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 (Python):
import sys
input = sys.stdin.readline
N, Q = map(int, input().split())
a = list(map(int, input().split()))

class SegTree:
  def segfunc(self, x, y):
    return max(x, y)
  def __init__(self, n, ide_ele, init_val):
    
    self.ide_ele = ide_ele
    
    self.num = 2 ** (n - 1).bit_length()
    self.seg = [self.ide_ele] * 2 * self.num
    
    for i in range(n):
      self.seg[i + self.num - 1] = init_val[i]    
    
    for i in range(self.num - 2, -1, -1) :
      self.seg[i] = self.segfunc(self.seg[2 * i + 1], self.seg[2 * i + 2]) 
  def update(self, k, x):
    k += self.num - 1
    self.seg[k] = x
    while k + 1:
      k = (k - 1) // 2
      self.seg[k] = self.segfunc(self.seg[k * 2 + 1], self.seg[k * 2 + 2]) 
  def query(self, p, q):
    if q <= p:
      return self.ide_ele
    p += self.num - 1
    q += self.num - 2
    res = self.ide_ele
    while q - p > 1:
      if p & 1 == 0:
        res = self.segfunc(res, self.seg[p])
      if q & 1 == 1:
        res = self.segfunc(res, self.seg[q])
        q -= 1
      p = p // 2
      q = (q - 1) // 2
    if p == q:
      res = self.segfunc(res, self.seg[p])
    else:
      res = self.segfunc(self.segfunc(res, self.seg[p]), self.seg[q])
    return res

seg = SegTree(N + 1, -1, [-1] + a)
for _ in range(Q):
  t, x, v = map(int, input().split())
  if t == 1: seg.update(x, v)
  if t == 2: print(seg.query(x, v + 1))
  if t == 3:
    ok = N + 1
    ng = x - 1
    while ok - ng > 1:
      m = (ok + ng) // 2
      if seg.query(x, m + 1) >= v: ok = m
      else: ng = m
    print(ok)

Solution (Python):
def segfunc(x,y):return max(x,y)
ide_ele=0
class segtree():
  def __init__(self,init_val,segfunc=segfunc,ide_ele=ide_ele):
    n=len(init_val)
    self.segfunc=segfunc
    self.ide_ele=ide_ele
    self.num=1<<(n-1).bit_length()
    self.tree=[ide_ele]*2*self.num
    for i in range(n):
      self.tree[self.num+i]=init_val[i]
    for i in range(self.num-1,0,-1):
      self.tree[i]=self.segfunc(self.tree[2*i], self.tree[2*i+1])
  def update(self,k,x):
    k+=self.num
    self.tree[k]=x
    while k>1:
      self.tree[k>>1]=self.segfunc(self.tree[k],self.tree[k^1])
      k>>=1
  def query(self,l,r):
    res=self.ide_ele
    l+=self.num
    r+=self.num+1
    while l<r:
      if l&1:
        res=self.segfunc(res,self.tree[l])
        l+=1
      if r&1:
        res=self.segfunc(res,self.tree[r-1])
      l>>=1
      r>>=1
    return res
def isok(mid,key):
  return st.query(x-1,mid-1)>=key
def b_search(x,key):
  ng=x-1
  ok=n+1
  while abs(ng-ok)>1:
    mid=(ng+ok)//2
    if isok(mid,key):ok=mid
    else:ng=mid
  return ok
n,q=map(int,input().split())
a=list(map(int,input().split()))
st=segtree(a)
ans=[]
for _ in range(q):
  c,x,y=map(int,input().split())
  if c==1:st.update(x-1,y)
  elif c==2:ans+=[st.query(x-1,y-1)]
  else:ans+=[b_search(x,y)]
print(*ans,sep='\n')

Solution (Python):
class SegTree:
    """ define what you want to do with 0 index, ex) size = tree_size, func = min or max, sta = default_value """
    
    def __init__(self,size,func,sta):
        self.n = size
        self.size = 1 << size.bit_length()
        self.func = func
        self.sta = sta
        self.tree = [sta]*(2*self.size)

    def build(self, list):
        """ set list and update tree"""
        for i,x in enumerate(list,self.size):
            self.tree[i] = x

        for i in range(self.size-1,0,-1):
            self.tree[i] = self.func(self.tree[i<<1],self.tree[i<<1 | 1])

    def set(self,i,x):
        i += self.size
        self.tree[i] = x
        while i > 1:
            i >>= 1
            self.tree[i] = self.func(self.tree[i<<1],self.tree[i<<1 | 1])

    
    def get(self,l,r):
        """ take the value of [l r) with func (min or max)"""
        l += self.size
        r += self.size
        res = self.sta

        while l < r:
            if l & 1:
                res = self.func(self.tree[l],res)
                l += 1
            if r & 1:
                res = self.func(self.tree[r-1],res)
            l >>= 1
            r >>= 1
        return res

    def max_right(self, l, x):
        """[l,r) が ok であるような最大の r を返す"""
        if l == self.n:
            return l
        l += self.size
        res = self.sta
        check = True
        while check or (l & -l) != l:
            check = False
            while l%2 == 0:
                l >>= 1
            if not self.func(res,self.tree[l]) < x:
                while l < self.size:
                    l <<= 1
                    if self.func(res,self.tree[l]) < x:
                        res = self.func(res,self.tree[l])
                        l += 1
                return l - self.size
            res = self.func(res,self.tree[l])
            l += 1
        return self.n

    def min_left(self, r, x):
        """(r が ok であるような最小の r を返す"""
        if r == 0:
            return 0
        r += self.size
        res = self.sta
        check = True
        while check and (r & -r) != r:
            check = False
            r -= 1
            while (r > 1 and r%2):
                r >>= 1
            if not self.func(res, self.tree[r]) < x:
                while r < self.size:
                    r = 2*r + 1
                    if self.func(res, self.tree[r]) < x:
                        res = self.func(res, self.tree[r])
                        r -= 1
                return r + 1 - self.size
            res = self.func(self.tree[r],res)
        return 0

n,q = map(int,input().split())
a = list(map(int,input().split()))

seg = SegTree(n,max,-1)
seg.build(a)
for _ in range(q):
    t,x,v = map(int,input().split())
    if t == 1:
        seg.set(x-1,v)
    elif t == 2:
        print(seg.get(x-1,v))
    else:
        print(seg.max_right(x-1,v)+1)

Solution (Python):
class segtree:
    
    sta = -1
    func = max

    def __init__(self,n):
        self.n = n
        self.size = 1 << n.bit_length()
        self.tree = [self.sta]*(2*self.size)

    def build(self, list):
        for i,x in enumerate(list,self.size):
            self.tree[i] = x

        for i in range(self.size-1,0,-1):
            self.tree[i] = self.func(self.tree[i<<1],self.tree[i<<1 | 1])

    def set(self,i,x):
        i += self.size
        self.tree[i] = x
        while i > 1:
            i >>= 1
            self.tree[i] = self.func(self.tree[i<<1],self.tree[i<<1 | 1])

    
    def get(self,l,r):
        l += self.size
        r += self.size
        res = self.sta

        while l < r:
            if l & 1:
                res = self.func(self.tree[l],res)
                l += 1
            if r & 1:
                res = self.func(self.tree[r-1],res)
            l >>= 1
            r >>= 1
        return res

    def max_right(self, l, x):
        if l == self.n:
            return l
        l += self.size
        res = self.sta
        check = True
        while check or (l & -l) != l:
            check = False
            while l%2 == 0:
                l >>= 1
            if not self.func(res,self.tree[l]) < x:
                while l < self.size:
                    l <<= 1
                    if self.func(res,self.tree[l]) < x:
                        res = self.func(res,self.tree[l])
                        l += 1
                return l - self.size
            res = self.func(res,self.tree[l])
            l += 1
        return self.n

n,q = map(int,input().split())
a = list(map(int,input().split()))

seg = segtree(n)
seg.build(a)
for _ in range(q):
    t,x,v = map(int,input().split())
    if t == 1:
        seg.set(x-1,v)
    elif t == 2:
        print(seg.get(x-1,v))
    else:
        print(seg.max_right(x-1,v)+1)

Solution (Python):
class segtree:
    
    sta = -1
    func = max

    def __init__(self,n):
        self.n = n
        self.tree = [self.sta]*(2*n)

    def build(self, list):
        for i,x in enumerate(list,self.n):
            self.tree[i] = x

        for i in range(self.n-1,0,-1):
            self.tree[i] = self.func(self.tree[i<<1],self.tree[i<<1 | 1])

    def set(self,i,x):
        i += self.n
        self.tree[i] = x
        while i > 1:
            i >>= 1
            self.tree[i] = self.func(self.tree[i<<1],self.tree[i<<1 | 1])

    
    def get(self,l,r):
        l += self.n
        r += self.n
        res = self.sta

        while l < r:
            if l & 1:
                res = self.func(self.tree[l],res)
                l += 1
            if r & 1:
                res = self.func(self.tree[r-1],res)
            l >>= 1
            r >>= 1
        return res

    def max_right(self, l, x):
        """[l,r) が ok であるような最大の r を返す"""
        N = self.n
    
        def is_ok(v):
            return v < x
    
        def get_max(a,b):
            if a is None:
                return b
            if b is None:
                return a
            return max(a,b)
        v = None
        i, node_l, node_w = l + N, l, 1
        while True:
            if not (i & 1):
                i >>= 1
                node_w <<= 1
                continue
            if node_l + node_w > N:
                break
            v1 = get_max(v, self.tree[i])
            if is_ok(v1):
                v, i = v1, (i + 1) >> 1
                node_l, node_w = node_l + node_w, node_w << 1
            else:
                break
        
        i, node_w = i << 1, node_w >> 1
        while node_w:
            if node_l + node_w > N:
                i, node_w = i << 1, node_w >> 1
                continue
            v1 = get_max(v, self.tree[i])
            if is_ok(v1):
                v, i = v1, (i + 1) << 1
                node_l, node_w = node_l + node_w, node_w >> 1
            else:
                i, node_w = i << 1, node_w >> 1
        i >>= 1
        return i - N


n,q = map(int,input().split())
a = list(map(int,input().split()))

seg = segtree(n)
seg.build(a)
for _ in range(q):
    t,x,v = map(int,input().split())
    if t == 1:
        seg.set(x-1,v)
    elif t == 2:
        print(seg.get(x-1,v))
    else:
        print(seg.max_right(x-1,v)+1)

Solution (Python):
class segtree:
    
    sta = -1
    func = max

    def __init__(self,n):
        self.n = n
        self.tree = [self.sta]*(2*n)

    def build(self, list):
        for i,x in enumerate(list,self.n):
            self.tree[i] = x

        for i in range(self.n-1,0,-1):
            self.tree[i] = self.func(self.tree[i<<1],self.tree[i<<1 | 1])

    def set(self,i,x):
        i += self.n
        self.tree[i] = x
        while i > 1:
            i >>= 1
            self.tree[i] = self.func(self.tree[i<<1],self.tree[i<<1 | 1])

    
    def get(self,l,r):
        l += self.n
        r += self.n
        res = self.sta

        while l < r:
            if l & 1:
                res = self.func(self.tree[l],res)
                l += 1
            if r & 1:
                res = self.func(self.tree[r-1],res)
            l >>= 1
            r >>= 1
        return res

    def max_right(self, l, x):
        """[l,r) が ok であるような最大の r を返す"""
        N = self.n
    
        def is_ok(v):
            return v < x
    
        def get_max(a,b):
            if a is None:
                return b
            if b is None:
                return a
            return max(a,b)
        v = None
        i, node_l, node_w = l + N, l, 1
        while True:
            if not (i & 1):
                i >>= 1
                node_w <<= 1
                continue
            if node_l + node_w > N:
                break
            v1 = get_max(v, self.tree[i])
            if is_ok(v1):
                v, i = v1, (i + 1) >> 1
                node_l, node_w = node_l + node_w, node_w << 1
            else:
                break
        
        i, node_w = i << 1, node_w >> 1
        while node_w:
            if node_l + node_w > N:
                i, node_w = i << 1, node_w >> 1
                continue
            v1 = get_max(v, self.tree[i])
            if is_ok(v1):
                v, i = v1, (i + 1) << 1
                node_l, node_w = node_l + node_w, node_w >> 1
            else:
                i, node_w = i << 1, node_w >> 1
        i >>= 1
        return i - N


n,q = map(int,input().split())
a = list(map(int,input().split()))

seg = segtree(n)
seg.build(a)
for _ in range(q):
    t,x,v = map(int,input().split())
    if t == 1:
        seg.set(x-1,v)
    elif t == 2:
        print(seg.get(x-1,v))
    else:
        print(seg.max_right(x-1,v)+1)

Solution (Python):
import sys

sys.setrecursionlimit(10**7)
def I(): return int(sys.stdin.readline().rstrip())
def MI(): return map(int,sys.stdin.readline().rstrip().split())
def LI(): return list(map(int,sys.stdin.readline().rstrip().split()))  


class SegTree:  
    def __init__(self,N,seg_func,unit):
        self.N = 1 << (N-1).bit_length()
        self.func = seg_func
        self.unit = unit
        self.tree = [self.unit]*(2*self.N)

    def build(self,init_value):  
        for i in range(len(init_value)):
            self.tree[i+self.N] = init_value[i]
        for i in range(self.N-1,0,-1):
            self.tree[i] = self.func(self.tree[i << 1],self.tree[i << 1 | 1])

    def set_val(self,i,x):  
        i += self.N
        self.tree[i] = x
        i >>= 1
        while i:
            self.tree[i] = self.func(self.tree[i << 1],self.tree[i << 1 | 1])
            i >>= 1

    def fold(self,L,R):  
        L += self.N
        R += self.N
        vL = self.unit
        vR = self.unit
        while L < R:
            if L & 1:
                vL = self.func(vL,self.tree[L])
                L += 1
            if R & 1:
                R -= 1
                vR = self.func(self.tree[R],vR)
            L >>= 1
            R >>= 1
        return self.func(vL,vR)

    def max_right(self,l,z):  
        if l >= self.N:
            return self.N
        l += self.N
        s = self.unit
        while True:
            while l % 2 == 0:
                l >>= 1
            if not z(self.func(s,self.tree[l])):
                while l < self.N:
                    l *= 2
                    if z(self.func(s, self.tree[l])):
                        s = self.func(s, self.tree[l])
                        l += 1
                return l - self.N
            s = self.func(s, self.tree[l])
            l += 1
            if l & -l == l:
                break
        return self.N

    def min_left(self,r,z):  
        if r <= 0:
            return 0
        r += self.N
        s = self.unit
        while True:
            r -= 1
            while r > 1 and r % 2:
                r >>= 1
            if not z(self.func(self.tree[r], s)):
                while r < self.N:
                    r = r*2+1
                    if z(self.func(self.tree[r], s)):
                        s = self.func(self.tree[r], s)
                        r -= 1
                return r + 1 - self.N
            s = self.func(self.tree[r], s)
            if r & -r == r:
                break
        return 0


def seg_func(x,y):
    return max(x,y)

unit = 0


N,Q = MI()
A = LI()
ST = SegTree(N,seg_func,unit)

ST.build(A)

for _ in range(Q):
    t,a,b = MI()
    if t == 1:
        ST.set_val(a-1,b)
    elif t == 2:
        print(ST.fold(a-1,b))
    else:
        def z(x):
            return x < b
        print(min(ST.max_right(a-1,z),N)+1)

Solution (Python):

class SegTree():
    def __init__(self, a, func, idem):
        
        self.n = 1 << len(a).bit_length()
        self.size = len(a)
        self.func = func
        self.idem = idem 
        self.node = [idem] * (2 * self.n - 1)
        for i in range(len(a) + self.n - 2, -1, -1):
            if i >= self.n - 1:
                self.node[i] = a[i - self.n + 1]
            else:
                self.node[i] = self.func(self.node[2 * i + 1], self.node[2 * i + 2])
            
    def get(self, x):
        return self.node[x + self.n - 1]

    def set(self, x, val):
        x += self.n - 1
        self.node[x] = val
        while x > 0:
            x = (x - 1) >> 1
            self.node[x] = self.func(self.node[2*x+1], self.node[2*x+2])
        return
    
    def prod(self, l, r):
        L, R = l + self.n, r + self.n
        s_l, s_r = self.idem, self.idem
        while L < R:
            if R & 1:
                R -= 1
                s_r = self.func(self.node[R-1], s_r)
            if L & 1:
                s_l = s_l = self.func(s_l, self.node[L-1])
                L += 1
            L >>= 1
            R >>= 1
        return self.func(s_l, s_r)

    def max_right(self, l, val):
        if l == self.size:
            return self.size
        L = l + self.n
        s = self.idem
        while 1:
            while L % 2 == 0:
                L >>= 1
            if not val(self.func(s, self.node[L - 1])):
                while L < self.n:
                    L <<= 1
                    if val(self.func(s, self.node[L-1])):
                        s = self.func(s, self.node[L-1])
                        L += 1
                return L - self.n
            s = self.func(s, self.node[L - 1])
            L += 1
            if (L & -L) == L:
                return self.size

    def min_left(self, r, val):
        if r == 0:
            return 0
        R = r + self.n
        s = self.idem
        while True:
            R -= 1
            while R > 1 and R % 2:
                R >>= 1
            if not val(self.func(self.node[R - 1], s)):
                while R < self.n:
                    R = (R << 1) + 1
                    if val(self.func(self.node[R - 1], s)):
                        s = self.func(self.node[R - 1], s)
                        R -= 1
                return R + 1 - self.n
            s = self.func(self.node[R - 1], s)
            if (R & -R) == R:
                return 0
            
import sys
input = sys.stdin.readline

N, Q = map(int, input().split())
A = list(map(int, input().split()))
st = SegTree(A, max, 0)
for _ in range(Q):
    t, x, y = map(int, input().split())
    if t == 1:
        st.set(x - 1, y)
    elif t == 2:
        print(st.prod(x - 1, y))
    else:
        print(st.max_right(x - 1, lambda z : z < y) + 1)

Solution (Python):
import sys
import numpy as np
import numba
from numba import njit, b1, i4, i8, f8

read = sys.stdin.buffer.read
readline = sys.stdin.buffer.readline
readlines = sys.stdin.buffer.readlines

@njit((numba.types.optional(i8), ) * 2, cache=True)
def seg_f(x, y):
    if x is None:
        return y
    if y is None:
        return x
    return max(x, y)


@njit((i8[:], ), cache=True)
def build(raw_data):
    N = len(raw_data)
    seg = np.empty(N + N, np.int64)
    seg[N:] = raw_data
    for i in range(N - 1, 0, -1):
        seg[i] = seg_f(seg[i << 1], seg[i << 1 | 1])
    return seg


@njit((i8[:], i8, i8), cache=True)
def set_val(seg, i, x):
    N = len(seg) // 2
    i += N
    seg[i] = x
    while i > 1:
        i >>= 1
        seg[i] = seg_f(seg[i << 1], seg[i << 1 | 1])


@njit((i8[:], i8, i8), cache=True)
def fold(seg, l, r):
    vl = vr = None
    N = len(seg) // 2
    l, r = l + N, r + N
    while l < r:
        if l & 1:
            vl = seg_f(vl, seg[l])
            l += 1
        if r & 1:
            r -= 1
            vr = seg_f(seg[r], vr)
        l, r = l >> 1, r >> 1
    return seg_f(vl, vr)


@njit((i8[:], i8, i8), cache=True)
def max_right(seg, l, x):
    """[l,r) が ok であるような最大の r を返す"""
    N = len(seg) // 2

    def is_ok(v):
        return v < x

    v = None
    i, node_l, node_w = l + N, l, 1
    while True:
        if not (i & 1):
            i >>= 1
            node_w <<= 1
            continue
        if node_l + node_w > N:
            break
        v1 = seg_f(v, seg[i])
        if is_ok(v1):
            v, i = v1, (i + 1) >> 1
            node_l, node_w = node_l + node_w, node_w << 1
        else:
            break
    
    i, node_w = i << 1, node_w >> 1
    while node_w:
        if node_l + node_w > N:
            i, node_w = i << 1, node_w >> 1
            continue
        v1 = seg_f(v, seg[i])
        if is_ok(v1):
            v, i = v1, (i + 1) << 1
            node_l, node_w = node_l + node_w, node_w >> 1
        else:
            i, node_w = i << 1, node_w >> 1
    i >>= 1
    return i - N

@njit((i8, i8[:], i8[:]), cache=True)
def main(N, A, query):
    seg = build(A)
    for i in range(0, len(query), 3):
        t, a, b = query[i:i + 3]
        if t == 1:
            set_val(seg, a - 1, b)
        elif t == 2:
            print(fold(seg, a - 1, b))
        elif t == 3:
            print(max_right(seg, a - 1, b) + 1)

N, Q = map(int, readline().split())
A = np.array(readline().split(), np.int64)
query = np.array(read().split(), np.int64)

main(N, A, query)

Solution (Python):
import types

_atcoder_code = """


__version__ = '0.0.1'
"""

atcoder = types.ModuleType('atcoder')
exec(_atcoder_code, atcoder.__dict__)

_atcoder__bit_code = """
def _ceil_pow2(n: int) -> int:
    x = 0
    while (1 << x) < n:
        x += 1

    return x


def _bsf(n: int) -> int:
    x = 0
    while n % 2 == 0:
        x += 1
        n //= 2

    return x
"""

atcoder._bit = types.ModuleType('atcoder._bit')
exec(_atcoder__bit_code, atcoder._bit.__dict__)


_atcoder_segtree_code = """
import typing




class SegTree:
    def __init__(self,
                 op: typing.Callable[[typing.Any, typing.Any], typing.Any],
                 e: typing.Any,
                 v: typing.Union[int, typing.List[typing.Any]]) -> None:
        self._op = op
        self._e = e

        if isinstance(v, int):
            v = [e] * v

        self._n = len(v)
        self._log = atcoder._bit._ceil_pow2(self._n)
        self._size = 1 << self._log
        self._d = [e] * (2 * self._size)

        for i in range(self._n):
            self._d[self._size + i] = v[i]
        for i in range(self._size - 1, 0, -1):
            self._update(i)

    def set(self, p: int, x: typing.Any) -> None:
        assert 0 <= p < self._n

        p += self._size
        self._d[p] = x
        for i in range(1, self._log + 1):
            self._update(p >> i)

    def get(self, p: int) -> typing.Any:
        assert 0 <= p < self._n

        return self._d[p + self._size]

    def prod(self, left: int, right: int) -> typing.Any:
        assert 0 <= left <= right <= self._n
        sml = self._e
        smr = self._e
        left += self._size
        right += self._size

        while left < right:
            if left & 1:
                sml = self._op(sml, self._d[left])
                left += 1
            if right & 1:
                right -= 1
                smr = self._op(self._d[right], smr)
            left >>= 1
            right >>= 1

        return self._op(sml, smr)

    def all_prod(self) -> typing.Any:
        return self._d[1]

    def max_right(self, left: int,
                  f: typing.Callable[[typing.Any], bool]) -> int:
        assert 0 <= left <= self._n
        assert f(self._e)

        if left == self._n:
            return self._n

        left += self._size
        sm = self._e

        first = True
        while first or (left & -left) != left:
            first = False
            while left % 2 == 0:
                left >>= 1
            if not f(self._op(sm, self._d[left])):
                while left < self._size:
                    left *= 2
                    if f(self._op(sm, self._d[left])):
                        sm = self._op(sm, self._d[left])
                        left += 1
                return left - self._size
            sm = self._op(sm, self._d[left])
            left += 1

        return self._n

    def min_left(self, right: int,
                 f: typing.Callable[[typing.Any], bool]) -> int:
        assert 0 <= right <= self._n
        assert f(self._e)

        if right == 0:
            return 0

        right += self._size
        sm = self._e

        first = True
        while first or (right & -right) != right:
            first = False
            right -= 1
            while right > 1 and right % 2:
                right >>= 1
            if not f(self._op(self._d[right], sm)):
                while right < self._size:
                    right = 2 * right + 1
                    if f(self._op(self._d[right], sm)):
                        sm = self._op(self._d[right], sm)
                        right -= 1
                return right + 1 - self._size
            sm = self._op(self._d[right], sm)

        return 0

    def _update(self, k: int) -> None:
        self._d[k] = self._op(self._d[2 * k], self._d[2 * k + 1])
"""

atcoder.segtree = types.ModuleType('atcoder.segtree')
exec(_atcoder_segtree_code, atcoder.segtree.__dict__)
atcoder.segtree.__dict__['atcoder'] = atcoder
atcoder.segtree.__dict__['atcoder._bit'] = atcoder._bit
SegTree = atcoder.segtree.SegTree



import sys




def main() -> None:
    n, q = map(int, sys.stdin.readline().split())
    a = list(map(int, sys.stdin.readline().split()))[::-1]

    segtree = SegTree(max, -1, a)

    for _ in range(q):
        t, x, y = map(int, sys.stdin.readline().split())
        if t == 1:
            segtree.set(n - x, y)
        elif t == 2:
            print(segtree.prod(n - y, n - x + 1))
        else:
            print(n - segtree.min_left(n - x + 1, lambda v: v < y) + 1)


if __name__ == '__main__':
    main()

Solution (Python):
import sys
import numpy as np
import numba
from numba import njit, b1, i4, i8, f8

read = sys.stdin.buffer.read
readline = sys.stdin.buffer.readline
readlines = sys.stdin.buffer.readlines

@njit((numba.types.optional(i8), ) * 2, cache=True)
def seg_f(x, y):
    if x is None:
        return y
    if y is None:
        return x
    return max(x, y)


@njit((i8[:], ), cache=True)
def build(raw_data):
    N = len(raw_data)
    seg = np.empty(N + N, np.int64)
    seg[N:] = raw_data
    for i in range(N - 1, 0, -1):
        seg[i] = seg_f(seg[i << 1], seg[i << 1 | 1])
    return seg


@njit((i8[:], i8, i8), cache=True)
def set_val(seg, i, x):
    N = len(seg) // 2
    i += N
    seg[i] = x
    while i > 1:
        i >>= 1
        seg[i] = seg_f(seg[i << 1], seg[i << 1 | 1])


@njit((i8[:], i8, i8), cache=True)
def fold(seg, l, r):
    vl = vr = None
    N = len(seg) // 2
    l, r = l + N, r + N
    while l < r:
        if l & 1:
            vl = seg_f(vl, seg[l])
            l += 1
        if r & 1:
            r -= 1
            vr = seg_f(seg[r], vr)
        l, r = l >> 1, r >> 1
    return seg_f(vl, vr)


@njit((i8[:], i8, i8), cache=True)
def max_right(seg, l, x):
    """とりあえず愚直"""
    N = len(seg) // 2

    def is_ok(v, x):
        return v >= x

    ng, ok = l, N + 1
    while ng + 1 < ok:
        mid = (ok + ng) // 2
        if is_ok(fold(seg, l, mid), x):
            ok = mid
        else:
            ng = mid
    return ok

@njit((i8, i8[:], i8[:]), cache=True)
def main(N, A, query):
    seg = build(A)
    for i in range(0, len(query), 3):
        t, a, b = query[i:i + 3]
        if t == 1:
            set_val(seg, a - 1, b)
        elif t == 2:
            print(fold(seg, a - 1, b))
        elif t == 3:
            print(max_right(seg, a - 1, b))

N, Q = map(int, readline().split())
A = np.array(readline().split(), np.int64)
query = np.array(read().split(), np.int64)

main(N, A, query)