>::new((P::as_u64() / i as u64) as u32);
        }
        res
    }

    /*
    #[test]
    fn test_modular_1_000_000_007() {
        type Mod = Modular<Mod_1_000_000_007>;
        assert_eq!(Mod::new(2).pow(1_000_000).as_u32(), 235_042_059)
    }

    #[test]
    fn test_modular_998_244_353() {
        type Mod = Modular<Mod_998_244_353>;
        assert_eq!(Mod::new(2).pow(1_000_000).as_u32(), 421_273_117);
    }
    */
}