import sys readline=sys.stdin.readline def Extended_Euclid(n,m): stack=[] while m: stack.append((n,m)) n,m=m,n%m if n>=0: x,y=1,0 else: x,y=-1,0 for i in range(len(stack)-1,-1,-1): n,m=stack[i] x,y=y,x-(n//m)*y return x,y class MOD: def __init__(self,p,e=None): self.p=p self.e=e if self.e==None: self.mod=self.p else: self.mod=self.p**self.e def Pow(self,a,n): a%=self.mod if n>=0: return pow(a,n,self.mod) else: assert math.gcd(a,self.mod)==1 x=Extended_Euclid(a,self.mod)[0] return pow(x,-n,self.mod) def Build_Fact(self,N): assert N>=0 self.factorial=[1] if self.e==None: for i in range(1,N+1): self.factorial.append(self.factorial[-1]*i%self.mod) else: self.cnt=[0]*(N+1) for i in range(1,N+1): self.cnt[i]=self.cnt[i-1] ii=i while ii%self.p==0: ii//=self.p self.cnt[i]+=1 self.factorial.append(self.factorial[-1]*ii%self.mod) self.factorial_inve=[None]*(N+1) self.factorial_inve[-1]=self.Pow(self.factorial[-1],-1) for i in range(N-1,-1,-1): ii=i+1 while ii%self.p==0: ii//=self.p self.factorial_inve[i]=(self.factorial_inve[i+1]*ii)%self.mod def Fact(self,N): if N<0: return 0 retu=self.factorial[N] if self.e!=None and self.cnt[N]: retu*=pow(self.p,self.cnt[N],self.mod)%self.mod retu%=self.mod return retu def Fact_Inve(self,N): if self.e!=None and self.cnt[N]: return None return self.factorial_inve[N] def Comb(self,N,K,divisible_count=False): if K<0 or K>N: return 0 retu=self.factorial[N]*self.factorial_inve[K]%self.mod*self.factorial_inve[N-K]%self.mod if self.e!=None: cnt=self.cnt[N]-self.cnt[N-K]-self.cnt[K] if divisible_count: return retu,cnt else: retu*=pow(self.p,cnt,self.mod) retu%=self.mod return retu def Primitive_Root(p): if p==2: return 1 if p==167772161: return 3 if p==469762049: return 3 if p==754974721: return 11 if p==998244353: return 3 if p==10**9+7: return 5 divisors=[2] pp=(p-1)//2 while pp%2==0: pp//=2 for d in range(3,pp+1,2): if d**2>pp: break if pp%d==0: divisors.append(d) while pp%d==0: pp//=d if pp>1: divisors.append(pp) primitive_root=2 while True: for d in divisors: if pow(primitive_root,(p-1)//d,p)==1: break else: return primitive_root primitive_root+=1 class Polynomial: def __init__(self,polynomial,max_degree=-1,eps=0,mod=0): self.max_degree=max_degree if self.max_degree!=-1 and len(polynomial)>self.max_degree+1: self.polynomial=polynomial[:self.max_degree+1] else: self.polynomial=polynomial self.mod=mod self.eps=eps def __eq__(self,other): if type(other)!=Polynomial: return False if len(self.polynomial)!=len(other.polynomial): return False for i in range(len(self.polynomial)): if self.epsself.max_degree+1: prod=prod[:self.max_degree+1] while prod and abs(prod[-1])<=self.eps: prod.pop() prod=Polynomial(prod,max_degree=self.max_degree,eps=self.eps,mod=self.mod) return prod def __truediv__(self,other): if type(other)==Polynomial: assert other.polynomial for n in range(len(other.polynomial)): if self.epsn for i in range(n): assert abs(self.polynomial[i])<=self.eps self_polynomial=self.polynomial[n:] other_polynomial=other.polynomial[n:] if self.mod: inve=MOD(self.mod).Pow(other_polynomial[0],-1) else: inve=1/other_polynomial[0] quot=[] for i in range(len(self_polynomial)-len(other_polynomial)+1): if self.mod: quot.append(self_polynomial[i]*inve%self.mod) else: quot.append(self_polynomial[i]*inve) for j in range(len(other_polynomial)): self_polynomial[i+j]-=other_polynomial[j]*quot[-1] if self.mod: self_polynomial[i+j]%=self.mod for i in range(max(0,len(self_polynomial)-len(other_polynomial)+1),len(self_polynomial)): if self.eps>bit,mod) U=[1] for _ in range(a): U.append(U[-1]*x%mod) for i in range(1<>bit,mod) U=[1] for _ in range(a): U.append(U[-1]*x%mod) for i in range(1<>bit,mod) U=[1] for _ in range(a): U.append(U[-1]*x%mod) for i in range(1<>bit,mod) U=[1] for _ in range(a): U.append(U[-1]*x%mod) for i in range(1<