import sys,bisect,string,math,time,functools,random,fractions from heapq import heappush,heappop,heapify from collections import deque,defaultdict,Counter from itertools import permutations,combinations,groupby rep=range;R=range def Golf():n,*t=map(int,open(0).read().split()) def I():return int(input()) def S_():return input() def IS():return input().split() def LS():return [i for i in input().split()] def MI():return map(int,input().split()) def LI():return [int(i) for i in input().split()] def LI_():return [int(i)-1 for i in input().split()] def NI(n):return [int(input()) for i in range(n)] def NI_(n):return [int(input())-1 for i in range(n)] def StoLI():return [ord(i)-97 for i in input()] def ItoS(n):return chr(n+97) def LtoS(ls):return ''.join([chr(i+97) for i in ls]) def RA():return map(int,open(0).read().split()) def GI(V,E,ls=None,Directed=False,index=1): org_inp=[];g=[[] for i in range(V)] FromStdin=True if ls==None else False for i in range(E): if FromStdin: inp=LI() org_inp.append(inp) else: inp=ls[i] if len(inp)==2: a,b=inp;c=1 else: a,b,c=inp if index==1:a-=1;b-=1 aa=(a,c);bb=(b,c);g[a].append(bb) if not Directed:g[b].append(aa) return g,org_inp def GGI(h,w,search=None,replacement_of_found='.',mp_def={'#':1,'.':0},boundary=1): #h,w,g,sg=GGI(h,w,search=['S','G'],replacement_of_found='.',mp_def={'#':1,'.':0},boundary=1) # sample usage mp=[boundary]*(w+2);found={} for i in R(h): s=input() for char in search: if char in s: found[char]=((i+1)*(w+2)+s.index(char)+1) mp_def[char]=mp_def[replacement_of_found] mp+=[boundary]+[mp_def[j] for j in s]+[boundary] mp+=[boundary]*(w+2) return h+2,w+2,mp,found def TI(n):return GI(n,n-1) def accum(ls): rt=[0] for i in ls:rt+=[rt[-1]+i] return rt def bit_combination(n,base=2): rt=[] for tb in R(base**n):s=[tb//(base**bt)%base for bt in R(n)];rt+=[s] return rt def gcd(x,y): if y==0:return x if x%y==0:return y while x%y!=0:x,y=y,x%y return y def YN(x):print(['NO','YES'][x]) def Yn(x):print(['No','Yes'][x]) def show(*inp,end='\n'): if show_flg:print(*inp,end=end) mo=10**9+7 inf=float('inf') FourNb=[(-1,0),(1,0),(0,1),(0,-1)];EightNb=[(-1,0),(1,0),(0,1),(0,-1),(1,1),(-1,-1),(1,-1),(-1,1)];compas=dict(zip('WENS',FourNb));cursol=dict(zip('LRUD',FourNb)) l_alp=string.ascii_lowercase #sys.setrecursionlimit(10**7) read=sys.stdin.buffer.read;readline=sys.stdin.buffer.readline;input=lambda:sys.stdin.readline().rstrip() ######################################################################################################################################################################## # Binary Indexed Tree # Bit.add(i,x) : add x at i-th value # Bit.sum(i) : get sum up to i-th value # Bit.l_bound(w) : get lower bound of index where w can be inserted class Bit: def __init__(self,n,init=None): self.size=n self.m=len(bin(self.size))-2 self.arr=[0]*(2**self.m+1) if init!=None: for i in range(len(init)): self.add(i,init[i]) def __str__(self): a=[self.sum(i+1)-self.sum(i) for i in range(self.size)] return str(a) def add(self,i,x): if i<0:return NotImplemented k=0 while i<=self.size: k+=1 self.arr[i]+=x i+=i&(-i) return def sum(self,i): if i<0:return NotImplemented rt=0 while i>0: rt+=self.arr[i] i-=i&(-i) return rt def __getitem__(self,key): return self.sum(key+1)-self.sum(key) def __setitem__(self,key,value): self.add(key,value-self.sum(key+1)+self.sum(key)) def l_bound(self,w): if w<=0: return 0 x=0 k=2**self.m while k>0: if x+k0: if x+k