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346 | namespace Poly
{
static const int MAX=4000010;
static const double PI=acos(-1);
static const ModInt<Mod> gn=332748118,gni=3,ZERO=0,ONE=1;
namespace Quadraticresidue
{
ModInt<Mod> I;
struct Complex
{
ModInt<Mod> x,y;
Complex(ModInt<Mod> X=0,ModInt<Mod> Y=0):x(X),y(Y) {}
inline friend bool operator == (Complex a,Complex b) {return a.x==b.x&&a.y==b.y;}
inline friend Complex operator * (Complex a,Complex b) {return Complex(a.x*b.x+a.y*b.y*I,a.x*b.y+a.y*b.x);}
};
inline Complex qpow(Complex a,int b) {Complex res(1,0); while(b) {if(b&1) res=res*a; a=a*a; b>>=1;} return res;}
inline ModInt<Mod> random(int l,int r)
{
static mt19937_64 sd(20070707);
static uniform_int_distribution<int> range(l,r);
return ModInt<Mod>(range(sd));
}
inline pair<ModInt<Mod>,ModInt<Mod>> Cipolla(ModInt<Mod> n)
{
if(!n) return make_pair(0,0);
ModInt<Mod> x,x1,x2;
do x=random(1,Mod-1); while(((x*x+Mod-n)^((Mod-1)>>1))==ONE);
I=x*x+Mod-n,x1=qpow(Complex(x,1),(Mod+1)>>1).x,x2=Mod-x1;
return make_pair(min(x1,x2),max(x1,x2));
}
}
using namespace Quadraticresidue;
struct poly
{
vector<ModInt<Mod>> a;
inline ModInt<Mod> &operator [] (int i) {return a[i];}
inline poly operator = (const poly &T) {this->a=T.a; return *this;}
inline int size() {return a.size();}
inline void resize(int N) {return a.resize(N),void();}
inline void reverse() {return std::reverse(a.begin(),a.end());}
inline void pin(int x) {return a.emplace_back(ModInt<Mod>(x)),void();}
inline void pout() {for(auto it:a) write(it.x,' ');}
};
inline int Sup(int N) {int K=1; while(K<N) K<<=1; return K;}
static const int GT=21,GR=31;
ModInt<Mod> omega[MAX]; poly zero,one;
inline void Prework(int N)
{
zero.a.emplace_back(ZERO),one.a.emplace_back(ONE);
int K=1; while((1<<K)<N) ++K; K=min(K-1,21ll);
omega[0]=1,omega[1<<K]=(ModInt<Mod>(GR))^(1<<(GT-K));
for(int i=K;i>=1;--i) omega[1<<(i-1)]=omega[1<<i]*omega[1<<i];
for(int i=1;i<(1<<K);++i) omega[i]=omega[i&(i-1)]*omega[i&(-i)];
}
inline void NTT(poly &F,int typ)
{
ModInt<Mod> U,V;
int N=F.size();
if(typ==1)
{
for(int mid=N>>1;mid>=1;mid>>=1)
for(int i=0,k=0;i<N;i+=mid<<1,++k)
for(int j=0;j<mid;++j)
U=F[i+j],V=F[i+j+mid]*omega[k],
F[i+j]=U+V,F[i+j+mid]=U-V;
}
if(typ==-1)
{
for(int mid=1;mid<N;mid<<=1)
for(int i=0,k=0;i<N;i+=mid<<1,++k)
for(int j=0;j<mid;++j)
U=F[i+j],V=F[i+j+mid],
F[i+j]=U+V,F[i+j+mid]=(U-V)*omega[k];
ModInt<Mod> Ninv=ONE/N;
for(int i=0;i<N;++i) F[i]*=Ninv;
reverse(F.a.begin()+1,F.a.end());
}
}
inline poly operator * (poly a,ModInt<Mod> b) {for(int i=0;i<a.size();++i) a[i]*=b; return a;}
inline poly operator / (poly a,ModInt<Mod> b) {for(int i=0;i<a.size();++i) a[i]/=b; return a;}
inline poly operator - (poly a) {for(int i=0;i<a.size();++i) a[i]=-a[i]; return a;}
inline poly operator + (poly a,poly b)
{
int N=max(a.size(),b.size());
a.resize(N),b.resize(N);
for(int i=0;i<N;++i) a[i]+=b[i];
return a;
}
inline poly operator - (poly a,poly b)
{
int N=max(a.size(),b.size());
a.resize(N),b.resize(N);
for(int i=0;i<N;++i) a[i]-=b[i];
return a;
}
inline poly operator * (poly a,poly b)
{
int K=a.size()+b.size()-1,N=Sup(K);
a.resize(N),b.resize(N);
NTT(a,1),NTT(b,1);
for(int i=0;i<N;++i) a[i]*=b[i];
NTT(a,-1);
a.resize(K);
return a;
}
inline poly operator >> (poly a,int d)
{
int N=a.size(); poly b;
b.resize(N+d);
for(int i=0;i<N;++i) b[i+d]=a[i];
return b;
}
inline poly operator << (poly a,int d)
{
int N=a.size(); if(N<=d) return zero;
poly b; b.resize(N-d);
for(int i=0;i<N-d;++i) b[i]=a[i+d];
return b;
}
inline poly Dvt(poly a)
{
int N=a.size()-1;
for(int i=0;i<N;++i) a[i]=a[i+1]*(i+1);
return a.resize(N),a;
}
inline poly Itg(poly a)
{
int N=a.size()+1; a.resize(N);
for(int i=N;i>=1;--i) a[i]=a[i-1]/i;
return a[0]=0,a;
}
inline poly Inv(poly a)
{
int K=a.size(),N=Sup(K);
poly b,c,d; a.resize(N),b.resize(N);
b[0]=ONE/a[0];
for(int len=2;len<=N;len<<=1)
{
c.resize(len),d.resize(len);
for(int i=0;i<len;++i) c[i]=a[i];
for(int i=0;i<(len>>1);++i) d[i]=b[i];
NTT(c,1),NTT(d,1);
for(int i=0;i<len;++i) c[i]=c[i]*d[i];
NTT(c,-1); for(int i=1;i<(len>>1);++i) c[i]=0; c[0]=Mod-1;
NTT(c,1); for(int i=0;i<len;++i) d[i]=d[i]*c[i];
NTT(d,-1); for(int i=len>>1;i<len;++i) b[i]=-d[i];
}
b.resize(K);
return b;
}
inline poly operator / (poly a,poly b)
{
int N=a.size(),M=b.size();
poly c=a,d=b;
c.reverse(),d.reverse();
d.resize(N-M+1),d=Inv(d);
c=c*d;c.resize(N-M+1),c.reverse();
return c;
}
inline poly operator % (poly a,poly b)
{
int M=b.size();
poly q=a/b; b=b*q;
poly r=a-b; r.resize(M-1);
return r;
}
inline pair<poly,poly> Div(poly a,poly b)
{
int M=b.size();
poly q=a/b; b=b*q;
poly r=a-b; r.resize(M-1);
return make_pair(q,r);
}
namespace DivideConquer
{
poly F[20][8],G[20][8];
inline void brute(poly &a,poly &b,int l,int r,void work(poly &a,poly &b,int i))
{
work(a,b,l);
for(int i=l+1;i<=r;++i)
{
for(int j=l;j<i;++j) a[i]+=a[j]*b[i-j];
work(a,b,i);
}
}
inline void Divide(poly &a,poly &b,int l,int r,int dep,void work(poly &a,poly &b,int i))
{
if(r-l+1<=32) return brute(a,b,l,r,work);
if(l>=r) return;
int coef=1<<((dep-1)*3);
poly tmp; tmp.resize(coef<<1);
for(int i=0;;++i)
{
int L=l+i*coef,R=min(r,L+coef-1);
if(i)
{
for(int j=0;j<(coef<<1);++j) tmp[j]=0;
for(int j=0;j<i;++j) for(int k=0;k<(coef<<1);++k) tmp[k]+=F[dep][j][k]*G[dep][i-j][k];
NTT(tmp,-1);
for(int j=L;j<=R;++j) a[j]+=tmp[j-L+coef];
}
Divide(a,b,L,R,dep-1,work);
if(R==r) return;
for(int j=0;j<(coef<<1);++j) F[dep][i][j]=0;
for(int j=L;j<=R;++j) F[dep][i][j-L]=a[j];
NTT(F[dep][i],1);
}
}
inline void solve(poly &a,poly &b,int N,void work(poly &a,poly &b,int i))
{
if(N<=128) return brute(a,b,0,N-1,work);
int len=1,dep=0;
while(len<N) len<<=3,++dep;
len>>=3;
for(int i=1;i<=dep;++i)
{
int coef=1<<((i-1)*3),mix=min((N-1)/coef,7ll);
for(int j=1;j<=mix;++j)
{
int L=(j-1)*coef+1,R=min(N-1,(j+1)*coef-1);
F[i][j-1].resize(coef<<1),G[i][j].resize(coef<<1);
for(int k=0;k<(coef<<1);++k) G[i][j][k]=0;
for(int k=L;k<=R;++k) G[i][j][k-L+1]=b[k];
NTT(G[i][j],1);
}
}
Divide(a,b,0,N-1,dep,work);
}
}
using namespace DivideConquer;
inline void workln(poly &a,poly &b,int i)
{
if(!i) a[i]=0;
else a[i]=i*b[i]-a[i];
}
inline poly Ln(poly a)
{
int N=a.size();
poly b; b.resize(N);
solve(b,a,N,workln);
b[0]=0; for(int i=1;i<N;++i) b[i]*=ONE/i;
return b;
}
inline void workexp(poly &a,poly &b,int i)
{
if(!i) a[i]=1;
else a[i]*=ONE/i;
}
inline poly Exp(poly a)
{
int N=a.size();
for(int i=0;i<N;++i) a[i]=a[i]*i;
poly b; b.resize(N);
solve(b,a,N,workexp);
return b;
}
inline void PrePow(char *s,int &k1,int &k2,int &k3)
{
int len=strlen(s+1);
for(int i=1;i<=len;++i) k1=((k1*10)+(s[i]^48))%Mod,k2=((k2*10)+(s[i]^48))%(Mod-1);
for(int i=1;i<=min(7ll,len);++i) k3=(k3*10)+(s[i]^48);
}
inline poly Pow(poly a,ModInt<Mod> b)
{
int N=a.size();
a=Exp(Ln(a)*b),a.resize(N);
return a;
}
inline poly Pow(poly a,char *s)
{
int N=a.size(),lst=N;
ModInt<Mod> k1,k2,k3;
PrePow(s,k1.x,k2.x,k3.x);
poly b; b.resize(N);
if(a[0]==0&&k3>=N) return b;
for(int i=0;i<N;++i) if(a[i]!=0) {lst=i; break;}
if(lst*k1>=N) return b;
ModInt<Mod> inv=ONE/a[lst],p=a[lst]^(k2.x);
for(int i=0;i<N-lst;++i) a[i]=a[i+lst]*inv;
for(int i=N-lst;i<N;++i) a[i]=0;
a=Ln(a); for(int i=0;i<N;++i) a[i]*=k1;
a=Exp(a),lst=lst*k1.x; for(int i=N-1;i>=lst;--i) b[i]=a[i-lst]*p;
return b;
}
inline poly Sqr(poly a,int typ)
{
int K=a.size(),N=Sup(K);
poly f,g,h,tmp,res,ans;
f.resize(N),g.resize(N),h.resize(N),res.resize(1);
f[0]=g[0]=Cipolla(a[0]).first,h[0]=ONE/g[0];
for(int len=2;len<=N;len<<=1)
{
tmp.resize(len),ans.resize(len);
for(int i=0;i<(len>>1);++i) tmp[i]=h[i],res[i]=g[i]*g[i];
NTT(res,-1); for(int i=0;i<(len>>1);++i) ans[i+(len>>1)]=res[i]-a[i]-a[i+(len>>1)],ans[i]=0;
NTT(ans,1),NTT(tmp,1); for(int i=0;i<len;++i) ans[i]*=tmp[i]*(ONE/2);
NTT(ans,-1); for(int i=len>>1;i<len;++i) f[i]=-ans[i];
if(len!=N||typ==-1)
{
res.resize(len); for(int i=0;i<len;++i) res[i]=f[i];
NTT(res,1); for(int i=0;i<len;++i) g[i]=res[i],res[i]*=tmp[i];
NTT(res,-1); res[0]=Mod-1; for(int i=1;i<(len>>1);++i) res[i]=0;
NTT(res,1); for(int i=0;i<len;++i) res[i]*=tmp[i];
NTT(res,-1); for(int i=len>>1;i<len;++i) h[i]=-res[i];
}
}
f.resize(K),h.resize(K);
if(typ==1) return f; else return h;
}
inline poly sin(poly a)
{
ModInt<Mod> E=((ModInt<Mod>(3))^((Mod-1)>>2));
return (Exp(a*E)-Exp(a*(-E)))*((E*2)^(Mod-2));
}
inline poly cos(poly a)
{
ModInt<Mod> E=((ModInt<Mod>(3))^((Mod-1)>>2));
return (Exp(a*E)+Exp(a*(-E)))*((Mod+1)>>1);
}
inline poly tan(poly a)
{
int N=a.size();
a=sin(a)*Inv(cos(a)),a.resize(N);
return a;
}
inline poly arcsin(poly a)
{
int N=a.size();
poly b=a*a; b.resize(N),b=-b,b[0]+=1;
a=Itg(Dvt(a)*Sqr(b,-1)); a.resize(N);
return a;
}
inline poly arctan(poly a)
{
int N=a.size();
poly b=a*a; b.resize(N),b[0]+=1;
a=Itg(Dvt(a)*Inv(b)); a.resize(N);
return a;
}
}
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