#include <bits/stdc++.h>
using namespace std;

//入力が必ず-mod<a<modの時.
template<const int mod>
struct modint{ //mod変更が不可能.
    public:
    long long v = 0;
    static void setmod(int m){} //飾り.
    static constexpr long long getmod(){return mod;}
    modint(){v = 0;} modint(int a){v = a<0?a+mod:a;} modint(long long a){v = a<0?a+mod:a;}
    modint(unsigned int a){v = a;} modint(unsigned long long a){v = a;}
    long long val()const{return v;}
 
    modint &operator=(const modint &b) = default;
    modint operator+()const{return (*this);}
    modint operator-()const{return modint(0)-(*this);}
    modint operator+(const modint b)const{return modint(v)+=b;}
    modint operator-(const modint b)const{return modint(v)-=b;}
    modint operator*(const modint b)const{return modint(v)*=b;}
    modint operator/(const modint b)const{return modint(v)/=b;}
    modint operator+=(const modint b){
        v += b.v; if(v >= mod) v -= mod;
        return *this;
    }
    modint operator-=(const modint b){
        v -= b.v; if(v < 0) v += mod; 
        return *this;
    }   
    modint operator*=(const modint b){v = v*b.v%mod; return *this;}
    modint operator/=(modint b){ //b!=0 mod素数が必須.
        if(b == 0) assert(false);
        int left = mod-2;
        while(left){if(left&1) *this *= b; b *= b; left >>= 1;}
        return *this;
    }
    modint operator++(){*this += 1; return *this;}
    modint operator--(){*this -= 1; return *this;}
    modint operator++(int){*this += 1; return *this;}
    modint operator--(int){*this -= 1; return *this;}
    bool operator==(const modint b)const{return v == b.v;}
    bool operator!=(const modint b)const{return v != b.v;}
    bool operator>(const modint b)const{return v > b.v;}
    bool operator>=(const modint b)const{return v >= b.v;}
    bool operator<(const modint b)const{return v < b.v;}
    bool operator<=(const modint b)const{return v <= b.v;}
    modint pow(long long n)const{
        modint ret = 1,p = v;
        if(n < 0) p = p.inv(),n = -n;
        while(n){
            if(n&1) ret *= p;
            p *= p; n >>= 1;
        }
        return ret;
    }
    modint inv()const{return modint(1)/v;} //素数mod必須.
};
 
template<int idx> //modが入力で与えられる場合. 
struct dynamic_modint{ //mod変更が可能 最初にsetmod必須 idxで複数個所持が可能.
    private:
    static int mod;
    public:
    long long v = 0;
    static constexpr long long getmod(){return mod;}
    static void setmod(int m){
        assert(m > 0);
        mod = m;
    }
    dynamic_modint(){v = 0;}
    dynamic_modint(int a){v = a<0?a+mod:a;} dynamic_modint(long long a){v = a<0?a+mod:a;}
    dynamic_modint(unsigned int a){v = a;}
    dynamic_modint(unsigned long long a){v = a;}
    long long val()const{return v;}
 
    dynamic_modint &operator=(const dynamic_modint &b) = default;
    dynamic_modint operator+()const{return (*this);}
    dynamic_modint operator-()const{return dynamic_modint(0)-(*this);}
    dynamic_modint operator+(const dynamic_modint b)const{return dynamic_modint(v)+=b;}
    dynamic_modint operator-(const dynamic_modint b)const{return dynamic_modint(v)-=b;}
    dynamic_modint operator*(const dynamic_modint b)const{return dynamic_modint(v)*=b;}
    dynamic_modint operator/(const dynamic_modint b)const{return dynamic_modint(v)/=b;}
    dynamic_modint operator+=(const dynamic_modint b){
        v += b.v; if(v >= mod) v -= mod;
        return *this;
    }
    dynamic_modint operator-=(const dynamic_modint b){
        v -= b.v; if(v < 0) v += mod; 
        return *this;
    }   
    dynamic_modint operator*=(const dynamic_modint b){v = v*b.v%mod; return *this;}
    dynamic_modint operator/=(dynamic_modint b){ //b!=0 mod素数が必須.
        if(b == 0) assert(false);
        int left = mod-2;
        while(left){if(left&1) *this *= b; b *= b; left >>= 1;}
        return *this;
    }
    dynamic_modint operator++(){*this += 1; return *this;}
    dynamic_modint operator--(){*this -= 1; return *this;}
    dynamic_modint operator++(int){*this += 1; return *this;}
    dynamic_modint operator--(int){*this -= 1; return *this;}
    bool operator==(const dynamic_modint b)const{return v == b.v;}
    bool operator!=(const dynamic_modint b)const{return v != b.v;}
    bool operator>(const dynamic_modint b)const{return v > b.v;}
    bool operator>=(const dynamic_modint b)const{return v >= b.v;}
    bool operator<(const dynamic_modint b)const{return v < b.v;}
    bool operator<=(const dynamic_modint b)const{return v <= b.v;}
    dynamic_modint pow(long long n)const{
        dynamic_modint ret = 1,p = v;
        if(n < 0) p = p.inv(),n = -n;
        while(n){
            if(n&1) ret *= p;
            p *= p; n >>= 1;
        }
        return ret;
    }
    dynamic_modint inv()const{return dynamic_modint(1)/v;} //素数mod必須.
};
template<int idx> int dynamic_modint<idx>::mod=998244353;
using mint = modint<1234567891>;
//using mint = modint<1000000007>;
//using mint = dynamic_modint<0>;

namespace to_fold{
__int128_t safemod(__int128_t a,long long m){a %= m; if(a < 0) a += m; return a;}
pair<long long,long long> invgcd(long long a,long long b){
    //return {gcd(a,b),x} (xa≡g(mod b))
    a = safemod(a,b);
    if(a == 0) return {b,0};
    long long x = 0,y = 1,memob = b;
    while(a){
        long long q = b/a;
        b -= a*q;
        swap(x,y); y -= q*x;
        swap(a,b);
    }
    if(x < 0) x += memob/b;
    return {b,x};
}
template<long long mod>
long long Garner(const vector<long long> &A,const vector<long long> &M){
    __int128_t mulM = 1,x = A.at(0)%M.at(0); //Mの要素のペア互いに素必須.
    for(int i=1; i<A.size(); i++){
        //assert(gcd(mulM,M.at(i-1)) == 1);
        mulM *= M.at(i-1); //2乗がオーバーフローする時__int128_t
        long long t = safemod((A.at(i)-x)*invgcd(mulM,M.at(i)).second,M.at(i));
        x += t*mulM;
    }
    return x%mod;
}
int countzero(unsigned long long x){
    if(x == 0) return 64;
    else return __popcount((x&-x)-1);
}
template<typename mint>
struct fftinfo{
    static bool First;
    static mint g,sum_e[30],sum_ie[30]; //sum_e[i]=Π[j=0~i-1]ies[j] * es[i],sum_ie[i]=Π[i=0~j-1]es[j] * ies[i].
    static mint divpow2[30]; //div[i] = 1/(2^i).
    static mint Zeta[30];
    fftinfo(){
        if(!First) return;
        First = false;
        const long long mod = mint::getmod();
        if(mod == 998244353) g = 3;
        else if(mod == 754974721) g = 11;
        else if(mod == 167772161) g = 3;
        else if(mod == 469762049) g = 3;
        else assert(false); //現状RE.
        mint es[30],ies[30]; //es[i]^(2^(2+i))=1.
        int cnt2 = countzero(mod-1);
        mint e = g.pow((mod-1)>>cnt2),ie = e.inv();
        for(int i=cnt2; i>=2; i--){ //e^(2^i)=1;
            es[i-2] = e,e *= e;
            ies[i-2] = ie,ie *= ie;
        }
        mint rot = 1;
        for(int i=0; i<=cnt2-2; i++) sum_e[i] = es[i]*rot,rot *= ies[i];
        rot = 1;
        for(int i=0; i<=cnt2-2; i++) sum_ie[i] = ies[i]*rot,rot *= es[i];
        mint div2n = 1,div2 = mint(1)/2;
        for(int i=0; i<30; i++) divpow2[i] = div2n,div2n *= div2;
        for(int i=0; i<=cnt2; i++) Zeta[i] = g.pow((mod-1)/(2<<i));
    }
};
template<typename mint> bool fftinfo<mint>::First=true;
template<typename mint> mint fftinfo<mint>::g;
template<typename mint> mint fftinfo<mint>::sum_e[30];
template<typename mint> mint fftinfo<mint>::sum_ie[30];
template<typename mint> mint fftinfo<mint>::divpow2[30];
template<typename mint> mint fftinfo<mint>::Zeta[30];
template<typename mint>
void NTT(vector<mint> &A){ //ACLを超参考にしてる.
    int n = A.size();
    assert((n&-n) == n);
    fftinfo<mint> info;
    int h = countzero(n);
    for(int ph=1; ph<=h; ph++){
        int w = 1<<(ph-1),p = 1<<(h-ph);
        mint rot = 1;
        for(int s=0; s<w; s++){
            int offset = s<<(h-ph+1);
            for(int i=0; i<p; i++){
                mint l = A.at(i+offset),r = A.at(i+offset+p)*rot;
                A.at(i+offset) = l+r;
                A.at(i+offset+p) = l-r;
            }
            rot *= info.sum_e[countzero(~(unsigned int)(s))];
        }
    }
}
template<typename mint>
void INTT(vector<mint> &A){
    int n = A.size();
    assert((n&-n) == n);
    fftinfo<mint> info;
    const unsigned int mod = mint::getmod();
    int h = countzero(n);
    for(int ph=h; ph>0; ph--){
        int w = 1<<(ph-1),p = 1<<(h-ph);
        mint irot = 1;
        for(int s=0; s<w; s++){
            int offset = s<<(h-ph+1);
            for(int i=0; i<p; i++){
                mint l = A.at(i+offset),r = A.at(i+offset+p);
                A.at(i+offset) = l+r;
                A.at(i+offset+p) = ((unsigned long long)(mod+(unsigned int)l.v-(unsigned int)r.v)*irot.v)%mod;
            }
            irot *= info.sum_ie[countzero(~(unsigned int)(s))];
        }
    }
    mint divn = info.divpow2[h];
    for(auto &a : A) a *= divn;
}
template<typename mint>
vector<mint> convolution(vector<mint> A,vector<mint> B){ //mintじゃないのを突っ込まないように!!!.
    int siza = A.size(),sizb = B.size(),sizc = siza+sizb-1,N = 1;
    if(siza == 0 || sizb == 0) return {};
    if(min(siza,sizb) <= 60){ //naive.
        vector<mint> ret(sizc);
        if(siza >= sizb){for(int i=0; i<siza; i++) for(int k=0; k<sizb; k++) ret.at(i+k) += A.at(i)*B.at(k);}
        else{for(int i=0; i<sizb; i++) for(int k=0; k<siza; k++) ret.at(i+k) += B.at(i)*A.at(k);}
        return ret;
    }
    while(N < sizc) N <<= 1;
    A.resize(N),B.resize(N);
    NTT(A); NTT(B);
    for(int i=0; i<N; i++) A.at(i) *= B.at(i);
    INTT(A); A.resize(sizc);
    return A;
}
vector<long long> convolution_ll(const vector<long long> &A,const vector<long long> &B){ //long longに収まる範囲.
    int siza = A.size(),sizb = B.size(),sizc = siza+sizb-1;
    if(siza == 0 || sizb == 0) return {};
    vector<long long> ret(sizc);
    if(min(siza,sizb) <= 200){ //naive 200はやばい?.
        vector<long long> ret(sizc);
        if(siza >= sizb){for(int i=0; i<siza; i++) for(int k=0; k<sizb; k++) ret.at(i+k) += A.at(i)*B.at(k);}
        else{for(int i=0; i<sizb; i++) for(int k=0; k<siza; k++) ret.at(i+k) += B.at(i)*A.at(k);}
        return ret;
    }
    const unsigned long long mod1 = 754974721,mod2 = 167772161,mod3 = 469762049;
    const unsigned long long m1m2 = mod1*mod2,m2m3 = mod2*mod3,m3m1 = mod3*mod1,m1m2m3 = mod1*mod2*mod3;
    const unsigned long long i1 = invgcd(m2m3,mod1).second,i2 = invgcd(m3m1,mod2).second,i3 = invgcd(m1m2,mod3).second;
    assert(sizc <= (1<<24));
    using mint1 = modint<mod1>;
    using mint2 = modint<mod2>;
    using mint3 = modint<mod3>;
    vector<mint1> a1(siza),b1(sizb);
    vector<mint2> a2(siza),b2(sizb);
    vector<mint3> a3(siza),b3(sizb);
    for(int i=0; i<siza; i++) a1.at(i) = A.at(i)%mod1;
    for(int i=0; i<sizb; i++) b1.at(i) = B.at(i)%mod1;
    vector<mint1> C1 = convolution(a1,b1);
    for(int i=0; i<siza; i++) a2.at(i) = A.at(i)%mod2;
    for(int i=0; i<sizb; i++) b2.at(i) = B.at(i)%mod2;
    vector<mint2> C2 = convolution(a2,b2);
    for(int i=0; i<siza; i++) a3.at(i) = A.at(i)%mod3;
    for(int i=0; i<sizb; i++) b3.at(i) = B.at(i)%mod3;
    vector<mint3> C3 = convolution(a3,b3);
    vector<unsigned long long> offset = {0,0,m1m2m3,2*m1m2m3,3*m1m2m3};
    for(int i=0; i<sizc; i++){
        unsigned long long x = 0;
        x += (C1.at(i).v*i1)%mod1*m2m3;
        x += (C2.at(i).v*i2)%mod2*m3m1;
        x += (C3.at(i).v*i3)%mod3*m1m2;
        long long diff = C1.at(i).v-((long long)x+(long long)mod1)%mod1;
        if(diff < 0) diff += mod1;
        x -= offset.at(diff%5);
        ret.at(i) = x;
    }
    return ret;
}
template<typename mint>
vector<mint> convolution_llmod(const vector<mint> &A,const vector<mint> &B){
    int siza = A.size(),sizb = B.size(),sizc = siza+sizb-1;
    if(siza == 0 || sizb == 0) return {};
    vector<mint> ret(sizc);
    if(min(siza,sizb) <= 200){
        for(int i=0; i<siza; i++) for(int k=0; k<sizb; k++) ret.at(i+k) += A.at(i)*B.at(k);
        return ret;
    }
    const long long mod1 = 754974721,mod2 = 167772161,mod3 = 469762049;
    assert(sizc <= (1<<24));
    using mint1 = modint<mod1>;
    using mint2 = modint<mod2>;
    using mint3 = modint<mod3>;
    vector<mint1> a1(siza),b1(sizb);
    vector<mint2> a2(siza),b2(sizb);
    vector<mint3> a3(siza),b3(sizb);
    for(int i=0; i<siza; i++) a1.at(i) = A.at(i).v%mod1;
    for(int i=0; i<sizb; i++) b1.at(i) = B.at(i).v%mod1;
    vector<mint1> C1 = convolution(a1,b1);
    for(int i=0; i<siza; i++) a2.at(i) = A.at(i).v%mod2;
    for(int i=0; i<sizb; i++) b2.at(i) = B.at(i).v%mod2;
    vector<mint2> C2 = convolution(a2,b2);
    for(int i=0; i<siza; i++) a3.at(i) = A.at(i).v%mod3;
    for(int i=0; i<sizb; i++) b3.at(i) = B.at(i).v%mod3;
    vector<mint3> C3 = convolution(a3,b3);
    for(int i=0; i<sizc; i++){
        vector<long long> A = {C1.at(i).v,C2.at(i).v,C3.at(i).v};
        vector<long long> M = {mod1,mod2,mod3};
        ret.at(i) = Garner<mint::getmod()>(A,M);
    }
    return ret;
}
vector<int> convolution_int(const vector<int> &A,const vector<int> &B){ //intに収まる範囲.
    if(A.size() == 0 || B.size() == 0) return {};
    vector<int> ret;
    if(min(A.size(),B.size()) <= 60){
        ret.resize(A.size()+B.size()-1);
        for(int i=0; i<A.size(); i++) for(int k=0; k<B.size(); k++) ret.at(i+k) += A.at(i)*B.at(k);
    }
    else{
        using mint1 = modint<998244353>;
        vector<mint1> X(A.size()),Y(B.size()),Z;
        for(int i=0; i<A.size(); i++) X.at(i) = A.at(i);
        for(int i=0; i<B.size(); i++) Y.at(i) = B.at(i);
        Z = convolution(X,Y);
        ret.resize(Z.size());
        for(int i=0; i<Z.size(); i++) ret.at(i) = Z.at(i).v;
    }
    return ret;
}
template<typename mint> 
void NTTdoubling(vector<mint> &A){ //NTTの原理を忘れているため何やってるのか意味が分からない NTT-friendly専用.
        //INTT->resize(2倍）->NTTの代わりにcopy->INTT->謎の操作->NTT->push sizeが小さい時は効率悪いらしいよ.
        int n = A.size();
        fftinfo<mint> info;
        vector<mint> B = A;
        INTT(B);
        mint rot = 1,zeta = info.Zeta[countzero(n)];
        for(auto &v : B) v *= rot,rot *= zeta;
        NTT(B); A.reserve(n<<1);
        for(auto &v : B) A.push_back(v); 
}
bool isNTTfriendly(long long mod){
    if(mod == 998244353 || mod == 754974721 || mod == 16777216 || mod == 469762049) return true;
    return false; //現状false 原子根求める機能を追加してから.
    int have2 = countzero(mod-1);
    return have2 >= 20;//とりあえず2^20でokとする;
}
}
using namespace to_fold;

template<typename T> //実質mintだけ?.
struct FormalPowerSeries:vector<T>{ //NTT-friendly素数だけ じゃなくてもいいけど全部書き直せ!.
    using vector<T>::vector;
    using fps = FormalPowerSeries;

    FormalPowerSeries(const vector<mint> &A){(*this) = A;}
    //重要なところは某のほぼパクリ.
    fps operator++(){*this += 1; return *this;}
    fps operator--(){*this -= 1; return *this;}
    fps operator++(int){*this += 1; return *this;}
    fps operator--(int){*this -= 1; return *this;}
    fps operator+(const fps &b) const {return fps(*this)+=b;}
    fps operator+(const T &b) const {return fps(*this)+=b;}
    fps operator-(const fps &b) const {return fps(*this)-=b;}
    fps operator-(const T &b) const {return fps(*this)-=b;}
    fps operator*(const fps &b){return fps(*this)*=b;}
    fps operator*(const T &b) const {return fps(*this)*=b;}
    fps operator/(const fps &b) const {return fps(*this)/=b;}
    fps operator%(const fps &b) const {return fps(*this)%=b;}
    fps operator>>(const unsigned int b) const {return fps(*this)>>=b;}
    fps operator<<(const unsigned int b) const {return fps(*this)<<=b;}
    fps operator-()const{ //-1倍;
        fps ret = (*this);
        for(auto &v : ret) v = -v;
        return ret;
    }
    bool operator==(const fps &b)const{
        if((*this).size() != b.size()) return false;
        for(int i=0; i<(*this).size(); i++) if((*this).at(i) != b.at(i)) return false;
        return true;
    }
    bool operator!=(const fps &b)const{return !((*this)==b);}
 
    fps &operator+=(const fps &b){ //Cix^i = (Ai+Bi)x^i. O(n).
        if((*this).size() < b.size()) (*this).resize(b.size(),0);
        for(int i=0; i<b.size(); i++) (*this).at(i) += b.at(i);
        return *this;
    }
    fps &operator+=(const T &b){ //x^0の係数に+b O(1).
        if((*this).size() == 0) (*this).resize(1);
        (*this).at(0) += b;
        return *this;
    }
    fps &operator-=(const fps &b){ //Cix^i = (Ai-Bi)x^i. O(n).
        int n = b.size();
        if((*this).size() < n) (*this).resize(n,0);
        for(int i=0; i<n; i++) (*this).at(i) -= b.at(i);
        return *this;
    }
    fps &operator-=(const T &b){ //x^0の係数に-b O(1).
        if((*this).size() == 0) (*this).resize(1);
        (*this).at(0) -= b;
        return *this;
    }
    fps &operator*=(const fps &b){ //C[i+k]x^(i+k) = Aix^i+Bkx^k. O(nlogn).
        vector<T> re;
        if(isNTTfriendly(mint::getmod())) re = convolution((*this),b); //NTT-friendlyならok 現在は4種以外認めない.
        else re = convolution_llmod((*this),b);
        (*this).resize(re.size());
        for(int i=0; i<re.size(); i++) (*this).at(i) = re.at(i);
        return *this;
    }
    fps &operator*=(const T &b){ //x^iの係数に*b O(n).
        for(auto &v : (*this)) v *= b; 
        return *this;
    }
    fps &operator/=(const T &b){ //x^iの係数に/b O(logmod+n).
        T mul = b.inv();
        for(auto &v : (*this)) v *= mul;
        return *this;
    }
    fps &operator>>=(const unsigned int &b){//b<0は対象外. 先頭b項を削除. O(n)
        if((*this).size() <= b) (*this).clear();
        else (*this).erase((*this).begin(),(*this).begin()+b);
        return *this;
    }    
    fps &operator<<=(const unsigned int &b){//b<0は対象外. 先頭b項に0を挿入. O(n)
        (*this).insert((*this).begin(),b,0);
        return *this;
    }   
    fps &operator%=(const fps &b){ //多項式の余り. O(nlogn)
        (*this) -= (*this)/b*b; del0();
        return (*this);
    }
    fps &operator/=(const fps &b){ //多項式としての除算 O(nlogn).
        assert(b.size() > 0); //分母の末尾0は駄目.
        T check = b.back(); assert(check != 0);
        del0(); //分子の末尾0は消して許容.
        if((*this).size() < b.size()){
            (*this).clear();
            return *this;
        }
        int n = (*this).size()-b.size()+1;
        if(b.size() <= 64){ //愚直.
            fps G(b);
            assert(G.size() > 0);
            T div = G.back().inv();
            for (auto &v : G) v *= div;
            int deg = (*this).size()-G.size()+1;
            fps Q(deg);
            for(int i=deg-1; i>=0; i--){
                Q[i] = (*this).at(i+G.size()-1);
                for(int k=0; k<G.size(); k++) (*this).at(i+k) -= Q.at(i)*G.at(k);
            }
            (*this) = Q*div; (*this).resize(n,0);
            return *this;
        }
        //rev(f)/rev(g)≡rev(ret)(mod x^n)らしい わお.
        return (*this) = ((*this).rev().prefix(n) * b.rev().inv(n)).prefix(n).rev();
    }
 
    fps multi_sparse(const fps &b)const{ //非0が少ない時の掛け算 愚直 O(n*|非0|+m).
        int n = (*this).size(),m = b.size();
        vector<pair<int,T>> P;
        for(int i=0; i<m; i++) if(b.at(i) != 0) P.push_back({i,b.at(i)});
        fps ret(n+m-1);
        for(int i=0; i<n; i++) for(auto [k,v] : P) ret.at(i+k) += (*this).at(i)*v;
        return ret;
    };
    fps inv_sparse(int deg=-1)const{ //非0が少ない時の1/fを返す O(deg*|非0|+m+logmod).
        int n = (*this).size();
        if(deg == -1) deg = n;
        assert((*this).at(0) != 0);
        T div = 1;
        vector<pair<int,T>> P;
        if((*this).at(0) != 1) div = T((*this).at(0)).inv();
        for(int i=1; i<n; i++)  if((*this).at(i) != 0) P.emplace_back(pair{i,(*this).at(i)*div});
 
        fps ret(deg); ret.at(0) = 1;
        for(int i=1; i<deg; i++) for(auto [k,v] : P) if(i >= k) ret.at(i) -= v*ret.at(i-k); //-xが+ret[i-1]に対応.
        if(div != 1) for(auto &v : ret) v *= div;
        return ret;
    }
    fps inv_sparse(const fps &b,int deg = -1)const{ //f/gを返す 1/fでは分母だがこれは分子に注意.
        int n = (*this).size(),m = b.size();
        if(deg == -1) deg = n;
        assert(b.at(0) != 0);
        T div = 1;
        vector<pair<int,T>> P;
        if(b.at(0) != 1) div = T(b.at(0)).inv();
        for(int i=1; i<m; i++) if(b.at(i) != 0) P.emplace_back(pair{i,b.at(i)*div});
 
        fps ret = (*this).prefix(deg);
        for(int i=1; i<deg; i++) for(auto [k,v] : P) if(i >= k) ret.at(i) -= v*ret.at(i-k);
        if(div != 1) for(auto &v : ret) v *= div;
        return ret;
    }
    fps log_sparse(int deg = -1){ //log(f)を返す O(N*非0).
        //logf = ∫(f'/f) inv,1/f*(f')がO(N*非0) 他はO(N).
        assert((*this).size()&&(*this).at(0)==1);
        if(deg == -1) deg = (*this).size();
        fps ret = (*this).diff();
        ret = ((*this).inv_sparse(deg)).multi_sparse(ret);
        return ret.inte().prefix(deg);
    }
    fps exp_sparse(int deg = -1)const{ //exp(f)を返す O(N*非0).
        //(expf)'=(f')*exp(f)より低次から決まる.
        //[x^0]expf=1より左辺のx^0の係数が求まる->expfのx^1の係数が求まる->...
        if(deg == -1) deg = (*this).size();
        fps ret(deg); 
        if((*this).size() == 0){
            if(deg > 0) ret.at(0) = 1;
            return ret;
        }
        assert((*this).at(0) == 0);
        if(deg == 1) return fps{1};
        ret.at(0) = 1; ret.at(1) = 1;
        const long long mod = mint::getmod();
        for(int i=2; i<deg; i++) ret.at(i) = (-ret.at(mod%i)*(mod/i));
 
        vector<pair<int,T>> P;
        for(int i=1; i<(*this).size(); i++) if((*this).at(i) != 0) P.emplace_back(pair{i-1,(*this).at(i)*i});
        for(int i=0; i<deg-1; i++){
            T now = 0;
            for(auto [k,v] : P){
                if(i < k) break;
                now += ret.at(i-k)*v;
            }
            ret.at(i+1) *= now;
        }
        return ret;
    }
    fps pow_sparse(long long K,int deg = -1)const{ //f^Kを返す O(N*非0).
        //f^K = Fとする fF'= K*F*f'を使う.
        //[x^0]f=1に因数分解 (ax^i)^Kは別で処理.
        //n次はΣ[i=0~n]fi*F'[n-i]=K*Σ[i=1~n+1]F[n+1-i]*i*fi.
        //左辺のi=n以外移項 f0=1より F'n=K*Σ[i=1~n+1]F[n+1-i]*i*fi-Σ[i=1~n]fi*(n+1-i)*F[n+1-i].
        //Fのn次まで分かっていればF'nが求まりFn+1も求まる F0=1からスタート.
        int n = (*this).size();
        if(deg == -1) deg = (*this).size();
        if(K == 0 || deg == 0){
            fps ret(deg);
            if(deg > 0) ret.at(0) = 1;
            return ret;
        }
        for(int t=0; t<n; t++){
            if((*this).at(t) != 0){
                T div = T(1)/(*this).at(t);
                vector<pair<int,T>> P;
                for(int i=t+1; i<n; i++) if((*this).at(i) != 0) P.emplace_back(pair{i-t,(*this).at(i)*div});
                fps ret(deg); ret.at(0) = 1;
                if(deg > 1) ret.at(1) = 1;
                const long long mod = mint::getmod();
                for(int i=2; i<deg; i++) ret.at(i) = (-ret.at(mod%i)*(mod/i));
 
                T mulK = K%mod;
                for(n=0; n<deg-1; n++){ //ここでnの役割が変わる注意.
                    T now = 0;
                    for(auto [i,v] : P){
                        if(i > n+1) break;
                        if(i > 0 && i<=n+1) now += mulK*ret.at(n+1-i)*i*v;
                        if(i > 0 && i <= n) now -= v*(n+1-i)*ret.at(n+1-i);
                    }
                    ret.at(n+1) *= now;
                }
                ret *= T((*this).at(t)).pow(K);
                return (ret<<(t*K)).prefix(deg);
            }
            if(K >= deg || (t+1)*K >= deg) break;
        }
        return fps(deg,0);
    }
 
    fps diff()const{ //微分 nx^(n-1) = (x^n)' O(n).
        int n = (*this).size();
        if(n <= 1) return {};
        fps ret(n-1);
        T multi = 1;
        for(int i=1; i<n; i++) ret.at(i-1) = (*this).at(i)*multi,multi++;
        return ret;
    }
    fps inte()const{ //積分. 1/(n+1)x^(n+1) = ∫x^n dx O(n).
        int n = (*this).size();
        fps ret(n+1);
        ret.at(0) = 0;
        if(n == 0) return ret;
        ret.at(1) = 1;
        const long long mod = mint::getmod();
        for(int i=2; i<=n; i++) ret.at(i) = (-ret.at(mod%i)*(mod/i)); //ここでret[i]=1/iになる.
        for(int i=0; i<n; i++) ret.at(i+1) *= (*this).at(i);
        return ret;
    }
 
    fps inv(int deg=-1)const{ //f*g=1 mod x^degとなるgを求める O(nlogn).
        if(isNTTfriendly(mint::getmod())){
            //g.pre(2m)<- (2g.pre(m)-f*g.pre(m)*g.pre(m))mod x^2mとする.
            //fgm≡1 mod x^mからfg2m≡1 mod x^2mが求まる ダブリング.
            //g2m<- gm-((fgm-1)*gm)を求める.
            if((*this).size() <= 100) return inv_sparse(deg);
            assert((*this).at(0) != 0);
            if(deg == -1) deg = (*this).size();
            fps ret(deg);
            ret.at(0) = T(1)/(*this).at(0);
            
            for(int i=1; i<deg; i<<=1){ //NTT-friendlyじゃないと駄目らしい.
                fps f(2*i),g(2*i);
                int n = min(2*i,(int)(*this).size());
                for(int k=0; k<n; k++) f.at(k) = (*this).at(k);
                for(int k=0; k<i; k++) g.at(k) = ret.at(k);
            
                NTT(f),NTT(g);
                for(int k=0; k<2*i; k++) f.at(k) *= g.at(k); //(fgm-1)を計算.
                INTT(f);
                for(int k=0; k<i; k++) f.at(k) = 0; //[m,2m)の項だけにする([0,m)は一致してない).
                NTT(f);
                for(int k=0; k<2*i; k++) f.at(k) *= g.at(k); //(fgm-1)*gmを計算.
                INTT(f);
                n = min(2*i,deg);
                for(int k=i; k<n; k++) ret.at(k) = -f.at(k); //-(fgm^2-gm)が入る.
            }
            return ret; //ね、簡単でしょ？.
        }
        else{
            //g.pre(2m)<- (2g.pre(m)-f*g.pre(m)*g.pre(m))mod x^2mとする.
            //fgm≡1 mod x^mからfg2m≡1 mod x^2mが求まる ダブリング.
            //こっちは愚直にやる NTT-friendlyじゃないならllmodのconvolutionに変更しないと使えない.
            if((*this).size() <= 200) return inv_sparse(deg);
            assert((*this).at(0) != 0);
            if(deg == -1) deg = (*this).size();
            fps ret(1); ret.at(0) = T(1)/(*this).at(0);
            for(int i=1; i<deg; i<<=1) ret = (ret+ret-ret*ret*((*this).prefix(i<<1))).prefix(i<<1);
            return ret.prefix(deg);
        }        
    }
    fps log(int deg=-1)const{ //logfを求める O(nlogn).
        //log(1-f) = -Σ[n=1~∞](f^n/n)と定義する [x^0]f=1&n<modが条件.
        //(logf)'=f'/f,log(fg)=logf+loggが成り立つ.
        assert((*this).size()&&(*this).at(0)==1);
        if(deg == -1) deg = (*this).size();
        return ((*this).diff()*(*this).inv(deg)).prefix(deg-1).inte(); //∫f'/fから求める.
    }
    fps exp(int deg=-1)const{ //expfを求める O(nlogn).
        if(isNTTfriendly(mint::getmod())){
            //expf=Σ[n=0~∞]f^n/n!と定める [x^0]f=0,n<modが条件.
            //(expf)'= f'expf,exp(f+g)=expf*expg,log(expf)=fを満たす (各々条件は忘れずに).
            //ニュートン法で求める g2m<-gm*(1-log(gm)+f) mod x^2nに更新 納得してない.
            assert((*this).size() == 0 || (*this)[0] == mint(0));
            if (deg == -1) deg = (*this).size();
            
            vector<T> divi; //invと衝突回避用 mintでiの逆元.
            const long long mod = mint::getmod();
            divi.resize(deg*2); divi.at(1) = 1;
            for(int i=2; i<deg*2; i++) divi.at(i) = ((-divi.at(mod%i)))*(mod/i);
            auto integral = [&](fps &f) -> void { //inplaceで積分.
                int n = f.size();
                f.insert(f.begin(),0);
                for(int i=1; i<=n; i++) f.at(i) *= divi.at(i); 
            };
            auto differential = [&](fps &f) -> void { //inplaceで微分.
                if(f.size() == 0) return;
                f.erase(f.begin());
                T multi = 0;
                for(int i=0; i<f.size(); i++) f.at(i) *= ++multi;
            };
        
            fps f({1}),g({1}),z1,z2({1,1}); //何やってるか本当に意味わからずコメント書けない 後で確認.
            if((*this).size() > 1) f.push_back((*this).at(1));
            else f.push_back(0);
            for(int m=2; m<deg; m<<=1){
                auto ff = f;
                ff.resize(2*m);
                NTT(ff); z1 = z2;
                fps z(m);
                for(int i=0; i<m; i++) z.at(i) = ff.at(i)*z1.at(i);
                INTT(z);
                for(int i=0; i<m/2; i++) z.at(i) = 0;
                NTT(z);
                for(int i=0; i<m; i++) z.at(i) *= -z1.at(i);
                INTT(z);
                for(int i=m/2; i<m; i++) g.emplace_back(z.at(i));
                z2 = g;
                z2.resize(2*m);
                NTT(z2);
                fps h = (*this).prefix(m);
                differential(h);
                h.emplace_back(T(0));
                NTT(h);
                for(int i=0; i<m; i++) h.at(i) *= ff.at(i);
                INTT(h);
                h -= f.diff();
                h.resize(2*m);
                for(int i=0; i<m-1; i++) h.at(m+i) = h.at(i),h.at(i) = 0;
                NTT(h);
                for(int i=0; i<2*m; i++) h.at(i) *= z2.at(i);
                INTT(h);
                h.pop_back();
                integral(h);
                int n = min((int)(*this).size(),2*m);
                for(int i=m; i<n; i++) h.at(i) += (*this).at(i);
                for(int i=0; i<m; i++) h.at(i) = 0;
                NTT(h);
                for(int i=0; i<2*m; i++) h.at(i) *= ff.at(i);
                INTT(h);
                for(int i=m; i<2*m; i++) f.emplace_back(h.at(i));
            }
            return f.prefix(deg);
        }
        else{
            //NTT-friendlyじゃないとK=10^5で1sec以上かかる 畳み込みが遅い.
            assert((*this).size() == 0 || (*this).at(0) == 0);
            if(deg == -1) deg = (*this).size();
            fps ret(1); ret.at(0) = 1;
            for(int i=1; i<deg; i<<=1){
                //ニュートン法 g2m<-gm*(1-log(gm)+f) mod x^2n 再掲.
                ret = (ret*((*this).prefix(2*i)+T(1)-ret.log(2*i))).prefix(2*i); //logの引数普段と違う.
            }
            return ret.prefix(deg);
        }
    }
    fps pow(long long K,int deg=-1)const{ //K乗を返すO(nlogn) mod10^9+7で3番目引数false.
        //f^k = exp(klog(f))を使う.
        //logfが計算できるように調整する.
        int n = (*this).size();
        if(deg == -1) deg = n;
        if(K == 0){ //0乗はx^0だけ係数1.
            fps ret(deg);
            if(deg > 0) ret.at(0) = 1;
            return ret;
        }
        for(int i=0; i<n; i++){
            if((*this).at(i) != 0){ //0じゃなかったらOK.
                const long long mod = mint::getmod();
                T div = T(1)/(*this).at(i); //*([x^i]f)^Kと *x^(i*k)の分は後回し.
                fps ret = ((((*this)*div)>>i).log(deg)*(K%mod)).exp(deg);
                ret *= T((*this).at(i)).pow(K); //[x^i]f^Kの分.
                ret = (ret<<(i*K)).prefix(deg); //*x^(i*k)の分.
                return ret;
            }
            if(K >= deg || (i+1)*K >= deg) break; //((i+1)*K)乗未満は0確定 int128回避用にK>=deg(degがllはやばい).
        }
        return fps(deg,0); //fの係数全て0なら係数全て0.
    }
 
    fps prefix(int siz)const{ //先頭siz項を返す なかったら0埋め. O(siz).
        fps ret((*this).begin(),(*this).begin()+min((int)(*this).size(),siz));
        if(ret.size() < siz) ret.resize(siz,0);
        return ret;
    }
    void del0(){ //末尾の0を消す O(n).
        while((*this).size() && (*this).back() == 0) (*this).pop_back();
    }
    fps rev()const{ //ひっくり返す O(n).
        fps ret(*this);
        reverse(ret.begin(),ret.end());
        return ret;
    }
    pair<fps,fps> getQR(const fps &b)const{ //多項式の商と余りを同時に得る O(nlogn).
        fps Q = (*this)/b,R = (*this)-Q*b;
        R.del0();
        return {Q,R};
    } 
 
    fps cumulativeNtimes(int N,T b,int deg=-1){ //1/(1-bx)^Nをdeg次まで返す 指定なしはN次まで.
        //負の二項定理を使う. 1/(1-x)^N=Σ[i=0~∞]((n+i-1) choose i)(bx^i);
        //fps{}.cumulativeNtime()で無理やり関数を呼び出す.
        assert(N <= 0); //N=0も駄目? {1}を返すべき所{0}になる.
        if(deg == -1) deg = N+1; 
        int Limit = N+deg; //Limit 必要なサイズ fac->x! facinv->1/x! inv->1/x.
        long long mod = mint::getmod(),invstart = min((int)mod-1,Limit);
        vector<T> FAC(Limit+1,1); for(int i=1; i<=Limit; i++) FAC.at(i) = FAC.at(i-1)*i;
        vector<T> FACinv(Limit+1); FACinv.at(invstart) = FAC.at(invstart).inv();
        for(int i=invstart-1; i>=0; i--) FACinv.at(i) = FACinv.at(i+1)*(i+1);
        auto nCr = [&](int n, int r) -> T {
            if(n < r || r < 0 || n < 0) return 0;
            return FAC.at(n)*FACinv.at(r)*FACinv.at(n-r);
        };
        fps ret(deg); T value = 1;
        for(int i=0; i<deg; i++) ret.at(i) = nCr(N+i-1,i)*value,value *= b;
        return ret; 
    }
    fps Taylorshift(long long C)const{ //Σ[i=0~n]Ai*(x+c)^iを返す O(nlogn).
        //Σ[i=0~n]x^i/i! Σ[k=0~n-i](A[i+k]*(i+k)!)*(c^k/k!)になる.
        //kの部分はXi=Ai*i!,Yi=c^i/i!として.Zi=ΣX[i+k]Yk.
        //これは[x^(i+n)]X*rev(Y)で求められる.
        const long long mod = mint::getmod();
        mint c = (C%mod+mod)%mod,pc = 1;
        if(c == 0) return (*this);
        int deg = (*this).size(); //deg 必要なサイズ fac->x! facinv->1/x! inv->1/x.
        long long invstart = min((int)mod-1,deg);
        vector<mint> fac(deg+1,1); for(int i=1; i<=deg; i++) fac.at(i) = fac.at(i-1)*i;
        vector<mint> facinv(deg+1); facinv.at(invstart) = fac.at(invstart).inv();
        for(int i=invstart-1; i>=0; i--) facinv.at(i) = facinv.at(i+1)*(i+1);
    
        fps X(deg),Y(deg);
        for(int i=0; i<deg; i++) X.at(i) = (*this).at(i)*fac.at(i),Y.at(deg-1-i) = pc*facinv.at(i),pc *= c;
        X *= Y;
        fps ret(deg);
        for(int i=0; i<deg; i++) ret.at(i) = X.at(i+deg-1)*facinv.at(i);
        return ret;
    }
};
using fps = FormalPowerSeries<mint>;

template<typename mint>
vector<mint> ProductPoly(vector<vector<mint>> Fs){ //ΠFiを返す 返り値の次数をdをしてO(dlog^2d). 
    int n = Fs.size();
    if(n == 0) return {1};
    if(n == 1) return Fs.at(0);
    auto f = [&](auto f,int l,int r) -> vector<mint> {
        if(l+1 == r) return Fs.at(l);
        return convolution(f(f,l,(l+r)/2),f(f,(l+r)/2,r));
    };
    return f(f,0,n);
}
fps ProductPoly(vector<fps> Fs){ //ΠFiを返す 返り値の次数をdをしてO(dlog^2d). 
    int n = Fs.size();
    if(n == 0) return {1};
    if(n == 1) return Fs.at(0);
    auto f = [&](auto f,int l,int r) -> fps {
        if(l+1 == r) return Fs.at(l);
        return f(f,l,(l+r)/2)*f(f,(l+r)/2,r);
    };
    return f(f,0,n);
}

mint BostanMori(long long N,fps P,fps Q){
    P.del0(); Q.del0();
    int K = Q.size();
    mint ret = 0;
    if(P.size() >= Q.size()){ //deg(P)>=deg(Q)の時.
        auto R = P/Q;
        P -= R*Q; P.del0();
        if(N < R.size()) ret += R.at(N); 
    }
    if(P.size() == 0) return ret;
    if(isNTTfriendly(mint::getmod())){ //NTT-friendly
        int n = 1;
        while(n < K) n <<= 1;
        P.resize(2*n); Q.resize(2*n);
        NTT(P),NTT(Q);
        vector<mint> S(n),T(n);
        vector<int> Bitrev(n); //ビットリバース.
        for(int i=0; i<n; i++){
            if(i&1) Bitrev.at(i) = n>>1;
            Bitrev.at(i) += (Bitrev.at(i>>1)>>1);
        }
        fftinfo<mint> info;
        mint dw = info.g.inv().pow((mint::getmod()-1)/(n<<1)),Div2 = mint(1)/2;
        while(N){
            mint div2 = Div2;
            S.resize(n),T.resize(n);
            for(int i=0; i<(n<<1); i+=2) T.at(i>>1) = Q.at(i)*Q.at(i+1); //Q(x)Q(-x)の偶数次.
            if(N&1){ //P(x)Q(-x)の奇数次.
                for(auto i : Bitrev){
                    S.at(i) = (P.at(i<<1)*Q.at((i<<1)+1)-P.at((i<<1)+1)*Q.at(i<<1))*div2;
                    div2 *= dw; //これ分からん.
                }
            }
            else{ //P(x)Q(-x)の偶数次.
                for(int i=0; i<(n<<1); i+=2) S.at(i>>1) = (P.at(i)*Q.at(i+1)+P.at(i+1)*Q.at(i))*div2;
            }
            swap(P,S),swap(Q,T);
            N >>= 1;
            if(N < n) break;
            NTTdoubling(P),NTTdoubling(Q);
        }
        INTT(P),INTT(Q);
        Q = Q.inv();
        for(int i=0; i<=N; i++) ret += P.at(i)*Q.at(N-i); //ret+=[x^N]P/Q
        return ret;
    }
    else{ //P(x)/Q(x)=P(x)Q(-x)/Q(x)Q(-x) 分母の奇数次は全部0になる.
        P.resize(K-1);
        bool skip = false;
        while(N){
            if(!skip){
                auto memo = Q;
                fps Q2 = Q;
                for(int i=1; i<Q2.size(); i+=2) Q2.at(i) = -Q2.at(i);
                P *= Q2,Q *= Q2; 
                for(int i=0; i<Q.size(); i+=2) Q.at(i>>1) = Q.at(i); //偶数次しかないので次数/2をする.
                Q.resize((Q.size()+1)>>1);
                if(memo == Q){
                    skip = true;
                    for(int i=1; i<Q.size(); i+=2) Q.at(i) = -Q.at(i);
                }    
            }
            else P *= Q;
            if(N&1){ //Nthが奇数の場合は分子の偶数次は要らない (分母が偶数次しかない).
                for(int i=1; i<P.size(); i+=2) P.at(i>>1) = P.at(i);
                P.resize(P.size()>>1);
            }
            else{   //Nthが偶数.
                for(int i=0; i<P.size(); i+=2) P.at(i>>1) = P.at(i);
                P.resize((P.size()+1)>>1);
            }
            N >>= 1; 
        }
        return ret+P.at(0);
    }
}

int main(){
    ios_base::sync_with_stdio(false);
    cin.tie(nullptr);

    long long N,M; cin >> N >> M;
    vector<fps> Fs,Gs; Fs.reserve(N*10),Gs.reserve(N*10);
    for(int i=0; i<N; i++){
        int a; cin >> a;
        fps f(a+1);
        f.at(0) = 1,f.at(a) = -1;
        Gs.emplace_back(f);
    }
    fps G = ProductPoly(Gs);
    cout << BostanMori(M,{1},G).v << endl;
}