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



Sample input:
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:
15
404
41511
4317767

Submission status: Accepted

Solution code:
"""Buggy! the output of practice_k becomes about 1/4."""
from typing import Callable, List, TypeVar

S = TypeVar("S")
F = TypeVar("F")


class LazySegmentTree:
    """Lazy Segment Tree
    References:
        https://github.com/atcoder/ac-library/blob/master/atcoder/lazysegtree.hpp
    """

    __slots__ = [
        "e",
        "op",
        "id",
        "mapping",
        "composition",
        "_n",
        "_log",
        "_size",
        "tree",
        "lazy",
    ]

    def __init__(
        self,
        a: List[S],
        e: S,
        op: Callable[[S, S], S],
        id_: F,
        mapping: Callable[[F, S], S],
        composition: Callable[[F, F], F],
    ) -> None:
        self.e = e
        self.op = op
        self.id = id_
        self.mapping = mapping
        self.composition = composition

        self._n = len(a)
        self._log = (self._n - 1).bit_length()
        self._size = 1 << self._log

        self.tree = [e] * self._size + a + [e] * (self._size - self._n)
        for i in range(self._size - 1, 0, -1):
            self._update(i)

        self.lazy = [id_] * self._size

    def _update(self, k: int) -> None:
        """Update the value of a[k]."""
        self.tree[k] = self.op(self.tree[2 * k], self.tree[2 * k + 1])

    def _apply_all(self, k: int, f: F) -> None:
        self.tree[k] = self.mapping(f, self.tree[k])
        if k < self._size:
            self.lazy[k] = self.composition(f, self.lazy[k])

    def _push(self, k: int) -> None:
        self._apply_all(2 * k, self.lazy[k])
        self._apply_all(2 * k + 1, self.lazy[k])
        self.lazy[k] = self.id

    def set(self, k: int, x: S) -> None:
        """Assign x to a[k] in O(log n)."""
        assert 0 <= k < self._n

        k += self._size
        for i in range(self._log, 0, -1):
            self._push(k >> i)
        self.tree[k] = x
        for i in range(1, self._log + 1):
            self._update(k >> i)
        while k:
            k >>= 1
            self._update(k)

    def get(self, k: int) -> S:
        """Return a[k] in O(1)."""
        assert 0 <= k < self._n

        k += self._size
        for i in range(self._log, 0, -1):
            self._push(k >> i)
        return self.tree[k]

    def prod(self, l: int, r: int) -> S:
        """Return op(a[l], ..., a[r - 1]). Return e, if l == r.
        Complexity: O(log n)
        """
        assert 0 <= l <= r <= self._n

        if l == r:
            return self.e

        l += self._size
        r += self._size
        for i in range(self._log, 0, -1):
            if ((l >> i) << i) != l:
                self._push(l >> i)
            if ((r >> i) << i) != r:
                self._push(r >> i)

        sml, smr = self.e, self.e
        while l < r:
            if l & 1:
                sml = self.op(sml, self.tree[l])
                l += 1
            if r & 1:
                r -= 1
                smr = self.op(self.tree[r], smr)
            l >>= 1
            r >>= 1
        return self.op(sml, smr)

    def prod_all(self) -> S:
        """Return op(a[0], ..., a[n - 1]. Return e if n == 0.
        Complexity: O(1)
        """
        return self.tree[1]

    def apply(self, k: int, f: F) -> None:
        """Apply a[p] = op_st(a[p], x) in O(log n)."""
        assert 0 <= k < self._n

        k += self._size
        for i in range(self._log, 0, -1):
            self._push(k >> i)
        self.tree[k] = self.mapping(f, self.tree[k])
        for i in range(1, self._log + 1):
            self._update(k >> i)

    def apply_range(self, l: int, r: int, f: F) -> None:
        """Apply a[i] = op_st(a[i], x) for all i = l..r-1 in O(log n)."""
        assert 0 <= l <= r <= self._n

        if l == r:
            return

        l += self._size
        r += self._size
        for i in range(self._log, 0, -1):
            if ((l >> i) << i) != l:
                self._push(l >> i)
            if ((r >> i) << i) != r:
                self._push((r - 1) >> i)

        l_tmp, r_tmp = l, r
        while l < r:
            if l & 1:
                self._apply_all(l, f)
                l += 1
            if r & 1:
                r -= 1
                self._apply_all(r, f)
            l >>= 1
            r >>= 1
        l, r = l_tmp, r_tmp

        for i in range(1, self._log + 1):
            if ((l >> i) << i) != l:
                self._update(l >> i)
            if ((r >> i) << i) != r:
                self._update((r - 1) >> i)

    def max_right(self, l: int, g: Callable[[S], bool]) -> int:
        """
        Return an index r satisfying both:
            1. r = l or f(op(a[l], a[l + 1], ..., a[r - 1])) = true
            2. r = n or f(op(a[l], a[l + 1], ..., a[r])) = false.

        If f is monotone, this is the maximum r satisfying:
            f(op(a[l], a[l + 1], ..., a[r - 1])) = true.

        Complexity: O(log n)
        """
        assert 0 <= l <= self._n
        assert g(self.e)

        if l == self._n:
            return self._n

        l += self._size
        for i in range(self._log, 0, -1):
            self._push(l >> i)
        sm = self.e

        while True:
            while not l & 1:
                l >>= 1

            if not g(self.op(sm, self.tree[l])):
                while l < self._size:
                    l *= 2
                    if g(self.op(sm, self.tree[l])):
                        sm = self.op(sm, self.tree[l])
                        l += 1
                return l - self._size

            sm = self.op(sm, self.tree[l])
            l += 1

            if (l & -l) == l:
                break

        return self._n

    def min_left(self, r: int, g: Callable[[S], bool]) -> int:
        """
        Return an index l satisfying both:
            1. l = r or f(op(a[l], a[l + 1], ..., a[r - 1])) = true
            2. l = 0 or f(op(a[l - 1], a[l + 1], ..., a[r - 1])) = false.
        If f is monotone, this is the minimum l satisfying:
            f(op(a[l], a[l + 1], ..., a[r - 1])) = true.

        Complexity: O(log n)
        """
        assert 0 <= r <= self._n
        assert g(self.e)

        if not r:
            return 0

        r += self._size
        for i in range(self._log, 0, -1):
            self._push((r - 1) >> i)
        sm = self.e

        while True:
            r -= 1
            while r > 1 and r & 1:
                r >>= 1

            if not g(self.op(self.tree[r], sm)):
                while r < self._size:
                    r = 2 * r + 1
                    if g(self.op(self.tree[r], sm)):
                        sm = self.op(self.tree[r], sm)
                        r -= 1
                return r + 1 - self._size

            sm = self.op(self.tree[r], sm)

            if (r & -r) == r:
                break

        return 0


def practice2_k():
    # https://atcoder.jp/contests/practice2/tasks/practice2_k
    import sys

    readline = sys.stdin.buffer.readline

    N, Q = map(int, readline().split())
    A = [(i << 32) + 1 for i in map(int, readline().split())]

    MOD = 998244353
    e = 0
    id_ = 1 << 32

    def op(s, t):
        sv, sn = s >> 32, s % (1 << 32)
        tv, tn = t >> 32, t % (1 << 32)
        return (((sv + tv) % MOD) << 32) + sn + tn

    def mapping(f, a):
        fb, fc = f >> 32, f % (1 << 32)
        av, an = a >> 32, a % (1 << 32)
        return (((fb * av + fc * an) % MOD) << 32) + an

    def composition(f, g):
        fb, fc = f >> 32, f % (1 << 32)
        gb, gc = g >> 32, g % (1 << 32)
        return ((fb * gb % MOD) << 32) + ((gc * fb + fc) % MOD)

    tree = LazySegmentTree(A, e, op, id_, mapping, composition)
    res = []
    for _ in range(Q):
        com, l, r, *bc = map(int, readline().split())
        if com:
            res.append(tree.prod(l, r) >> 32)
        else:
            B, C = bc
            tree.apply_range(l, r, (B << 32) + C)
    print("\n".join(map(str, res)))


if __name__ == "__main__":
    practice2_k()

Submission status: Accepted

Solution code:
class LazySegmentTree:


    __slots__ = ("n", "original_size", "log", "data", "lazy", "me", "oe", "fmm", "fmo", "foo")


    def __init__(self, length_or_list, monoid_identity, operator_identity, func_monoid_monoid, func_monoid_operator, func_operator_operator):
        self.me = monoid_identity
        self.oe = operator_identity
        self.fmm = func_monoid_monoid
        self.fmo = func_monoid_operator
        self.foo = func_operator_operator

        if isinstance(length_or_list, int):
            self.original_size = length_or_list
            self.log = (self.original_size - 1).bit_length()
            self.n = 1 << self.log
            self.data = [self.me] * (self.n * 2)
            self.lazy = [self.oe] * (self.n * 2)
        elif isinstance(length_or_list, list):
            self.original_size = len(length_or_list)
            self.log = (self.original_size - 1).bit_length()
            self.n = 1 << self.log
            self.data = [self.me] * self.n + length_or_list + [self.me] * (self.n - self.original_size)
            for i in range(self.n-1, 0, -1):
                self.data[i] = self.fmm(self.data[2*i], self.data[2*i+1])
            self.lazy = [self.oe] * (self.n * 2)
        else:
            raise TypeError(f"The argument 'length_or_list' must be an integer or a list, not {type(length_or_list).__name__}")
            

    def replace(self, index, value):
        if index < -self.original_size or self.original_size <= index:
            raise IndexError("LazySegmentTree index out of range")
        if index < 0:
            index += self.original_size

        index += self.n

        # propagation
        for shift in range(self.log, 0, -1):
            i = index >> shift
            self.lazy[2*i]   = self.foo(self.lazy[2*i],   self.lazy[i])
            self.lazy[2*i+1] = self.foo(self.lazy[2*i+1], self.lazy[i])
            self.data[i] = self.fmo(self.data[i], self.lazy[i])
            self.lazy[i] = self.oe

        # update
        self.data[index] = value
        self.lazy[index] = self.oe

        # recalculation
        i = index
        while i > 1:
            i //= 2
            self.data[i] = self.fmm( self.fmo(self.data[2*i], self.lazy[2*i]), self.fmo(self.data[2*i+1], self.lazy[2*i+1]) )
            self.lazy[i] = self.oe


    def __getitem__(self, index):
        if index < -self.original_size or self.original_size <= index:
            raise IndexError("LazySegmentTree index out of range")
        if index < 0:
            index += self.original_size

        index += self.n

        # propagation
        for shift in range(self.log, 0, -1):
            i = index >> shift
            self.lazy[2*i]   = self.foo(self.lazy[2*i],   self.lazy[i])
            self.lazy[2*i+1] = self.foo(self.lazy[2*i+1], self.lazy[i])
            self.data[i] = self.fmo(self.data[i], self.lazy[i])
            self.lazy[i] = self.oe

        # self.data[index] = self.fmo(self.data[index], self.lazy[index])
        # self.lazy[index] = self.oe

        return self.data[index]


    def effect(self, l, r, operator):
        if l < 0:
            l += self.original_size
        if r < 0:
            r += self.original_size
        if not 0 <= l <= r <= self.original_size:
            raise IndexError("LazySegmentTree index out of range")

        if l == r:
            return

        l += self.n
        r += self.n

        l0 = l
        r0 = r - 1
        while l0 % 2 == 0:
            l0 //= 2
        while r0 % 2 == 1:
            r0 //= 2
        
        # propagation
        for shift in range(self.log, 0, -1):
            i = l >> shift
            if i << shift != l:
                self.lazy[2*i]   = self.foo(self.lazy[2*i],   self.lazy[i])
                self.lazy[2*i+1] = self.foo(self.lazy[2*i+1], self.lazy[i])
                self.data[i] = self.fmo(self.data[i], self.lazy[i])
                self.lazy[i] = self.oe
            i = r >> shift
            if i << shift != r:
                self.lazy[2*i]   = self.foo(self.lazy[2*i],   self.lazy[i])
                self.lazy[2*i+1] = self.foo(self.lazy[2*i+1], self.lazy[i])
                self.data[i] = self.fmo(self.data[i], self.lazy[i])
                self.lazy[i] = self.oe

        # effect
        while l < r:
            if l % 2:
                self.lazy[l] = self.foo(self.lazy[l], operator)
                l += 1
            if r % 2:
                r -= 1
                self.lazy[r] = self.foo(self.lazy[r], operator)
            l //= 2
            r //= 2

        # recalculation
        i = l0
        while i > 1:
            i //= 2
            self.data[i] = self.fmm( self.fmo(self.data[2*i], self.lazy[2*i]), self.fmo(self.data[2*i+1], self.lazy[2*i+1]) )
            self.lazy[i] = self.oe
        i = r0
        while i > 1:
            i //= 2
            self.data[i] = self.fmm( self.fmo(self.data[2*i], self.lazy[2*i]), self.fmo(self.data[2*i+1], self.lazy[2*i+1]) )
            self.lazy[i] = self.oe
            
        
    def folded(self, l, r):
        if l < 0:
            l += self.original_size
        if r < 0:
            r += self.original_size
        if not 0 <= l <= r <= self.original_size:
            raise IndexError("LazySegmentTree index out of range")

        if l == r:
            return self.me 

        l += self.n
        r += self.n

        # propagation
        for shift in range(self.log, 0, -1):
            i = l >> shift
            if i << shift != l:
                self.lazy[2*i]   = self.foo(self.lazy[2*i],   self.lazy[i])
                self.lazy[2*i+1] = self.foo(self.lazy[2*i+1], self.lazy[i])
                self.data[i] = self.fmo(self.data[i], self.lazy[i])
                self.lazy[i] = self.oe
            i = r >> shift
            if i << shift != r:
                self.lazy[2*i]   = self.foo(self.lazy[2*i],   self.lazy[i])
                self.lazy[2*i+1] = self.foo(self.lazy[2*i+1], self.lazy[i])
                self.data[i] = self.fmo(self.data[i], self.lazy[i])
                self.lazy[i] = self.oe

        # fold
        left_folded = self.me
        right_folded = self.me
        while l < r:
            if l % 2:
                left_folded = self.fmm(left_folded, self.fmo(self.data[l], self.lazy[l]))
                l += 1
            if r % 2:
                r -= 1
                right_folded = self.fmm(self.fmo(self.data[r], self.lazy[r]), right_folded)
            l //= 2
            r //= 2
        return self.fmm(left_folded, right_folded)

    
    def max_right(self, l, f):
        if l < -self.original_size or self.original_size < l:
            raise IndexError("LazySegmentTree index out of range")
        if l < 0:
            l += self.original_size
        assert f(self.me)

        if l == self.original_size:
            return self.original_size

        l += self.n

        # propagation
        for shift in range(self.log, 0, -1):
            i = l >> shift
            self.lazy[2*i]   = self.foo(self.lazy[2*i],   self.lazy[i])
            self.lazy[2*i+1] = self.foo(self.lazy[2*i+1], self.lazy[i])
            self.data[i] = self.fmo(self.data[i], self.lazy[i])
            self.lazy[i] = self.oe

        left_folded = self.me
        while True:
            while l % 2 == 0:
                l //= 2
            if not f(self.fmm(left_folded, self.data[l])):
                while l < self.n:
                    i = l
                    self.lazy[2*i]   = self.foo(self.lazy[2*i],   self.lazy[i])
                    self.lazy[2*i+1] = self.foo(self.lazy[2*i+1], self.lazy[i])
                    self.data[i] = self.fmo(self.data[i], self.lazy[i])
                    self.lazy[i] = self.oe
                    l *= 2
                    if f(self.fmm(left_folded, self.data[l])):
                        left_folded = self.fmm(left_folded, self.data[l])
                        l += 1
                return l - self.n
            left_folded = self.fmm(left_folded, self.data[l])
            l += 1
            if l == l & -l:
                break
        return self.original_size

    def min_left(self, r, f):
        if r < -self.original_size or self.original_size < r:
            raise IndexError("LazySegmentTree index out of range")
        if r < 0:
            r += self.original_size
        assert f(self.me)

        if r == 0:
            return 0
        
        r += self.n

        # propagation
        for shift in range(self.log, 0, -1):
            i = r >> shift
            self.lazy[2*i]   = self.foo(self.lazy[2*i],   self.lazy[i])
            self.lazy[2*i+1] = self.foo(self.lazy[2*i+1], self.lazy[i])
            self.data[i] = self.fmo(self.data[i], self.lazy[i])
            self.lazy[i] = self.oe

        right_folded = self.me
        while True:
            r -= 1
            while (r > 1 and r % 2):
                r //= 2
            if not f(self.fmm(self.data[r], right_folded)):
                while r < self.n:
                    i = r
                    self.lazy[2*i]   = self.foo(self.lazy[2*i],   self.lazy[i])
                    self.lazy[2*i+1] = self.foo(self.lazy[2*i+1], self.lazy[i])
                    self.data[i] = self.fmo(self.data[i], self.lazy[i])
                    self.lazy[i] = self.oe
                    r = 2 * r + 1
                    if f(self.fmm(self.data[r], right_folded)):
                        right_folded = self.fmm(self.data[r], right_folded)
                        r -= 1
                return r + 1 - self.n
            right_folded = self.fmm(self.data[r], right_folded)
            if r == r & -r:
                break
        return 0


MOD = 998244353
mask32 = (1 << 32) - 1

# K - Range Affine Range Sum
def yosupo():
    # Monoid は (値 << 32) + 要素数 にする。
    # Operator も (b << 32) + c    
    import sys
    input = sys.stdin.buffer.readline

    def fmm(m1, m2):
        m = m1 + m2
        return (((m >> 32) % MOD) << 32) + (m & mask32)
    
    def fmo(m1, o1):
        val = m1 >> 32
        cnt = m1 & mask32
        b = o1 >> 32
        c = o1 & mask32
        return (((b * val + c * cnt) % MOD) << 32) + cnt

    def foo(o1, o2):
        b1 = o1 >> 32
        c1 = o1 & mask32
        b2 = o2 >> 32
        c2 = o2 & mask32
        b = b1 * b2 % MOD
        c = (c1 * b2 + c2) % MOD
        return (b << 32) + c

    N, Q = map(int, input().split())
    monoid_data = [(A << 32) + 1 for A in map(int, input().split())]
    lst = LazySegmentTree(monoid_data, 1, 1 << 32, fmm, fmo, foo)
    ans = []
    for _ in range(Q):
        q, *k = map(int, input().split())
        if q == 0:
            o = (k[2] << 32) + k[3]
            lst.effect(k[0], k[1], o)
        else:
            ans.append(lst.folded(k[0], k[1]) >> 32)
    print('\n'.join(map(str, ans)))

# L - Lazy Segment Tree
def atc2():
    # Monoid:   ((0の数), (1の数), (転倒数))
    # Operator: 反転するか? (1 or 0) 

    import sys
    input = sys.stdin.buffer.readline

    N, Q = map(int, input().split())
    monoid_data = [(0, 1, 0) if A == b'1' else (1, 0, 0) for A in input().split()]

    def fmm(m1, m2):
        return (m1[0] + m2[0], m1[1] + m2[1], m1[2] + m2[2] + m1[1] * m2[0])

    def fmo(m1, o1):
        if o1:
            return (m1[1], m1[0], m1[0] * m1[1] - m1[2])
        else:
            return m1 

    def foo(o1, o2):
        return o1 ^ o2

    lst = LazySegmentTree(monoid_data, (0, 0, 0), 0, fmm, fmo, foo)
    ans = []
    for _ in range(Q):
        T, L, R = map(int, input().split())
        if T == 1:
            lst.effect(L-1, R, 1)
        else:
            ans.append(lst.folded(L-1, R)[2])
    print('\n'.join(map(str, ans)))

if __name__ == "__main__":
    yosupo()