from copy import deepcopy class Modulo_Matrix(): __slots__=("ele","row","col","size") #入力 def __init__(self,M): """ 行列 M の定義 M: 行列 ※ Mod: 法はグローバル変数から指定 """ self.ele=[[x%Mod for x in X] for X in M] R=len(M) if R!=0: C=len(M[0]) else: C=0 self.row=R self.col=C self.size=(R,C) #出力 def __str__(self): return "["+"\n".join(map(str,self.ele))+"]" def __repr__(self): return str(self) # 零行列, 単位行列 @classmethod def Zero_Matrix(cls, row, col): return Modulo_Matrix([[0] * col for _ in range(row)]) @classmethod def Identity_Matrix(cls, N): return Modulo_Matrix([[1 if i==j else 0 for j in range(N)] for i in range(N)]) #+,- def __pos__(self): return self def __neg__(self): return self.__scale__(-1) #加法 def __add__(self,other): M=self.ele; N=other.ele L=[[0]*self.col for _ in range(self.row)] for i in range(self.row): Li,Mi,Ni=L[i],M[i],N[i] for j in range(self.col): Li[j]=Mi[j]+Ni[j] return Modulo_Matrix(L) def __iadd__(self,other): M=self.ele; N=other.ele for i in range(self.row): Mi,Ni=M[i],N[i] for j in range(self.col): Mi[j]+=Ni[j] Mi[j]%=Mod return self #減法 def __sub__(self,other): M=self.ele; N=other.ele L=[[0]*self.col for _ in range(self.row)] for i in range(self.row): Li,Mi,Ni=L[i],M[i],N[i] for j in range(self.col): Li[j]=Mi[j]-Ni[j] return Modulo_Matrix(L) def __isub__(self,other): M=self.ele; N=other.ele for i in range(self.row): Mi,Ni=M[i],N[i] for j in range(self.col): Mi[j]-=Ni[j] Mi[j]%=Mod return self #乗法 def __mul__(self, other): if isinstance(other, Modulo_Matrix): assert self.col == other.row, f"左側の列と右側の行が一致しません (left: {self.col}, right:{other.row})." A = self.ele; B = other.ele C = [[0] * other.col for _ in range(self.row)] for i in range(self.row): Ai = A[i] Ci = C[i] for k in range(self.col): a_ik = Ai[k] Bk = B[k] for j in range(other.col): Ci[j] = (Ci[j] + a_ik * Bk[j]) % Mod return Modulo_Matrix(C) elif isinstance(other,int): return self.__scale__(other) def __rmul__(self,other): if isinstance(other,int): return self.__scale__(other) def inverse(self): assert self.row==self.col,"正方行列ではありません." M=self N=M.row R=[[1 if i==j else 0 for j in range(N)] for i in range(N)] T=deepcopy(M.ele) for j in range(N): if T[j][j]==0: for i in range(j+1,N): if T[i][j]: break else: assert 0, "正則行列ではありません" T[j],T[i]=T[i],T[j] R[j],R[i]=R[i],R[j] Tj,Rj=T[j],R[j] inv=pow(Tj[j], -1, Mod) for k in range(N): Tj[k]*=inv; Tj[k]%=Mod Rj[k]*=inv; Rj[k]%=Mod for i in range(N): if i==j: continue c=T[i][j] Ti,Ri=T[i],R[i] for k in range(N): Ti[k]-=Tj[k]*c; Ti[k]%=Mod Ri[k]-=Rj[k]*c; Ri[k]%=Mod return Modulo_Matrix(R) #スカラー倍 def __scale__(self,r): M=self.ele r%=Mod L=[[(r*M[i][j])%Mod for j in range(self.col)] for i in range(self.row)] return Modulo_Matrix(L) #累乗 def __pow__(self, n): assert self.row==self.col, "正方行列ではありません." sgn = 1 if n >= 0 else -1 n = abs(n) C = Modulo_Matrix.Identity_Matrix(self.row) tmp = self while n: if n & 1: C = C * tmp tmp = tmp * tmp n >>= 1 return C if sgn == 1 else C.inverse() #等号 def __eq__(self,other): return self.ele==other.ele #不等号 def __neq__(self,other): return not(self==other) #転置 def transpose(self): return Modulo_Matrix(list(map(list,zip(*self.ele)))) #行基本変形 def row_reduce(self): M=self (R,C)=M.size T=[] for i in range(R): U=[] for j in range(C): U.append(M.ele[i][j]) T.append(U) I=0 for J in range(C): if T[I][J]==0: for i in range(I+1,R): if T[i][J]!=0: T[i],T[I]=T[I],T[i] break if T[I][J]!=0: u=T[I][J] u_inv=pow(u, -1, Mod) for j in range(C): T[I][j]*=u_inv T[I][j]%=Mod for i in range(R): if i!=I: v=T[i][J] for j in range(C): T[i][j]-=v*T[I][j] T[i][j]%=Mod I+=1 if I==R: break return Modulo_Matrix(T) #列基本変形 def column_reduce(self): M=self (R,C)=M.size T=[] for i in range(R): U=[] for j in range(C): U.append(M.ele[i][j]) T.append(U) J=0 for I in range(R): if T[I][J]==0: for j in range(J+1,C): if T[I][j]!=0: for k in range(R): T[k][j],T[k][J]=T[k][J],T[k][j] break if T[I][J]!=0: u=T[I][J] u_inv=pow(u, -1, Mod) for i in range(R): T[i][J]*=u_inv T[i][J]%=Mod for j in range(C): if j!=J: v=T[I][j] for i in range(R): T[i][j]-=v*T[i][J] T[i][j]%=Mod J+=1 if J==C: break return Modulo_Matrix(T) #行列の階数 def rank(self): M=self.row_reduce() (R,C)=M.size T=M.ele rnk=0 for i in range(R): f=False for j in range(C): if T[i][j]!=0: f=True break if f: rnk+=1 else: break return rnk # 単射 ? def is_injection(self): return self.rank() == self.col # 全射 ? def is_surjective(self): return self.rank() == self.row # 全単射 ? def is_bijection(self): return self.col == self.row == self.rank() #行の結合 def row_union(self,other): return Modulo_Matrix(self.ele+other.ele) #列の結合 def column_union(self,other): E=[] for i in range(self.row): E.append(self.ele[i]+other.ele[i]) return Modulo_Matrix(E) def __getitem__(self,index): if isinstance(index, int): return self.ele[index] else: return self.ele[index[0]][index[1]] def __setitem__(self,index,val): assert isinstance(index,tuple) and len(index)==2 self.ele[index[0]][index[1]]=val #======================== #===入力 MA,NA,S=map(int,input().split()) MB,NB,T=map(int,input().split()) K=int(input()) #===定数の設定 Mod=998244353 rho_A=(MA*pow(NA,Mod-2,Mod))%Mod rho_B=(MB*pow(NB,Mod-2,Mod))%Mod #===Aについての行列 U=[[0]*(S+T+1) for _ in range(S+T+1)] for y in range(S+T+1): for x in range(S+T+1): if x==0: U[y][x]=1 if y==0 else 0 elif x==S+T: U[y][x]=1 if y==S+T else 0 else: if yx: V[y][x]=0 else: if y==0: V[y][x]=pow(rho_B,x-y,Mod) else: V[y][x]=(pow(rho_B,x-y,Mod)*(1-rho_B))%Mod #===行列の計算 E=pow(Modulo_Matrix(V)*Modulo_Matrix(U),K) #===結果の出力 print(E[S+T,T]) print(E[0,T])