#pragma once
#include"FPS_base.hpp"
#include<atcoder/convolution.hpp>
#include"../math/ceil_pow2.hpp"
#include"../math/garner.hpp"
/**
* @brief 形式的冪級数(ModInt)
*/
template<typename Mint>
struct _FPS{
template<typename T>
T operator()(const T& _s,const T& _t){
if(_s.size()==0||_t.size()==0)return T();
const size_t sz=_s.size()+_t.size()-1;
if((Mint::get_mod()&((1<<ceil_pow2(sz))-1))==1){
std::vector<atcoder::static_modint<Mint::get_mod()>>s(_s.size()),t(_t.size());
for(size_t i=0;i<_s.size();++i)s[i]=_s[i].value();
for(size_t i=0;i<_t.size();++i)t[i]=_t[i].value();
std::vector<atcoder::static_modint<Mint::get_mod()>> _v=atcoder::convolution(s,t);
T v(_v.size());
for (size_t i=0;i<_v.size();++i)v[i]=_v[i].val();
return v;
}else{
std::vector<atcoder::static_modint<1224736769>>s1(_s.size()),t1(_t.size());
std::vector<atcoder::static_modint<1045430273>>s2(_s.size()),t2(_t.size());
std::vector<atcoder::static_modint<1007681537>>s3(_s.size()),t3(_t.size());
for(size_t i=0;i<_s.size();++i){
s1[i]=_s[i].value();
s2[i]=_s[i].value();
s3[i]=_s[i].value();
}
for(size_t i=0;i<_t.size();++i){
t1[i]=_t[i].value();
t2[i]=_t[i].value();
t3[i]=_t[i].value();
}
auto v1=atcoder::convolution(s1,t1);
auto v2=atcoder::convolution(s2,t2);
auto v3=atcoder::convolution(s3,t3);
T v(sz);
for(size_t i=0;i<sz;++i){
v[i]=garner(std::vector<long long>{v1[i].val(),v2[i].val(),v3[i].val()},std::vector<long long>{1224736769,1045430273,1007681537,(long long)Mint::get_mod()});
}
return v;
}
}
template<typename T>
T fact(const T& s){
return s.fact();
}
template<typename T>
T pow(const T& s,long long i){
return s.pow(i);
}
};
template<typename Mint>using fps=FPS_BASE<Mint,_FPS<Mint>>;
#line 2 "math/FPS_base.hpp"
#include<vector>
#include<tuple>
#include<iostream>
#include<cmath>
#include<type_traits>
#include<cassert>
/**
* @brief 形式的冪級数(BASE)
*/
template<typename T,typename F>
struct FPS_BASE:std::vector<T>{
using std::vector<T>::vector;
using P=FPS_BASE<T,F>;
F fft;
FPS_BASE(){}
inline P operator +(T x)const noexcept{return P(*this)+=x;}
inline P operator -(T x)const noexcept{return P(*this)-=x;}
inline P operator *(T x)const noexcept{return P(*this)*=x;}
inline P operator /(T x)const noexcept{return P(*this)/=x;}
inline P operator <<(int x)noexcept{return P(*this)<<=x;}
inline P operator >>(int x)noexcept{return P(*this)>>=x;}
inline P operator +(const P& x)const noexcept{return P(*this)+=x;}
inline P operator -(const P& x)const noexcept{return P(*this)-=x;}
inline P operator -()const noexcept{return P(1,T(0))-=P(*this);}
inline P operator *(const P& x)const noexcept{return P(*this)*=x;}
inline P operator /(const P& x)const noexcept{return P(*this)/=x;}
inline P operator %(const P& x)const noexcept{return P(*this)%=x;}
bool operator ==(P x){
for(int i=0;i<(int)max((*this).size(),x.size());++i){
if(i>=(int)(*this).size()&&x[i]!=T())return 0;
if(i>=(int)x.size()&&(*this)[i]!=T())return 0;
if(i<(int)min((*this).size(),x.size()))if((*this)[i]!=x[i])return 0;
}
return 1;
}
P &operator +=(T x){
if(this->size()==0)this->resize(1,T(0));
(*this)[0]+=x;
return (*this);
}
P &operator -=(T x){
if(this->size()==0)this->resize(1,T(0));
(*this)[0]-=x;
return (*this);
}
P &operator *=(T x){
for(int i=0;i<(int)this->size();++i){
(*this)[i]*=x;
}
return (*this);
}
P &operator /=(T x){
if(std::is_same<T,long long>::value){
for(int i=0;i<(int)this->size();++i){
(*this)[i]/=x;
}
return (*this);
}
return (*this)*=(T(1)/x);
}
P &operator <<=(int x){
P ret(x,T(0));
ret.insert(ret.end(),begin(*this),end(*this));
return (*this)=ret;
}
P &operator >>=(int x){
if((int)(*this).size()<=x)return (*this)=P();
P ret;
ret.insert(ret.end(),begin(*this)+x,end(*this));
return (*this)=ret;
}
P &operator +=(const P& x){
if(this->size()<x.size())this->resize(x.size(),T(0));
for(int i=0;i<(int)x.size();++i){
(*this)[i]+=x[i];
}
return (*this);
}
P &operator -=(const P& x){
if(this->size()<x.size())this->resize(x.size(),T(0));
for(int i=0;i<(int)x.size();++i){
(*this)[i]-=x[i];
}
return (*this);
}
P &operator *=(const P& x){
return (*this)=F()(*this,x);
}
P &operator /=(P x){
if(this->size()<x.size()) {
this->clear();
return (*this);
}
const int n=this->size()-x.size()+1;
return (*this) = (rev().pre(n)*x.rev().inv(n)).pre(n).rev(n);
}
P &operator %=(const P& x){
return ((*this)-=(*this)/x*x);
}
inline void print(){
for(int i=0;i<(int)(*this).size();++i)std::cerr<<(*this)[i]<<" \n"[i==(int)(*this).size()-1];
if((int)(*this).size()==0)std::cerr<<'\n';
}
inline P& shrink(){while((*this).back()==0)(*this).pop_back();return (*this);}
inline P pre(int sz)const{
return P(begin(*this),begin(*this)+std::min((int)this->size(),sz));
}
P rev(int deg=-1){
P ret(*this);
if(deg!=-1)ret.resize(deg,T(0));
reverse(begin(ret),end(ret));
return ret;
}
P inv(int deg=-1){
assert((*this)[0]!=T(0));
const int n=deg==-1?this->size():deg;
P ret({T(1)/(*this)[0]});
for(int i=1;i<n;i<<=1){
ret*=(-ret*pre(i<<1)+2).pre(i<<1);
}
return ret.pre(n);
}
inline P dot(const P& x){
P ret(*this);
for(int i=0;i<int(min(this->size(),x.size()));++i){
ret[i]*=x[i];
}
return ret;
}
P diff(){
if((int)(*this).size()<=1)return P();
P ret(*this);
for(int i=0;i<(int)ret.size();i++){
ret[i]*=i;
}
return ret>>1;
}
P integral(){
P ret(*this);
for(int i=0;i<(int)ret.size();i++){
ret[i]/=i+1;
}
return ret<<1;
}
P log(int deg=-1){
assert((*this)[0]==T(1));
const int n=deg==-1?this->size():deg;
return (diff()*inv(n)).pre(n-1).integral();
}
P exp(int deg=-1){
assert((*this)[0]==T(0));
const int n=deg==-1?this->size():deg;
P ret({T(1)});
for(int i=1;i<n;i<<=1){
ret=ret*(pre(i<<1)+1-ret.log(i<<1)).pre(i<<1);
}
return ret.pre(n);
}
P pow(int c,int deg=-1){
const int n=deg==-1?this->size():deg;
long long i=0;
P ret(*static_cast<P*>(this));
while(i!=(int)this->size()&&ret[i]==0)i++;
if(i==(int)this->size())return P(n,0);
if(i*c>=n)return P(n,0);
T k=ret[i];
return ((((ret>>i)/k).log()*c).exp()*(k.pow(c))<<(i*c)).pre(n);
// const int n=deg==-1?this->size():deg;
// long long i=0;
// P ret(*this);
// while(i!=(int)this->size()&&ret[i]==0)i++;
// if(i==(int)this->size())return P(n,0);
// if(i*c>=n)return P(n,0);
// T k=ret[i];
// return ((((ret>>i)/k).log()*c).exp()*(k.pow(c))<<(i*c)).pre(n);
// P x(*this);
// P ret(1,1);
// while(c) {
// if(c&1){
// ret*=x;
// if(~deg)ret=ret.pre(deg);
// }
// x*=x;
// if(~deg)x=x.pre(deg);
// c>>=1;
// }
// return ret;
}
P sqrt(int deg=-1){
const int n=deg==-1?this->size():deg;
if((*this)[0]==T(0)) {
for(int i=1;i<(int)this->size();i++) {
if((*this)[i]!=T(0)) {
if(i&1)return{};
if(n-i/2<=0)break;
auto ret=(*this>>i).sqrt(n-i/2)<<(i/2);
if((int)ret.size()<n)ret.resize(n,T(0));
return ret;
}
}
return P(n,0);
}
P ret({T(1)});
for(int i=1;i<n;i<<=1){
ret=(ret+pre(i<<1)*ret.inv(i<<1)).pre(i<<1)/T(2);
}
return ret.pre(n);
}
P shift(int c){
const int n=this->size();
P f(*this),g(n,0);
for(int i=0;i<n;++i)f[i]*=F().fact(T(i));
for(int i=0;i<n;++i)g[i]=F().pow(T(c),i)/F().fact(T(i));
g=g.rev();
f*=g;
f>>=n-1;
for(int i=0;i<n;++i)f[i]/=F().fact(T(i));
return f;
}
T eval(T x){
T res=0;
for(int i=(int)this->size()-1;i>=0;--i){
res*=x;
res+=(*this)[i];
}
return res;
}
P mul(const std::vector<std::pair<int,T>>& x){
int mx=0;
for(auto [s,t]:x){
if(mx<s)mx=s;
}
P res((int)this->size()+mx);
for(int i=0;i<(int)this->size();++i){
for(auto [s,t]:x){
res[i+s]+=(*this)[i]*t;
}
}
return res;
}
P div(const std::vector<std::pair<int,T>>& x){
P res(*this);
T cnt=0;
for(auto [s,t]:x){
if(s==0)cnt+=t;
}
cnt=cnt.inv();
for(int i=0;i<(int)this->size();++i){
for(auto [s,t]:x){
if(s==0)continue;
if(i>=s)res[i]-=res[i-s]*t*cnt;
}
}
res*=cnt;
return res;
}
static P interpolation(const std::vector<T>&x,const std::vector<T>& y){
const int n=x.size();
std::vector<std::pair<P,P>>a(n*2-1);
std::vector<P> b(n*2-1);
for(int i=0;i<n;++i)a[i+n-1]=std::make_pair(P{1},P{T()-x[i],1});
for(int i=n-2;i>=0;--i)a[i]={a[2*i+1].first*a[2*i+2].second+a[2*i+2].first*a[2*i+1].second,a[2*i+1].second*a[2*i+2].second};
auto d=(a[0].first).multipoint_eval(x);
for(int i=0;i<n;++i)b[i+n-1]=P{T(y[i]/d[i])};
for(int i=n-2;i>=0;--i)b[i]=b[2*i+1]*a[2*i+2].second+b[2*i+2]*a[2*i+1].second;
return b[0];
}
static P interpolation(const std::vector<T>& y){
const int n=y.size();
std::vector<std::pair<P,P>>a(n*2-1);
std::vector<P>b(n*2-1);
for(int i=0;i<n;++i)a[i+n-1]=std::make_pair(P{1},P{T()-i,1});
for(int i=n-2;i>=0;--i)a[i]={a[2*i+1].first*a[2*i+2].second+a[2*i+2].first*a[2*i+1].second,a[2*i+1].second*a[2*i+2].second};
for(int i=0;i<n;++i){
T tmp=F().fact(T(i))*F().pow(T(-1),i)*F().fact(T(n-1-i));
b[i+n-1]=P{T(y[i]/tmp)};
}
for(int i=n-2;i>=0;--i)b[i]=b[2*i+1]*a[2*i+2].second+b[2*i+2]*a[2*i+1].second;
return b[0];
}
std::vector<T> multipoint_eval(const std::vector<T>&x){
const int n=x.size();
P* v=new P[2*n-1];
for(int i=0;i<n;i++)v[i+n-1]={T()-x[i],T(1)};
for(int i=n-2;i>=0;i--){v[i]=v[i*2+1]*v[i*2+2];}
v[0]=P(*this)%v[0];v[0].shrink();
for(int i=1;i<n*2-1;i++){
v[i]=v[(i-1)/2]%v[i];
v[i].shrink();
}
std::vector<T>res(n);
for(int i=0;i<n;i++)res[i]=v[i+n-1][0];
return res;
}
P slice(int s,int e,int k){
P res;
for(int i=s;i<e;i+=k)res.push_back((*this)[i]);
return res;
}
T nth_term(P q,int64_t x){
if(x==0)return (*this)[0]/q[0];
P p(*this);
P q2=q;
for(int i=1;i<(int)q2.size();i+=2)q2[i]*=-1;
q*=q2;
p*=q2;
return p.slice(x%2,p.size(),2).nth_term(q.slice(0,q.size(),2),x/2);
}
P gcd(P q){
return *this==P()?q:(q%(*this).shrink()).gcd(*this);
}
//(*this)(t(x))
P manipulate(P t,int deg){
P s=P(*this);
if(deg==0)return P();
if((int)t.size()==1)return P{s.eval(t[0])};
int k=std::min((int)::sqrt(deg/(::log2(deg)+1))+1,(int)t.size());
int b=deg/k+1;
P t2=t.pre(k);
std::vector<P>table(s.size()/2+1,P{1});
for(int i=1;i<(int)table.size();i++){
table[i]=((table[i-1])*t2).pre(deg);
}
auto f=[&](auto f,auto l,auto r,int deg)->P{
if(r-l==1)return P{*l};
auto m=l+(r-l)/2;
return f(f,l,m,deg)+(table[m-l]*f(f,m,r,deg)).pre(deg);
};
P ans=P();
P tmp=f(f,s.begin(),s.end(),deg);
P tmp2=P{1};
T tmp3=T(1);
int tmp5=-1;
P tmp6=t2.diff();
if(tmp6==P()){
for(int i=0;i<b;++i){
if(tmp.size()==0)break;
ans+=(tmp2*tmp[0]).pre(deg)/tmp3;
tmp=tmp.diff();
tmp2=(tmp2*(t-t2)).pre(deg);
tmp3*=T(i+1);
}
}else{
while(t2[++tmp5]==T());
P tmp4=(tmp6>>(tmp5-1)).inv(deg);
for(int i=0;i<b;++i){
ans+=(tmp*tmp2).pre(deg)/tmp3;
tmp=((tmp.diff()>>(tmp5-1))*tmp4).pre(deg);
tmp2=(tmp2*(t-t2)).pre(deg);
tmp3*=T(i+1);
}
}
return ans;
}
//(*this)(t(x))
P manipulate2(P t,int deg){
P ans=P();
P s=(*this).rev();
for(int i=0;i<(int)s.size();++i){
ans=(ans*t+s[i]).pre(deg);
}
return ans;
}
P find_linear_recurrence()const{
const int n=this->size();
P b={T(-1)},c={T(-1)};
T y=T(1);
for(int i=1;i<=n;++i){
int l=c.size(),m=b.size();
T x=0;
for(int j=0;j<l;++j)x+=c[j]*(*this)[i-l+j];
b.emplace_back(0);
m++;
if(x==T(0))continue;
T freq=x/y;
if(l<m){
auto tmp=c;
c<<=m-l;
c-=b*freq;
b=tmp;
y=x;
}else{
c-=(b*freq)<<(l-m);
}
}
return c;
}
static P stirling_second(int n){
P a(n+1,0),b(n+1,0);
for(int i=0;i<=n;++i){
a[i]=F().pow(T(i),n)/F().fact(T(i));
b[i]=(i%2?T(-1):T(1))/F().fact(T(i));
}
return (a*b).pre(n+1);
}
void debug(){
for(int i=0;i<(int)(*this).size();++i)std::cerr<<(*this)[i]<<" \n"[i==(int)(*this).size()-1];
}
};
#line 3 "math/FPS_mint.hpp"
#include<atcoder/convolution.hpp>
#line 1 "math/ceil_pow2.hpp"
int ceil_pow2(int n) {
int x = 0;
while ((1U << x) < (unsigned int)(n)) x++;
return x;
}
#line 1 "math/mod_pow.hpp"
/**
* @brief (x^y)%mod
*/
long long mod_pow(long long x,long long y,long long mod){
long long ret=1;
while(y>0) {
if(y&1)(ret*=x)%=mod;
(x*=x)%=mod;
y>>=1;
}
return ret;
}
#line 4 "math/garner.hpp"
/**
*
* @brief ガーナーのアルゴリズム
*
*/
long long garner(const std::vector<long long>&a,const std::vector<long long>&mods){
const int sz=a.size();
long long coeffs[sz+1]={1,1,1,1};
long long constants[sz+1]={};
for(int i=0;i<sz;i++){
long long v=(mods[i]+a[i]-constants[i])%mods[i]*mod_pow(coeffs[i],mods[i]-2,mods[i])%mods[i];
for(int j=i+1;j<sz+1;j++) {
constants[j]=(constants[j]+coeffs[j]*v)%mods[j];
coeffs[j]=(coeffs[j]*mods[i])%mods[j];
}
}
return constants[sz];
}
#line 6 "math/FPS_mint.hpp"
/**
* @brief 形式的冪級数(ModInt)
*/
template<typename Mint>
struct _FPS{
template<typename T>
T operator()(const T& _s,const T& _t){
if(_s.size()==0||_t.size()==0)return T();
const size_t sz=_s.size()+_t.size()-1;
if((Mint::get_mod()&((1<<ceil_pow2(sz))-1))==1){
std::vector<atcoder::static_modint<Mint::get_mod()>>s(_s.size()),t(_t.size());
for(size_t i=0;i<_s.size();++i)s[i]=_s[i].value();
for(size_t i=0;i<_t.size();++i)t[i]=_t[i].value();
std::vector<atcoder::static_modint<Mint::get_mod()>> _v=atcoder::convolution(s,t);
T v(_v.size());
for (size_t i=0;i<_v.size();++i)v[i]=_v[i].val();
return v;
}else{
std::vector<atcoder::static_modint<1224736769>>s1(_s.size()),t1(_t.size());
std::vector<atcoder::static_modint<1045430273>>s2(_s.size()),t2(_t.size());
std::vector<atcoder::static_modint<1007681537>>s3(_s.size()),t3(_t.size());
for(size_t i=0;i<_s.size();++i){
s1[i]=_s[i].value();
s2[i]=_s[i].value();
s3[i]=_s[i].value();
}
for(size_t i=0;i<_t.size();++i){
t1[i]=_t[i].value();
t2[i]=_t[i].value();
t3[i]=_t[i].value();
}
auto v1=atcoder::convolution(s1,t1);
auto v2=atcoder::convolution(s2,t2);
auto v3=atcoder::convolution(s3,t3);
T v(sz);
for(size_t i=0;i<sz;++i){
v[i]=garner(std::vector<long long>{v1[i].val(),v2[i].val(),v3[i].val()},std::vector<long long>{1224736769,1045430273,1007681537,(long long)Mint::get_mod()});
}
return v;
}
}
template<typename T>
T fact(const T& s){
return s.fact();
}
template<typename T>
T pow(const T& s,long long i){
return s.pow(i);
}
};
template<typename Mint>using fps=FPS_BASE<Mint,_FPS<Mint>>;
形式的冪級数(ModInt)
math/kth_root.hpp