#line 2 "cpplib/util/template.hpp"
#pragma GCC optimize("Ofast")
#pragma GCC optimize("unroll-loops")
#pragma GCC target("avx2")
#include<bits/stdc++.h>
using namespace std;
struct __INIT__{__INIT__(){cin.tie(0);ios::sync_with_stdio(false);cout<<fixed<<setprecision(15);}}__INIT__;
typedef long long lint;
#define INF (1LL<<60)
#define IINF (1<<30)
#define EPS (1e-10)
#define endl ('\n')
typedef vector<lint> vec;
typedef vector<vector<lint>> mat;
typedef vector<vector<vector<lint>>> mat3;
typedef vector<string> svec;
typedef vector<vector<string>> smat;
template<typename T>using V=vector<T>;
template<typename T>using VV=V<V<T>>;
template<typename T>inline void output(T t){bool f=0;for(auto i:t){cout<<(f?" ":"")<<i;f=1;}cout<<endl;}
template<typename T>inline void output2(T t){for(auto i:t)output(i);}
template<typename T>inline void debug(T t){bool f=0;for(auto i:t){cerr<<(f?" ":"")<<i;f=1;}cerr<<endl;}
template<typename T>inline void debug2(T t){for(auto i:t)output(i);}
#define loop(n) for(long long _=0;_<(long long)(n);++_)
#define _overload4(_1,_2,_3,_4,name,...) name
#define __rep(i,a) repi(i,0,a,1)
#define _rep(i,a,b) repi(i,a,b,1)
#define repi(i,a,b,c) for(long long i=(long long)(a);i<(long long)(b);i+=c)
#define rep(...) _overload4(__VA_ARGS__,repi,_rep,__rep)(__VA_ARGS__)
#define _overload3_rev(_1,_2,_3,name,...) name
#define _rep_rev(i,a) repi_rev(i,0,a)
#define repi_rev(i,a,b) for(long long i=(long long)(b)-1;i>=(long long)(a);--i)
#define rrep(...) _overload3_rev(__VA_ARGS__,repi_rev,_rep_rev)(__VA_ARGS__)

// #define rep(i,...) for(auto i:range(__VA_ARGS__)) 
// #define rrep(i,...) for(auto i:reversed(range(__VA_ARGS__)))
// #define repi(i,a,b) for(lint i=lint(a);i<(lint)(b);++i)
// #define rrepi(i,a,b) for(lint i=lint(b)-1;i>=lint(a);--i)
// #define irep(i) for(lint i=0;;++i)
// inline vector<long long> range(long long n){if(n<=0)return vector<long long>();vector<long long>v(n);iota(v.begin(),v.end(),0LL);return v;}
// inline vector<long long> range(long long a,long long b){if(b<=a)return vector<long long>();vector<long long>v(b-a);iota(v.begin(),v.end(),a);return v;}
// inline vector<long long> range(long long a,long long b,long long c){if((b-a+c-1)/c<=0)return vector<long long>();vector<long long>v((b-a+c-1)/c);for(int i=0;i<(int)v.size();++i)v[i]=i?v[i-1]+c:a;return v;}
// template<typename T>inline T reversed(T v){reverse(v.begin(),v.end());return v;}
#define all(n) begin(n),end(n)
template<typename T,typename E>bool chmin(T& s,const E& t){bool res=s>t;s=min<T>(s,t);return res;}
template<typename T,typename E>bool chmax(T& s,const E& t){bool res=s<t;s=max<T>(s,t);return res;}
const vector<lint> dx={1,0,-1,0,1,1,-1,-1};
const vector<lint> dy={0,1,0,-1,1,-1,1,-1};
#define SUM(v) accumulate(all(v),0LL)
template<typename T,typename ...Args>auto make_vector(T x,int arg,Args ...args){if constexpr(sizeof...(args)==0)return vector<T>(arg,x);else return vector(arg,make_vector<T>(x,args...));}
#define extrep(v,...) for(auto v:__MAKE_MAT__({__VA_ARGS__}))
#define bit(n,a) ((n>>a)&1)
vector<vector<long long>> __MAKE_MAT__(vector<long long> v){if(v.empty())return vector<vector<long long>>(1,vector<long long>());long long n=v.back();v.pop_back();vector<vector<long long>> ret;vector<vector<long long>> tmp=__MAKE_MAT__(v);for(auto e:tmp)for(long long i=0;i<n;++i){ret.push_back(e);ret.back().push_back(i);}return ret;}
using graph=vector<vector<int>>;
template<typename T>using graph_w=vector<vector<pair<int,T>>>;
template<typename T,typename E>ostream& operator<<(ostream& out,pair<T,E>v){out<<"("<<v.first<<","<<v.second<<")";return out;}
constexpr inline long long powll(long long a,long long b){long long res=1;while(b--)res*=a;return res;}
#line 5 "cpplib/math/mod_int.hpp"

/**
 * @brief ModInt
 */

template<int MOD>
struct mod_int {
    using mint=mod_int<MOD>;
    using u64 = std::uint_fast64_t;
    u64 a;
    constexpr mod_int(const long long x = 0)noexcept:a(x>=0?x%get_mod():get_mod()-(-x)%get_mod()){}
    constexpr u64 &value()noexcept{return a;}
    constexpr const u64 &value() const noexcept {return a;}
    constexpr mint operator+(const mint rhs)const noexcept{return mint(*this) += rhs;}
    constexpr mint operator-(const mint rhs)const noexcept{return mint(*this)-=rhs;}
    constexpr mint operator*(const mint rhs) const noexcept {return mint(*this) *= rhs;}
    constexpr mint operator/(const mint rhs) const noexcept {return mint(*this) /= rhs;}
    constexpr mint &operator+=(const mint rhs) noexcept {
        a += rhs.a;
        if (a >= get_mod())a -= get_mod();
        return *this;
    }
    constexpr mint &operator-=(const mint rhs) noexcept {
        if (a<rhs.a)a += get_mod();
        a -= rhs.a;
        return *this;
    }
    constexpr mint &operator*=(const mint rhs) noexcept {
        a = a * rhs.a % get_mod();
        return *this;
    }
    constexpr mint operator++(int) noexcept {
        a += 1;
        if (a >= get_mod())a -= get_mod();
        return *this;
    }
    constexpr mint operator--(int) noexcept {
        if (a<1)a += get_mod();
        a -= 1;
        return *this;
    }
    constexpr mint &operator/=(mint rhs) noexcept {
        u64 exp=get_mod()-2;
        while (exp) {
            if (exp % 2) {
                *this *= rhs;
            }
            rhs *= rhs;
            exp /= 2;
        }
        return *this;
    }
    constexpr bool operator==(mint x) noexcept {
        return a==x.a;
    }
    constexpr bool operator!=(mint x) noexcept {
        return a!=x.a;
    }
    constexpr bool operator<(mint x) noexcept {
        return a<x.a;
    }
    constexpr bool operator>(mint x) noexcept {
        return a>x.a;
    }
    constexpr bool operator<=(mint x) noexcept {
        return a<=x.a;
    }
    constexpr bool operator>=(mint x) noexcept {
        return a>=x.a;
    }
    constexpr static int root(){
        mint root = 2;
        while(root.pow((get_mod()-1)>>1).a==1)root++;
        return root.a;
    }
    constexpr mint pow(long long n)const{
        long long x=a;
        mint ret = 1;
        while(n>0) {
            if(n&1)(ret*=x);
            (x*=x)%=get_mod();
            n>>=1;
        }
        return ret;
    }
    constexpr mint inv(){
        return pow(get_mod()-2);
    }
    static std::vector<mint> fac;
    static std::vector<mint> ifac;
    static bool init;
    constexpr static int mx=10000001;
    void build()const{
        init=0;
        fac.resize(mx);
        ifac.resize(mx);
        fac[0]=1,ifac[0]=1;
        for(int i=1;i<mx;i++)fac[i]=fac[i-1]*i;
        ifac[mx-1]=fac[mx-1].inv();
        for(int i=mx-2;i>=0;i--)ifac[i]=ifac[i+1]*(i+1);
    }
    mint comb(long long b){
        if(init)build();
        if(a<0||b<0)return 0;
        if(a==0&&b==0)return 1;
        if((long long)a<b)return 0;
        return fac[a]*ifac[a-b]*ifac[b];
    }
    mint fact()const{
        if(init)build();
        return fac[a];
    }
    mint fact_inv()const{
        if(init)build();
        return ifac[a];
    }
    friend std::ostream& operator<<(std::ostream& lhs, const mint& rhs) noexcept {
        lhs << rhs.a;
        return lhs;
    }
    friend std::istream& operator>>(std::istream& lhs,mint& rhs) noexcept {
        lhs >> rhs.a;
        return lhs;
    }
    constexpr static u64 get_mod(){
        return MOD;
    }
};
template<int MOD>std::vector<mod_int<MOD>> mod_int<MOD>::fac;
template<int MOD>std::vector<mod_int<MOD>> mod_int<MOD>::ifac;
template<int MOD>bool mod_int<MOD>::init=1;
#line 3 "cpplib/math/mod_int998244353.hpp"
using mint=mod_int<998'244'353>;

/**
 * @brief ModInt(998'244'353)
 */
#line 6 "cpplib/math/FPS_base.hpp"
#include<type_traits>
#line 8 "cpplib/math/FPS_base.hpp"

/**
 * @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;
    }
    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);
    }
    
    //(*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 "cpplib/math/FPS_mint.hpp"
//#include"../util/ACL.hpp"

#include <algorithm>
#include <array>

#ifdef _MSC_VER
#include <intrin.h>
#endif

namespace atcoder {

namespace internal {

// @param n `0 <= n`
// @return minimum non-negative `x` s.t. `n <= 2**x`
int ceil_pow2(int n) {
    int x = 0;
    while ((1U << x) < (unsigned int)(n)) x++;
    return x;
}

// @param n `1 <= n`
// @return minimum non-negative `x` s.t. `(n & (1 << x)) != 0`
int bsf(unsigned int n) {
#ifdef _MSC_VER
    unsigned long index;
    _BitScanForward(&index, n);
    return index;
#else
    return __builtin_ctz(n);
#endif
}

}  // namespace internal

}  // namespace atcoder



#include <utility>

#ifdef _MSC_VER
#include <intrin.h>
#endif

namespace atcoder {

namespace internal {

// @param m `1 <= m`
// @return x mod m
constexpr long long safe_mod(long long x, long long m) {
    x %= m;
    if (x < 0) x += m;
    return x;
}

// Fast modular multiplication by barrett reduction
// Reference: https://en.wikipedia.org/wiki/Barrett_reduction
// NOTE: reconsider after Ice Lake
struct barrett {
    unsigned int _m;
    unsigned long long im;

    // @param m `1 <= m < 2^31`
    barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}

    // @return m
    unsigned int umod() const { return _m; }

    // @param a `0 <= a < m`
    // @param b `0 <= b < m`
    // @return `a * b % m`
    unsigned int mul(unsigned int a, unsigned int b) const {
        // [1] m = 1
        // a = b = im = 0, so okay

        // [2] m >= 2
        // im = ceil(2^64 / m)
        // -> im * m = 2^64 + r (0 <= r < m)
        // let z = a*b = c*m + d (0 <= c, d < m)
        // a*b * im = (c*m + d) * im = c*(im*m) + d*im = c*2^64 + c*r + d*im
        // c*r + d*im < m * m + m * im < m * m + 2^64 + m <= 2^64 + m * (m + 1) < 2^64 * 2
        // ((ab * im) >> 64) == c or c + 1
        unsigned long long z = a;
        z *= b;
#ifdef _MSC_VER
        unsigned long long x;
        _umul128(z, im, &x);
#else
        unsigned long long x =
            (unsigned long long)(((unsigned __int128)(z)*im) >> 64);
#endif
        unsigned int v = (unsigned int)(z - x * _m);
        if (_m <= v) v += _m;
        return v;
    }
};

// @param n `0 <= n`
// @param m `1 <= m`
// @return `(x ** n) % m`
constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
    if (m == 1) return 0;
    unsigned int _m = (unsigned int)(m);
    unsigned long long r = 1;
    unsigned long long y = safe_mod(x, m);
    while (n) {
        if (n & 1) r = (r * y) % _m;
        y = (y * y) % _m;
        n >>= 1;
    }
    return r;
}

// Reference:
// M. Forisek and J. Jancina,
// Fast Primality Testing for Integers That Fit into a Machine Word
// @param n `0 <= n`
constexpr bool is_prime_constexpr(int n) {
    if (n <= 1) return false;
    if (n == 2 || n == 7 || n == 61) return true;
    if (n % 2 == 0) return false;
    long long d = n - 1;
    while (d % 2 == 0) d /= 2;
    constexpr long long bases[3] = {2, 7, 61};
    for (long long a : bases) {
        long long t = d;
        long long y = pow_mod_constexpr(a, t, n);
        while (t != n - 1 && y != 1 && y != n - 1) {
            y = y * y % n;
            t <<= 1;
        }
        if (y != n - 1 && t % 2 == 0) {
            return false;
        }
    }
    return true;
}
template <int n> constexpr bool is_prime = is_prime_constexpr(n);

// @param b `1 <= b`
// @return pair(g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g
constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) {
    a = safe_mod(a, b);
    if (a == 0) return {b, 0};

    // Contracts:
    // [1] s - m0 * a = 0 (mod b)
    // [2] t - m1 * a = 0 (mod b)
    // [3] s * |m1| + t * |m0| <= b
    long long s = b, t = a;
    long long m0 = 0, m1 = 1;

    while (t) {
        long long u = s / t;
        s -= t * u;
        m0 -= m1 * u;  // |m1 * u| <= |m1| * s <= b

        // [3]:
        // (s - t * u) * |m1| + t * |m0 - m1 * u|
        // <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u)
        // = s * |m1| + t * |m0| <= b

        auto tmp = s;
        s = t;
        t = tmp;
        tmp = m0;
        m0 = m1;
        m1 = tmp;
    }
    // by [3]: |m0| <= b/g
    // by g != b: |m0| < b/g
    if (m0 < 0) m0 += b / s;
    return {s, m0};
}

// Compile time primitive root
// @param m must be prime
// @return primitive root (and minimum in now)
constexpr int primitive_root_constexpr(int m) {
    if (m == 2) return 1;
    if (m == 167772161) return 3;
    if (m == 469762049) return 3;
    if (m == 754974721) return 11;
    if (m == 998244353) return 3;
    int divs[20] = {};
    divs[0] = 2;
    int cnt = 1;
    int x = (m - 1) / 2;
    while (x % 2 == 0) x /= 2;
    for (int i = 3; (long long)(i)*i <= x; i += 2) {
        if (x % i == 0) {
            divs[cnt++] = i;
            while (x % i == 0) {
                x /= i;
            }
        }
    }
    if (x > 1) {
        divs[cnt++] = x;
    }
    for (int g = 2;; g++) {
        bool ok = true;
        for (int i = 0; i < cnt; i++) {
            if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) {
                ok = false;
                break;
            }
        }
        if (ok) return g;
    }
}
template <int m> constexpr int primitive_root = primitive_root_constexpr(m);

}  // namespace internal

}  // namespace atcoder


#include <cassert>
#include <numeric>
#include <type_traits>

namespace atcoder {

namespace internal {

#ifndef _MSC_VER
template <class T>
using is_signed_int128 =
    typename std::conditional<std::is_same<T, __int128_t>::value ||
                                  std::is_same<T, __int128>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using is_unsigned_int128 =
    typename std::conditional<std::is_same<T, __uint128_t>::value ||
                                  std::is_same<T, unsigned __int128>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using make_unsigned_int128 =
    typename std::conditional<std::is_same<T, __int128_t>::value,
                              __uint128_t,
                              unsigned __int128>;

template <class T>
using is_integral = typename std::conditional<std::is_integral<T>::value ||
                                                  is_signed_int128<T>::value ||
                                                  is_unsigned_int128<T>::value,
                                              std::true_type,
                                              std::false_type>::type;

template <class T>
using is_signed_int = typename std::conditional<(is_integral<T>::value &&
                                                 std::is_signed<T>::value) ||
                                                    is_signed_int128<T>::value,
                                                std::true_type,
                                                std::false_type>::type;

template <class T>
using is_unsigned_int =
    typename std::conditional<(is_integral<T>::value &&
                               std::is_unsigned<T>::value) ||
                                  is_unsigned_int128<T>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using to_unsigned = typename std::conditional<
    is_signed_int128<T>::value,
    make_unsigned_int128<T>,
    typename std::conditional<std::is_signed<T>::value,
                              std::make_unsigned<T>,
                              std::common_type<T>>::type>::type;

#else

template <class T> using is_integral = typename std::is_integral<T>;

template <class T>
using is_signed_int =
    typename std::conditional<is_integral<T>::value && std::is_signed<T>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using is_unsigned_int =
    typename std::conditional<is_integral<T>::value &&
                                  std::is_unsigned<T>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using to_unsigned = typename std::conditional<is_signed_int<T>::value,
                                              std::make_unsigned<T>,
                                              std::common_type<T>>::type;

#endif

template <class T>
using is_signed_int_t = std::enable_if_t<is_signed_int<T>::value>;

template <class T>
using is_unsigned_int_t = std::enable_if_t<is_unsigned_int<T>::value>;

template <class T> using to_unsigned_t = typename to_unsigned<T>::type;

}  // namespace internal

}  // namespace atcoder

#include <cassert>
#include <numeric>
#include <type_traits>

#ifdef _MSC_VER
#include <intrin.h>
#endif

namespace atcoder {

namespace internal {

struct modint_base {};
struct static_modint_base : modint_base {};

template <class T> using is_modint = std::is_base_of<modint_base, T>;
template <class T> using is_modint_t = std::enable_if_t<is_modint<T>::value>;

}  // namespace internal

template <int m, std::enable_if_t<(1 <= m)>* = nullptr>
struct static_modint : internal::static_modint_base {
    using mint = static_modint;

  public:
    static constexpr int mod() { return m; }
    static mint raw(int v) {
        mint x;
        x._v = v;
        return x;
    }

    static_modint() : _v(0) {}
    template <class T, internal::is_signed_int_t<T>* = nullptr>
    static_modint(T v) {
        long long x = (long long)(v % (long long)(umod()));
        if (x < 0) x += umod();
        _v = (unsigned int)(x);
    }
    template <class T, internal::is_unsigned_int_t<T>* = nullptr>
    static_modint(T v) {
        _v = (unsigned int)(v % umod());
    }
    static_modint(bool v) { _v = ((unsigned int)(v) % umod()); }

    unsigned int val() const { return _v; }

    mint& operator++() {
        _v++;
        if (_v == umod()) _v = 0;
        return *this;
    }
    mint& operator--() {
        if (_v == 0) _v = umod();
        _v--;
        return *this;
    }
    mint operator++(int) {
        mint result = *this;
        ++*this;
        return result;
    }
    mint operator--(int) {
        mint result = *this;
        --*this;
        return result;
    }

    mint& operator+=(const mint& rhs) {
        _v += rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    mint& operator-=(const mint& rhs) {
        _v -= rhs._v;
        if (_v >= umod()) _v += umod();
        return *this;
    }
    mint& operator*=(const mint& rhs) {
        unsigned long long z = _v;
        z *= rhs._v;
        _v = (unsigned int)(z % umod());
        return *this;
    }
    mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }

    mint operator+() const { return *this; }
    mint operator-() const { return mint() - *this; }

    mint pow(long long n) const {
        assert(0 <= n);
        mint x = *this, r = 1;
        while (n) {
            if (n & 1) r *= x;
            x *= x;
            n >>= 1;
        }
        return r;
    }
    mint inv() const {
        if (prime) {
            assert(_v);
            return pow(umod() - 2);
        } else {
            auto eg = internal::inv_gcd(_v, m);
            assert(eg.first == 1);
            return eg.second;
        }
    }

    friend mint operator+(const mint& lhs, const mint& rhs) {
        return mint(lhs) += rhs;
    }
    friend mint operator-(const mint& lhs, const mint& rhs) {
        return mint(lhs) -= rhs;
    }
    friend mint operator*(const mint& lhs, const mint& rhs) {
        return mint(lhs) *= rhs;
    }
    friend mint operator/(const mint& lhs, const mint& rhs) {
        return mint(lhs) /= rhs;
    }
    friend bool operator==(const mint& lhs, const mint& rhs) {
        return lhs._v == rhs._v;
    }
    friend bool operator!=(const mint& lhs, const mint& rhs) {
        return lhs._v != rhs._v;
    }

  private:
    unsigned int _v;
    static constexpr unsigned int umod() { return m; }
    static constexpr bool prime = internal::is_prime<m>;
};

template <int id> struct dynamic_modint : internal::modint_base {
    using mint = dynamic_modint;

  public:
    static int mod() { return (int)(bt.umod()); }
    static void set_mod(int m) {
        assert(1 <= m);
        bt = internal::barrett(m);
    }
    static mint raw(int v) {
        mint x;
        x._v = v;
        return x;
    }

    dynamic_modint() : _v(0) {}
    template <class T, internal::is_signed_int_t<T>* = nullptr>
    dynamic_modint(T v) {
        long long x = (long long)(v % (long long)(mod()));
        if (x < 0) x += mod();
        _v = (unsigned int)(x);
    }
    template <class T, internal::is_unsigned_int_t<T>* = nullptr>
    dynamic_modint(T v) {
        _v = (unsigned int)(v % mod());
    }
    dynamic_modint(bool v) { _v = ((unsigned int)(v) % mod()); }

    unsigned int val() const { return _v; }

    mint& operator++() {
        _v++;
        if (_v == umod()) _v = 0;
        return *this;
    }
    mint& operator--() {
        if (_v == 0) _v = umod();
        _v--;
        return *this;
    }
    mint operator++(int) {
        mint result = *this;
        ++*this;
        return result;
    }
    mint operator--(int) {
        mint result = *this;
        --*this;
        return result;
    }

    mint& operator+=(const mint& rhs) {
        _v += rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    mint& operator-=(const mint& rhs) {
        _v += mod() - rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    mint& operator*=(const mint& rhs) {
        _v = bt.mul(_v, rhs._v);
        return *this;
    }
    mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }

    mint operator+() const { return *this; }
    mint operator-() const { return mint() - *this; }

    mint pow(long long n) const {
        assert(0 <= n);
        mint x = *this, r = 1;
        while (n) {
            if (n & 1) r *= x;
            x *= x;
            n >>= 1;
        }
        return r;
    }
    mint inv() const {
        auto eg = internal::inv_gcd(_v, mod());
        assert(eg.first == 1);
        return eg.second;
    }

    friend mint operator+(const mint& lhs, const mint& rhs) {
        return mint(lhs) += rhs;
    }
    friend mint operator-(const mint& lhs, const mint& rhs) {
        return mint(lhs) -= rhs;
    }
    friend mint operator*(const mint& lhs, const mint& rhs) {
        return mint(lhs) *= rhs;
    }
    friend mint operator/(const mint& lhs, const mint& rhs) {
        return mint(lhs) /= rhs;
    }
    friend bool operator==(const mint& lhs, const mint& rhs) {
        return lhs._v == rhs._v;
    }
    friend bool operator!=(const mint& lhs, const mint& rhs) {
        return lhs._v != rhs._v;
    }

  private:
    unsigned int _v;
    static internal::barrett bt;
    static unsigned int umod() { return bt.umod(); }
};
template <int id> internal::barrett dynamic_modint<id>::bt = 998244353;

using modint998244353 = static_modint<998244353>;
using modint1000000007 = static_modint<1000000007>;
using modint = dynamic_modint<-1>;

namespace internal {

template <class T>
using is_static_modint = std::is_base_of<internal::static_modint_base, T>;

template <class T>
using is_static_modint_t = std::enable_if_t<is_static_modint<T>::value>;

template <class> struct is_dynamic_modint : public std::false_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint<id>> : public std::true_type {};

template <class T>
using is_dynamic_modint_t = std::enable_if_t<is_dynamic_modint<T>::value>;

}  // namespace internal

}  // namespace atcoder

#include <cassert>
#include <type_traits>
#include <vector>

namespace atcoder {

namespace internal {

template <class mint, internal::is_static_modint_t<mint>* = nullptr>
void butterfly(std::vector<mint>& a) {
    static constexpr int g = internal::primitive_root<mint::mod()>;
    int n = int(a.size());
    int h = internal::ceil_pow2(n);

    static bool first = true;
    static mint sum_e[30];  // sum_e[i] = ies[0] * ... * ies[i - 1] * es[i]
    if (first) {
        first = false;
        mint es[30], ies[30];  // es[i]^(2^(2+i)) == 1
        int cnt2 = bsf(mint::mod() - 1);
        mint e = mint(g).pow((mint::mod() - 1) >> cnt2), ie = e.inv();
        for (int i = cnt2; i >= 2; i--) {
            // e^(2^i) == 1
            es[i - 2] = e;
            ies[i - 2] = ie;
            e *= e;
            ie *= ie;
        }
        mint now = 1;
        for (int i = 0; i <= cnt2 - 2; i++) {
            sum_e[i] = es[i] * now;
            now *= ies[i];
        }
    }
    for (int ph = 1; ph <= h; ph++) {
        int w = 1 << (ph - 1), p = 1 << (h - ph);
        mint now = 1;
        for (int s = 0; s < w; s++) {
            int offset = s << (h - ph + 1);
            for (int i = 0; i < p; i++) {
                auto l = a[i + offset];
                auto r = a[i + offset + p] * now;
                a[i + offset] = l + r;
                a[i + offset + p] = l - r;
            }
            now *= sum_e[bsf(~(unsigned int)(s))];
        }
    }
}

template <class mint, internal::is_static_modint_t<mint>* = nullptr>
void butterfly_inv(std::vector<mint>& a) {
    static constexpr int g = internal::primitive_root<mint::mod()>;
    int n = int(a.size());
    int h = internal::ceil_pow2(n);

    static bool first = true;
    static mint sum_ie[30];  // sum_ie[i] = es[0] * ... * es[i - 1] * ies[i]
    if (first) {
        first = false;
        mint es[30], ies[30];  // es[i]^(2^(2+i)) == 1
        int cnt2 = bsf(mint::mod() - 1);
        mint e = mint(g).pow((mint::mod() - 1) >> cnt2), ie = e.inv();
        for (int i = cnt2; i >= 2; i--) {
            // e^(2^i) == 1
            es[i - 2] = e;
            ies[i - 2] = ie;
            e *= e;
            ie *= ie;
        }
        mint now = 1;
        for (int i = 0; i <= cnt2 - 2; i++) {
            sum_ie[i] = ies[i] * now;
            now *= es[i];
        }
    }

    for (int ph = h; ph >= 1; ph--) {
        int w = 1 << (ph - 1), p = 1 << (h - ph);
        mint inow = 1;
        for (int s = 0; s < w; s++) {
            int offset = s << (h - ph + 1);
            for (int i = 0; i < p; i++) {
                auto l = a[i + offset];
                auto r = a[i + offset + p];
                a[i + offset] = l + r;
                a[i + offset + p] =
                    (unsigned long long)(mint::mod() + l.val() - r.val()) *
                    inow.val();
            }
            inow *= sum_ie[bsf(~(unsigned int)(s))];
        }
    }
}

}  // namespace internal

template <class mint, internal::is_static_modint_t<mint>* = nullptr>
std::vector<mint> convolution(std::vector<mint> a, std::vector<mint> b) {
    int n = int(a.size()), m = int(b.size());
    if (!n || !m) return {};
    if (std::min(n, m) <= 60) {
        if (n < m) {
            std::swap(n, m);
            std::swap(a, b);
        }
        std::vector<mint> ans(n + m - 1);
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) {
                ans[i + j] += a[i] * b[j];
            }
        }
        return ans;
    }
    int z = 1 << internal::ceil_pow2(n + m - 1);
    a.resize(z);
    internal::butterfly(a);
    b.resize(z);
    internal::butterfly(b);
    for (int i = 0; i < z; i++) {
        a[i] *= b[i];
    }
    internal::butterfly_inv(a);
    a.resize(n + m - 1);
    mint iz = mint(z).inv();
    for (int i = 0; i < n + m - 1; i++) a[i] *= iz;
    return a;
}

template <unsigned int mod = 998244353,
          class T,
          std::enable_if_t<internal::is_integral<T>::value>* = nullptr>
std::vector<T> convolution(const std::vector<T>& a, const std::vector<T>& b) {
    int n = int(a.size()), m = int(b.size());
    if (!n || !m) return {};

    using mint = static_modint<mod>;
    std::vector<mint> a2(n), b2(m);
    for (int i = 0; i < n; i++) {
        a2[i] = mint(a[i]);
    }
    for (int i = 0; i < m; i++) {
        b2[i] = mint(b[i]);
    }
    auto c2 = convolution(move(a2), move(b2));
    std::vector<T> c(n + m - 1);
    for (int i = 0; i < n + m - 1; i++) {
        c[i] = c2[i].val();
    }
    return c;
}

std::vector<long long> convolution_ll(const std::vector<long long>& a,
                                      const std::vector<long long>& b) {
    int n = int(a.size()), m = int(b.size());
    if (!n || !m) return {};

    static constexpr unsigned long long MOD1 = 754974721;  // 2^24
    static constexpr unsigned long long MOD2 = 167772161;  // 2^25
    static constexpr unsigned long long MOD3 = 469762049;  // 2^26
    static constexpr unsigned long long M2M3 = MOD2 * MOD3;
    static constexpr unsigned long long M1M3 = MOD1 * MOD3;
    static constexpr unsigned long long M1M2 = MOD1 * MOD2;
    static constexpr unsigned long long M1M2M3 = MOD1 * MOD2 * MOD3;

    static constexpr unsigned long long i1 =
        internal::inv_gcd(MOD2 * MOD3, MOD1).second;
    static constexpr unsigned long long i2 =
        internal::inv_gcd(MOD1 * MOD3, MOD2).second;
    static constexpr unsigned long long i3 =
        internal::inv_gcd(MOD1 * MOD2, MOD3).second;

    auto c1 = convolution<MOD1>(a, b);
    auto c2 = convolution<MOD2>(a, b);
    auto c3 = convolution<MOD3>(a, b);

    std::vector<long long> c(n + m - 1);
    for (int i = 0; i < n + m - 1; i++) {
        unsigned long long x = 0;
        x += (c1[i] * i1) % MOD1 * M2M3;
        x += (c2[i] * i2) % MOD2 * M1M3;
        x += (c3[i] * i3) % MOD3 * M1M2;
        // B = 2^63, -B <= x, r(real value) < B
        // (x, x - M, x - 2M, or x - 3M) = r (mod 2B)
        // r = c1[i] (mod MOD1)
        // focus on MOD1
        // r = x, x - M', x - 2M', x - 3M' (M' = M % 2^64) (mod 2B)
        // r = x,
        //     x - M' + (0 or 2B),
        //     x - 2M' + (0, 2B or 4B),
        //     x - 3M' + (0, 2B, 4B or 6B) (without mod!)
        // (r - x) = 0, (0)
        //           - M' + (0 or 2B), (1)
        //           -2M' + (0 or 2B or 4B), (2)
        //           -3M' + (0 or 2B or 4B or 6B) (3) (mod MOD1)
        // we checked that
        //   ((1) mod MOD1) mod 5 = 2
        //   ((2) mod MOD1) mod 5 = 3
        //   ((3) mod MOD1) mod 5 = 4
        long long diff =
            c1[i] - internal::safe_mod((long long)(x), (long long)(MOD1));
        if (diff < 0) diff += MOD1;
        static constexpr unsigned long long offset[5] = {
            0, 0, M1M2M3, 2 * M1M2M3, 3 * M1M2M3};
        x -= offset[diff % 5];
        c[i] = x;
    }

    return c;
}

}  // namespace atcoder

#line 1 "cpplib/math/ceil_pow2.hpp"
int ceil_pow2(int n) {
    int x = 0;
    while ((1U << x) < (unsigned int)(n)) x++;
    return x;
}
#line 1 "cpplib/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 "cpplib/math/garner.hpp"

/**
 * 
 * @brief ガーナーのアルゴリズム
 *
 */

long long garner(std::vector<long long>a,std::vector<long long>mods){
    const int sz=3;
    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[3];
}
#line 7 "cpplib/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,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 4 "code.cpp"

int main(){
    lint n;
    cin>>n;
    fps<mint>b(n*2+2);
    rep(i,1,2*n+2){
        b[i]=mint(i).fact_inv();
    }
    b=(b>>1).inv();
    rep(i,2*n+1)b[i]*=mint(i).fact();
    vector<mint>ans(n*2+1);
    mint tmp=mint(1)/2;
    rep(i,1,n+1){
        ans[i*2]=b[i*2]*mint(2).pow(2*i)*tmp*mint(2*i).fact_inv()*(i%2?1:-1)*mint(2).pow(i-1);
    }
    output(ans);
}