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.

The type of i-th query is represented by T_i.

T_i=1: You are given two integers X_i,V_i. Replace the value of A_{X_i} with V_i.

T_i=2: You are given two integers L_i,R_i. Calculate the maximum value among A_{L_i},A_{L_i+1},\cdots,A_{R_i}.

T_i=3: You are given two integers X_i,V_i. Calculate the minimum j such that X_i \leq j \leq N, V_i \leq A_j. If there is no such j, answer j=N+1 instead.

Constraints

1 \leq N \leq 2 \times 10^5

0 \leq A_i \leq 10^9

1 \leq Q \leq 2 \times 10^5

1 \leq T_i \leq 3

1 \leq X_i \leq N, 0 \leq V_i \leq 10^9 (T_i=1,3)

1 \leq L_i \leq R_i \leq N (T_i=2)

All values in Input are integer.

Input

Input is given from Standard Input in the following format:

N Q
A_1 A_2 \cdots A_N
First query
Second query
\vdots
Q-th query

Each query is given in the following format:

If T_i=1,3,

T_i X_i V_i

If T_i=2,

T_i L_i R_i

Output

For each query with T_i=2, 3, print the answer.

Sample Input 1

5 5
1 2 3 2 1
2 1 5
3 2 3
1 3 1
2 2 4
3 1 3

Sample Output 1

3
3
2
6

First query: Print 3, which is the maximum of (A_1,A_2,A_3,A_4,A_5)=(1,2,3,2,1).

Second query: Since 3>A_2, j=2 does not satisfy the condition．Since 3 \leq A_3, print j=3.

Third query: Replace the value of A_3 with 1. It becomes A=(1,2,1,2,1).

Fourth query: Print 2, which is the maximum of (A_2,A_3,A_4)=(2,1,2).

Fifth query: Since there is no j that satisfies the condition, print j=N+1=6.

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

use std::cmp::max;
use std::marker::PhantomData;

use std::{
    fmt,
    iter::{Product, Sum},
    ops::{
        Add, AddAssign, BitAnd, BitAndAssign, BitOr, BitOrAssign, BitXor, BitXorAssign, Div,
        DivAssign, Mul, MulAssign, Not, Rem, RemAssign, Shl, ShlAssign, Shr, ShrAssign, Sub,
        SubAssign,
    },
};

// Skipped:
//
// - `is_signed_int_t<T>`   (probably won't be used directly in `modint.rs`)
// - `is_unsigned_int_t<T>` (probably won't be used directly in `modint.rs`)
// - `to_unsigned_t<T>`     (not used in `fenwicktree.rs`)

/// Corresponds to `std::is_integral` in C++.
// We will remove unnecessary bounds later.
//
// Maybe we should rename this to `PrimitiveInteger` or something, as it probably won't be used in the
// same way as the original ACL.
pub trait Integral:
    'static
    + Send
    + Sync
    + Copy
    + Ord
    + Not<Output = Self>
    + Add<Output = Self>
    + Sub<Output = Self>
    + Mul<Output = Self>
    + Div<Output = Self>
    + Rem<Output = Self>
    + AddAssign
    + SubAssign
    + MulAssign
    + DivAssign
    + RemAssign
    + Sum
    + Product
    + BitOr<Output = Self>
    + BitAnd<Output = Self>
    + BitXor<Output = Self>
    + BitOrAssign
    + BitAndAssign
    + BitXorAssign
    + Shl<Output = Self>
    + Shr<Output = Self>
    + ShlAssign
    + ShrAssign
    + fmt::Display
    + fmt::Debug
    + fmt::Binary
    + fmt::Octal
{
    fn zero() -> Self;
    fn one() -> Self;
    fn min_value() -> Self;
    fn max_value() -> Self;
}

macro_rules! impl_integral {
    ($($ty:ty),*) => {
        $(
            impl Integral for $ty {
                #[inline]
                fn zero() -> Self {
                    0
                }

                #[inline]
                fn one() -> Self {
                    1
                }

                #[inline]
                fn min_value() -> Self {
                    Self::min_value()
                }

                #[inline]
                fn max_value() -> Self {
                    Self::max_value()
                }
            }
        )*
    };
}

impl_integral!(i8, i16, i32, i64, i128, isize, u8, u16, u32, u64, u128, usize);


#[allow(dead_code)]
pub(crate) fn ceil_pow2(n: u32) -> u32 {
    32 - n.saturating_sub(1).leading_zeros()
}


// TODO Should I split monoid-related traits to another module?
pub trait Monoid {
    type S: Copy;
    fn identity() -> Self::S;
    fn binary_operation(a: Self::S, b: Self::S) -> Self::S;
}
pub struct Max<S>(PhantomData<fn() -> S>);

// TODO We should not restrict to integral
impl<S> Monoid for Max<S>
where
    S: Integral,
{
    type S = S;

    fn identity() -> Self::S {
        S::min_value()
    }

    fn binary_operation(a: Self::S, b: Self::S) -> Self::S {
        max(a, b)
    }
}

impl<M: Monoid> Segtree<M> {
    pub fn new(n: usize) -> Segtree<M> {
        vec![M::identity(); n].into()
    }
}
impl<M: Monoid> From<Vec<M::S>> for Segtree<M> {
    fn from(v: Vec<M::S>) -> Self {
        let n = v.len();
        let log = ceil_pow2(n as u32) as usize;
        let size = 1 << log;
        let mut d = vec![M::identity(); 2 * size];
        d[size..(size + n)].clone_from_slice(&v);
        let mut ret = Segtree { n, size, log, d };
        for i in (1..size).rev() {
            ret.update(i);
        }
        ret
    }
}
impl<M: Monoid> Segtree<M> {
    pub fn set(&mut self, mut p: usize, x: M::S) {
        assert!(p < self.n);
        p += self.size;
        self.d[p] = x;
        for i in 1..=self.log {
            self.update(p >> i);
        }
    }

    pub fn get(&self, p: usize) -> M::S {
        assert!(p < self.n);
        self.d[p + self.size]
    }

    pub fn prod(&self, mut l: usize, mut r: usize) -> M::S {
        assert!(l <= r && r <= self.n);
        let mut sml = M::identity();
        let mut smr = M::identity();
        l += self.size;
        r += self.size;

        while l < r {
            if l & 1 != 0 {
                sml = M::binary_operation(sml, self.d[l]);
                l += 1;
            }
            if r & 1 != 0 {
                r -= 1;
                smr = M::binary_operation(self.d[r], smr);
            }
            l >>= 1;
            r >>= 1;
        }

        M::binary_operation(sml, smr)
    }

    pub fn all_prod(&self) -> M::S {
        self.d[1]
    }

    pub fn max_right<F>(&self, mut l: usize, f: F) -> usize
    where
        F: Fn(M::S) -> bool,
    {
        assert!(l <= self.n);
        assert!(f(M::identity()));
        if l == self.n {
            return self.n;
        }
        l += self.size;
        let mut sm = M::identity();
        while {
            // do
            while l % 2 == 0 {
                l >>= 1;
            }
            if !f(M::binary_operation(sm, self.d[l])) {
                while l < self.size {
                    l *= 2;
                    let res = M::binary_operation(sm, self.d[l]);
                    if f(res) {
                        sm = res;
                        l += 1;
                    }
                }
                return l - self.size;
            }
            sm = M::binary_operation(sm, self.d[l]);
            l += 1;
            // while
            {
                let l = l as isize;
                (l & -l) != l
            }
        } {}
        self.n
    }

    pub fn min_left<F>(&self, mut r: usize, f: F) -> usize
    where
        F: Fn(M::S) -> bool,
    {
        assert!(r <= self.n);
        assert!(f(M::identity()));
        if r == 0 {
            return 0;
        }
        r += self.size;
        let mut sm = M::identity();
        while {
            // do
            r -= 1;
            while r > 1 && r % 2 == 1 {
                r >>= 1;
            }
            if !f(M::binary_operation(self.d[r], sm)) {
                while r < self.size {
                    r = 2 * r + 1;
                    let res = M::binary_operation(self.d[r], sm);
                    if f(res) {
                        sm = res;
                        r -= 1;
                    }
                }
                return r + 1 - self.size;
            }
            sm = M::binary_operation(self.d[r], sm);
            // while
            {
                let r = r as isize;
                (r & -r) != r
            }
        } {}
        0
    }

    fn update(&mut self, k: usize) {
        self.d[k] = M::binary_operation(self.d[2 * k], self.d[2 * k + 1]);
    }
}

// Maybe we can use this someday
// ```
// for i in 0..=self.log {
//     for j in 0..1 << i {
//         print!("{}\t", self.d[(1 << i) + j]);
//     }
//     println!();
// }
// ```

#[derive(Default)]
pub struct Segtree<M>
where
    M: Monoid,
{
    // variable name is _n in original library
    n: usize,
    size: usize,
    log: usize,
    d: Vec<M::S>,
}

use std::io::Read;

fn main() {
    let mut buf = String::new();
    std::io::stdin().read_to_string(&mut buf).unwrap();
    let mut input = buf.split_whitespace();

    let n: usize = input.next().unwrap().parse().unwrap();
    let q: usize = input.next().unwrap().parse().unwrap();
    let mut segtree = Segtree::<Max<i32>>::new(n+1);
    for i in 1..=n {
        segtree.set(i, input.next().unwrap().parse().unwrap());
    }
    for _ in 0..q {
        match input.next().unwrap().parse().unwrap() {
            1 => {
                let x = input.next().unwrap().parse().unwrap();
                let v = input.next().unwrap().parse().unwrap();
                segtree.set(x, v);
            }
            2 => {
                let l = input.next().unwrap().parse().unwrap();
                let r: usize = input.next().unwrap().parse().unwrap();
                println!("{}", segtree.prod(l, r+1));
            }
            3 => {
                let x = input.next().unwrap().parse().unwrap();
                let v = input.next().unwrap().parse().unwrap();
                println!("{}", segtree.max_right(x, |a| a < v))
            }
            _ => {}
        }
    }
}