Problem:
Score : 100 points

Problem Statement

You are given an array a_0, a_1, ..., a_{N-1} of length N. Process Q queries of the following types.

0 l r b c: For each i = l, l+1, \dots, {r - 1}, set a_i \gets b \times a_i + c.

1 l r: Print \sum_{i = l}^{r - 1} a_i \bmod 998244353.

Constraints

1 \leq N, Q \leq 500000

0 \leq a_i, c < 998244353

1 \leq b < 998244353

0 \leq l < r \leq N

All values in Input are integer.

Input

Input is given from Standard Input in the following format:

N Q
a_0 a_1 ... a_{N - 1}
\textrm{Query}_0
\textrm{Query}_1
:
\textrm{Query}_{Q - 1}

Output

For each query of the latter type, print the answer.

Sample Input 1

5 7
1 2 3 4 5
1 0 5
0 2 4 100 101
1 0 3
0 1 3 102 103
1 2 5
0 2 5 104 105
1 0 5

Sample Output 1

15
404
41511
4317767

Solution in Rust (1.42.0) (judge verdict: Accepted)

#![allow(unused_imports, unused_macros)]

use kyoproio::*;
use std::{
    collections::*,
    io::{self, prelude::*},
    iter,
    mem::{replace, swap},
};

fn main() -> io::Result<()> {
    std::thread::Builder::new()
        .stack_size(64 * 1024 * 1024)
        .spawn(|| {
            let stdin = io::stdin();
            let stdout = io::stdout();
            run(KInput::new(stdin.lock()), io::BufWriter::new(stdout.lock()))
        })?
        .join()
        .unwrap()
}

fn run<I: Input, O: Write>(mut kin: I, mut out: O) -> io::Result<()> {
    macro_rules! output { ($($args:expr),+) => { write!(&mut out, $($args),+)?; }; }
    macro_rules! outputln {
        ($($args:expr),+) => { output!($($args),+); outputln!(); };
        () => { output!("\n"); if cfg!(debug_assertions) { out.flush()?; } }
    }

    //   a2(a1x + b1) + b2
    // = a1a2x + a2b1 + b2
    // (a1, b1) <- (a1a2, a2b1 + b2)

    //   (ax1 + b) + (ax2 + b) + ... + (axk + b)
    // = a(x1 + x2 + ... + xk) + kb

    let (n, q): (usize, usize) = kin.input();
    let a: Vec<_> = kin.iter::<i64>().take(n).map(|x| Add(x, 1)).collect();
    let mut lst = LazySegTree::from_slice(&a, |f: &Affine, a: &Add| Add((f.0 * a.0 + f.1 * a.1) % P, a.1));
    for _ in 0..q {
        if kin.input::<char>() == '0' {
            let (l, r, b, c): (usize, usize, i64, i64) = kin.input();
            lst.apply(l, r, &Affine(b, c));
        } else {
            let (l, r): (usize, usize) = kin.input();
            outputln!("{}", lst.prod(l, r).0);
        }
    }

    Ok(())
}

const P: i64 = 998244353;
#[derive(Debug, Clone)]
struct Add(i64, i64);
impl Monoid for Add {
    fn id() -> Self {
        Self(0, 1)
    }
    fn op(&self, other: &Self) -> Self {
        Self((self.0 + other.0) % P, self.1 + other.1)
    }
}

#[derive(Debug, Clone)]
struct Affine(i64, i64);
impl Monoid for Affine {
    fn id() -> Self {
        Self(1, 0)
    }
    fn op(&self, other: &Self) -> Self {
        Self(self.0 * other.0 % P, (self.0 * other.1 + self.1) % P)
    }
}

pub trait Monoid {
    fn id() -> Self;
    fn op(&self, other: &Self) -> Self;
}

pub struct LazySegTree<S, F, Map> {
    ss: Box<[S]>,
    fs: Box<[F]>,
    map: Map,
}

impl<S: Monoid, F: Monoid, Map: Fn(&F, &S) -> S> LazySegTree<S, F, Map> {
    pub fn new(n: usize, map: Map) -> Self {
        use std::iter::repeat_with;
        let len = 2 * n.next_power_of_two();
        Self {
            ss: repeat_with(S::id).take(len).collect(), 
            fs: repeat_with(F::id).take(len).collect(), 
            map, 
        }
    }
    fn len(&self) -> usize {
        self.ss.len() / 2
    }
    fn propagate(&mut self, i: usize) {
        let h = 8 * std::mem::size_of::<usize>() as u32 - i.leading_zeros();
        for k in (1..h).rev() {
            let p = i >> k;
            let l = 2 * p;
            let r = 2 * p + 1;
            self.ss[l] = (self.map)(&self.fs[p], &self.ss[l]);
            self.ss[r] = (self.map)(&self.fs[p], &self.ss[r]);
            self.fs[l] = self.fs[p].op(&self.fs[l]);
            self.fs[r] = self.fs[p].op(&self.fs[r]);
            self.fs[p] = F::id();
        }
    }
    pub fn prod(&mut self, l: usize, r: usize) -> S {
        assert!(l <= r);
        assert!(r <= self.len());
        let mut l = l + self.len();
        let mut r = r + self.len();
        let ln = l >> l.trailing_zeros();
        let rn = r >> r.trailing_zeros();
        self.propagate(ln);
        self.propagate(rn - 1);
        let mut lv = S::id();
        let mut rv = S::id();
        while l < r {
            if l % 2 == 1 {
                lv = lv.op(&self.ss[l]);
                l += 1;
            }
            if r % 2 == 1 {
                r -= 1;
                rv = rv.op(&self.ss[r]);
            }
            l /= 2;
            r /= 2;
        }
        lv.op(&rv)
    }
    pub fn set(&mut self, i: usize, v: S) {
        let mut i = i + self.len();
        self.propagate(i);
        self.ss[i] = v;
        while i > 1 {
            i /= 2;
            self.ss[i] = self.ss[2 * i].op(&self.ss[2 * i + 1]);
        }
    }
    pub fn apply(&mut self, l: usize, r: usize, f: &F) {
        assert!(l <= r);
        assert!(r <= self.len());
        let mut li = l + self.len();
        let mut ri = r + self.len();
        let ln = li >> li.trailing_zeros();
        let rn = ri >> ri.trailing_zeros();
        self.propagate(ln);
        self.propagate(rn - 1);
        while li < ri {
            if li % 2 == 1 {
                self.fs[li] = f.op(&self.fs[li]);
                self.ss[li] = (self.map)(f, &self.ss[li]);
                li += 1;
            }
            if ri % 2 == 1 {
                ri -= 1;
                self.fs[ri] = f.op(&self.fs[ri]);
                self.ss[ri] = (self.map)(f, &self.ss[ri]);
            }
            li /= 2;
            ri /= 2;
        }
        let mut l = (l + self.len()) / 2;
        let mut r = (r + self.len() - 1) / 2;
        while l > 0 {
            if l < ln {
                self.ss[l] = self.ss[2 * l].op(&self.ss[2 * l + 1]);
            }
            if /*l != r && */r < rn - 1 {
                self.ss[r] = self.ss[2 * r].op(&self.ss[2 * r + 1]);
            }
            l /= 2;
            r /= 2;
        }
    }
}

impl<S: Monoid + Clone, F: Monoid, Map: Fn(&F, &S) -> S> LazySegTree<S, F, Map> {
    pub fn from_slice(a: &[S], map: Map) -> Self {
        use std::iter::repeat_with;
        let n = a.len();
        let len = 2 * n.next_power_of_two();
        let mut ss: Vec<_> = repeat_with(S::id).take(n).collect();
        ss.extend_from_slice(a);
        for i in (1..n).rev() {
            ss[i] = ss[2 * i].op(&ss[2 * i + 1]);
        }
        Self {
            ss: ss.into(), 
            fs: repeat_with(F::id).take(len).collect(), 
            map, 
        }
    }
}

// -----------------------------------------------------------------------------
pub mod kyoproio {
    use std::io::prelude::*;
    pub trait Input {
        fn str(&mut self) -> &str;
        fn input<T: InputParse>(&mut self) -> T {
            T::input(self)
        }
        fn iter<T: InputParse>(&mut self) -> Iter<T, Self> {
            Iter(self, std::marker::PhantomData)
        }
        fn seq<T: InputParse, B: std::iter::FromIterator<T>>(&mut self, n: usize) -> B {
            self.iter().take(n).collect()
        }
    }
    pub struct KInput<R> {
        src: R,
        buf: String,
        pos: usize,
    }
    impl<R: BufRead> KInput<R> {
        pub fn new(src: R) -> Self {
            Self {
                src,
                buf: String::with_capacity(1024),
                pos: 0,
            }
        }
    }
    impl<R: BufRead> Input for KInput<R> {
        fn str(&mut self) -> &str {
            loop {
                if self.pos >= self.buf.len() {
                    self.pos = 0;
                    self.buf.clear();
                    if self.src.read_line(&mut self.buf).expect("io error") == 0 {
                        return &self.buf;
                    }
                }
                let range = self.pos
                    ..self.buf[self.pos..]
                        .find(|c: char| c.is_ascii_whitespace())
                        .map(|i| i + self.pos)
                        .unwrap_or_else(|| self.buf.len());
                self.pos = range.end + 1;
                if range.end > range.start {
                    return &self.buf[range];
                }
            }
        }
    }
    pub struct Iter<'a, T, I: ?Sized>(&'a mut I, std::marker::PhantomData<*const T>);
    impl<'a, T: InputParse, I: Input + ?Sized> Iterator for Iter<'a, T, I> {
        type Item = T;
        fn next(&mut self) -> Option<T> {
            Some(self.0.input())
        }
    }
    pub trait InputParse: Sized {
        fn input<I: Input + ?Sized>(src: &mut I) -> Self;
    }
    impl InputParse for Vec<u8> {
        fn input<I: Input + ?Sized>(src: &mut I) -> Self {
            src.str().as_bytes().to_owned()
        }
    }
    macro_rules! from_str_impl {
        { $($T:ty)* } => {
            $(impl InputParse for $T {
                fn input<I: Input + ?Sized>(src: &mut I) -> Self {
                    src.str().parse::<$T>().expect("parse error")
                }
            })*
        }
    }
    from_str_impl! {
        String char bool f32 f64 isize i8 i16 i32 i64 i128 usize u8 u16 u32 u64 u128
    }
    macro_rules! tuple_impl {
        ($H:ident $($T:ident)*) => {
            impl<$H: InputParse, $($T: InputParse),*> InputParse for ($H, $($T),*) {
                fn input<I: Input + ?Sized>(src: &mut I) -> Self {
                    ($H::input(src), $($T::input(src)),*)
                }
            }
            tuple_impl!($($T)*);
        };
        () => {}
    }
    tuple_impl!(A B C D E F G);
    #[macro_export]
    macro_rules! kdbg {
        ($($v:expr),*) => {
            if cfg!(debug_assertions) { dbg!($($v),*) } else { ($($v),*) }
        }
    }
}