#line 2 "/Users/noya2/Desktop/Noya2_library/template/template.hpp"
using namespace std;

#include<bits/stdc++.h>
#line 1 "/Users/noya2/Desktop/Noya2_library/template/inout_old.hpp"
namespace noya2 {

template <typename T, typename U>
ostream &operator<<(ostream &os, const pair<T, U> &p){
    os << p.first << " " << p.second;
    return os;
}
template <typename T, typename U>
istream &operator>>(istream &is, pair<T, U> &p){
    is >> p.first >> p.second;
    return is;
}

template <typename T>
ostream &operator<<(ostream &os, const vector<T> &v){
    int s = (int)v.size();
    for (int i = 0; i < s; i++) os << (i ? " " : "") << v[i];
    return os;
}
template <typename T>
istream &operator>>(istream &is, vector<T> &v){
    for (auto &x : v) is >> x;
    return is;
}

void in() {}
template <typename T, class... U>
void in(T &t, U &...u){
    cin >> t;
    in(u...);
}

void out() { cout << "\n"; }
template <typename T, class... U, char sep = ' '>
void out(const T &t, const U &...u){
    cout << t;
    if (sizeof...(u)) cout << sep;
    out(u...);
}

template<typename T>
void out(const vector<vector<T>> &vv){
    int s = (int)vv.size();
    for (int i = 0; i < s; i++) out(vv[i]);
}

struct IoSetup {
    IoSetup(){
        cin.tie(nullptr);
        ios::sync_with_stdio(false);
        cout << fixed << setprecision(15);
        cerr << fixed << setprecision(7);
    }
} iosetup_noya2;

} // namespace noya2
#line 1 "/Users/noya2/Desktop/Noya2_library/template/const.hpp"
namespace noya2{

const int iinf = 1'000'000'007;
const long long linf = 2'000'000'000'000'000'000LL;
const long long mod998 =  998244353;
const long long mod107 = 1000000007;
const long double pi = 3.14159265358979323;
const vector<int> dx = {0,1,0,-1,1,1,-1,-1};
const vector<int> dy = {1,0,-1,0,1,-1,-1,1};
const string ALP = "ABCDEFGHIJKLMNOPQRSTUVWXYZ";
const string alp = "abcdefghijklmnopqrstuvwxyz";
const string NUM = "0123456789";

void yes(){ cout << "Yes\n"; }
void no(){ cout << "No\n"; }
void YES(){ cout << "YES\n"; }
void NO(){ cout << "NO\n"; }
void yn(bool t){ t ? yes() : no(); }
void YN(bool t){ t ? YES() : NO(); }

} // namespace noya2
#line 1 "/Users/noya2/Desktop/Noya2_library/template/utils.hpp"
namespace noya2{

unsigned long long inner_binary_gcd(unsigned long long a, unsigned long long b){
    if (a == 0 || b == 0) return a + b;
    int n = __builtin_ctzll(a); a >>= n;
    int m = __builtin_ctzll(b); b >>= m;
    while (a != b) {
        int mm = __builtin_ctzll(a - b);
        bool f = a > b;
        unsigned long long c = f ? a : b;
        b = f ? b : a;
        a = (c - b) >> mm;
    }
    return a << min(n, m);
}

template<typename T> T gcd_fast(T a, T b){ return static_cast<T>(inner_binary_gcd(abs(a),abs(b))); }

long long sqrt_fast(long long n) {
    if (n <= 0) return 0;
    long long x = sqrt(n);
    while ((x + 1) * (x + 1) <= n) x++;
    while (x * x > n) x--;
    return x;
}

template<typename T> T floor_div(const T n, const T d) {
    assert(d != 0);
    return n / d - static_cast<T>((n ^ d) < 0 && n % d != 0);
}

template<typename T> T ceil_div(const T n, const T d) {
    assert(d != 0);
    return n / d + static_cast<T>((n ^ d) >= 0 && n % d != 0);
}

template<typename T> void uniq(vector<T> &v){
    sort(v.begin(),v.end());
    v.erase(unique(v.begin(),v.end()),v.end());
}

template <typename T, typename U> inline bool chmin(T &x, U y) { return (y < x) ? (x = y, true) : false; }

template <typename T, typename U> inline bool chmax(T &x, U y) { return (x < y) ? (x = y, true) : false; }

template<typename T> inline bool range(T l, T x, T r){ return l <= x && x < r; }

} // namespace noya2
#line 8 "/Users/noya2/Desktop/Noya2_library/template/template.hpp"

#define rep(i,n) for (int i = 0; i < (int)(n); i++)
#define repp(i,m,n) for (int i = (m); i < (int)(n); i++)
#define reb(i,n) for (int i = (int)(n-1); i >= 0; i--)
#define all(v) (v).begin(),(v).end()

using ll = long long;
using ld = long double;
using uint = unsigned int;
using ull = unsigned long long;
using pii = pair<int,int>;
using pll = pair<ll,ll>;
using pil = pair<int,ll>;
using pli = pair<ll,int>;

namespace noya2{

/*　~ (. _________ . /)　*/

}

using namespace noya2;


#line 2 "c.cpp"

#line 2 "/Users/noya2/Desktop/Noya2_library/utility/modint.hpp"

#line 2 "/Users/noya2/Desktop/Noya2_library/math/prime.hpp"

#line 4 "/Users/noya2/Desktop/Noya2_library/math/prime.hpp"

namespace noya2 {

constexpr ll safe_mod(ll x, ll m) {
    x %= m;
    if (x < 0) x += m;
    return x;
}

constexpr ll pow_mod_constexpr(ll x, ll n, int m) {
    if (m == 1) return 0;
    uint _m = (uint)(m);
    ull r = 1;
    ull y = safe_mod(x, m);
    while (n) {
        if (n & 1) r = (r * y) % _m;
        y = (y * y) % _m;
        n >>= 1;
    }
    return r;
}

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;
    ll d = n - 1;
    while (d % 2 == 0) d /= 2;
    constexpr ll bases[3] = {2, 7, 61};
    for (ll a : bases) {
        ll t = d;
        ll 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_flag = is_prime_constexpr(n);

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};
    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; 
        auto tmp = s;
        s = t;
        t = tmp;
        tmp = m0;
        m0 = m1;
        m1 = tmp;
    }
    if (m0 < 0) m0 += b / s;
    return {s, m0};
}

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; (ll)(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_flag = primitive_root_constexpr(m);

} // namespace noya2
#line 4 "/Users/noya2/Desktop/Noya2_library/utility/modint.hpp"

namespace noya2{

struct barrett {
    uint _m;
    ull  im;
    explicit barrett(uint m) : _m(m), im((ull)(-1) / m + 1) {}
    uint umod() const { return _m; }
    uint mul(uint a, uint b) const {
        ull z = a;
        z *= b;
        ull x = ull((__uint128_t(z) * im) >> 64);
        uint v = (uint)(z - x * _m);
        if (_m <= v) v += _m;
        return v;
    }
};

template <int m>
struct static_modint {
    using mint = static_modint;
  public:
    static constexpr int mod() { return m; }
    static mint raw(int v) {
        mint x;
        x._v = v;
        return x;
    }
    constexpr static_modint() : _v(0) {}
    template<signed_integral T>
    constexpr static_modint(T v){
        ll x = (ll)(v % (ll)(umod()));
        if (x < 0) x += umod();
        _v = (uint)(x);
    }
    template<unsigned_integral T>
    constexpr static_modint(T v){
        _v = (uint)(v % umod());
    }
    constexpr 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;
    }
    constexpr mint& operator+=(const mint& rhs) {
        _v += rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    constexpr mint& operator-=(const mint& rhs) {
        _v -= rhs._v;
        if (_v >= umod()) _v += umod();
        return *this;
    }
    constexpr mint& operator*=(const mint& rhs) {
        ull z = _v;
        z *= rhs._v;
        _v = (uint)(z % umod());
        return *this;
    }
    constexpr mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }
    constexpr mint operator+() const { return *this; }
    constexpr mint operator-() const { return mint() - *this; }
    constexpr mint pow(ll n) const {
        assert(0 <= n);
        mint x = *this, r = 1;
        while (n) {
            if (n & 1) r *= x;
            x *= x;
            n >>= 1;
        }
        return r;
    }
    constexpr mint inv() const {
        if (prime) {
            assert(_v);
            return pow(umod() - 2);
        } else {
            auto eg = inv_gcd(_v, m);
            assert(eg.first == 1);
            return eg.second;
        }
    }
    friend constexpr mint operator+(const mint& lhs, const mint& rhs) {
        return mint(lhs) += rhs;
    }
    friend constexpr mint operator-(const mint& lhs, const mint& rhs) {
        return mint(lhs) -= rhs;
    }
    friend constexpr mint operator*(const mint& lhs, const mint& rhs) {
        return mint(lhs) *= rhs;
    }
    friend constexpr mint operator/(const mint& lhs, const mint& rhs) {
        return mint(lhs) /= rhs;
    }
    friend constexpr bool operator==(const mint& lhs, const mint& rhs) {
        return lhs._v == rhs._v;
    }
    friend constexpr bool operator!=(const mint& lhs, const mint& rhs) {
        return lhs._v != rhs._v;
    }
    friend std::ostream &operator<<(std::ostream &os, const mint& p) {
        return os << p.val();
    }
    friend std::istream &operator>>(std::istream &is, mint &a) {
        long long t; is >> t;
        a = mint(t);
        return (is);
    }

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


template <int id> struct dynamic_modint {
    using mint = dynamic_modint;
  public:
    static int mod() { return (int)(bt.umod()); }
    static void set_mod(int m) {
        assert(1 <= m);
        bt = barrett(m);
    }
    static mint raw(int v) {
        mint x;
        x._v = v;
        return x;
    }

    dynamic_modint() : _v(0) {}
    template<signed_integral T>
    dynamic_modint(T v){
        ll x = (ll)(v % (ll)(mod()));
        if (x < 0) x += mod();
        _v = (uint)(x);
    }
    template<unsigned_integral T>
    dynamic_modint(T v){
        _v = (uint)(v % mod());
    }
    uint 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 = noya2::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;
    }
    friend std::ostream &operator<<(std::ostream &os, const mint& p) {
        return os << p.val();
    }
    friend std::istream &operator>>(std::istream &is, mint &a) {
        long long t; is >> t;
        a = mint(t);
        return (is);
    }

  private:
    unsigned int _v;
    static barrett bt;
    static unsigned int umod() { return bt.umod(); }
};
template <int id> noya2::barrett dynamic_modint<id>::bt(998244353);

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

template<typename T>
concept Modint = requires (T &a){
    T::mod();
    a.inv();
    a.val();
    a.pow(declval<int>());
};

} // namespace noya2
#line 4 "c.cpp"
using mint = modint998244353;
#line 2 "/Users/noya2/Desktop/Noya2_library/math/binomial.hpp"

namespace noya2 {

template<typename mint>
struct binomial {
    binomial(int len = 300000){ extend(len); }
    static mint fact(int n){
        if (n < 0) return 0;
        while (n >= (int)_fact.size()) extend();
        return _fact[n];
    }
    static mint ifact(int n){
        if (n < 0) return 0;
        while (n >= (int)_fact.size()) extend();
        return _ifact[n];
    }
    static mint inv(int n){
        return ifact(n) * fact(n-1);
    }
    static mint C(int n, int r){
        if (!(0 <= r && r <= n)) return 0;
        return fact(n) * ifact(r) * ifact(n-r);
    }
    static mint P(int n, int r){
        if (!(0 <= r && r <= n)) return 0;
        return fact(n) * ifact(n-r);
    }
    inline mint operator()(int n, int r) { return C(n, r); }
    template<class... Cnts> static mint M(const Cnts&... cnts){
        return multinomial(0,1,cnts...);
    }
  private:
    static mint multinomial(const int& sum, const mint& div_prod){
        if (sum < 0) return 0;
        return fact(sum) * div_prod;
    }
    template<class... Tail> static mint multinomial(const int& sum, const mint& div_prod, const int& n1, const Tail&... tail){
        if (n1 < 0) return 0;
        return multinomial(sum+n1,div_prod*ifact(n1),tail...);
    }
    static vector<mint> _fact, _ifact;
    static void extend(int len = -1){
        if (_fact.empty()){
            _fact = _ifact = {1,1};
        }
        int siz = _fact.size();
        if (len == -1) len = siz * 2;
        len = min<int>(len, mint::mod()-1);
        if (len < siz) return ;
        _fact.resize(len+1), _ifact.resize(len+1);
        for (int i = siz; i <= len; i++) _fact[i] = _fact[i-1] * i;
        _ifact[len] = _fact[len].inv();
        for (int i = len; i > siz; i--) _ifact[i-1] = _ifact[i] * i;
    }
};
template<typename T>
std::vector<T>binomial<T>::_fact = vector<T>(2,T(1));
template<typename T>
std::vector<T>binomial<T>::_ifact = vector<T>(2,T(1));

} // namespace noya2
#line 2 "/Users/noya2/Desktop/Noya2_library/fps/fps_ntt.hpp"

#line 2 "/Users/noya2/Desktop/Noya2_library/fps/formal_power_series.hpp"

#line 4 "/Users/noya2/Desktop/Noya2_library/fps/formal_power_series.hpp"

namespace noya2{

template<typename T>
concept Field = requires (T a, T b){
    a + b; a - b; a / b; a * b;
    T(0); T(1);
};

template<class Info>
concept Fps_Info = requires {
    typename Info::value_type;
    requires Field<typename Info::value_type>;
    {Info::multiply(declval<vector<typename Info::value_type>>(),declval<vector<typename Info::value_type>>())} -> convertible_to<vector<typename Info::value_type>>;
    {Info::inv(declval<vector<typename Info::value_type>>(),declval<int>())} -> convertible_to<vector<typename Info::value_type>>;
    {Info::integral(declval<vector<typename Info::value_type>>())} -> convertible_to<vector<typename Info::value_type>>;
};

template<Fps_Info Info>
struct FormalPowerSeries : vector<typename Info::value_type> {
    using T = typename Info::value_type;
    using vector<T>::vector;
    using vector<T>::operator=;
    using FPS = FormalPowerSeries;
    FormalPowerSeries (const vector<T> &init_ = {}){ (*this) = init_; }
    void shrink(){ while (!(*this).empty() && (*this).back() == T(0)) (*this).pop_back(); }
    FPS &operator+=(const T &r){
        if ((*this).empty()) (*this).resize(1);
        (*this)[0] += r;
        return *this;
    }
    FPS &operator-=(const T &r){
        if ((*this).empty()) (*this).resize(1);
        (*this)[0] -= r;
        return *this;
    }
    FPS &operator*=(const T &r){
        for (auto &x : *this) x *= r;
        return *this;
    }
    FPS &operator/=(const T &r){
        (*this) *= T(1)/r;
        return *this;
    }
    FPS &operator<<=(const int &d){
        (*this).insert((*this).begin(),d,T(0));
        return *this;
    }
    FPS &operator>>=(const int &d){
        if ((int)(*this).size() <= d) (*this).clear();
        else (*this).erase((*this).begin(),(*this).begin()+d);
        return *this;
    }
    FPS &operator+=(const FPS &r){
        if ((*this).size() < r.size()) (*this).resize(r.size());
        for (int i = 0; i < (int)(r.size()); i++) (*this)[i] += r[i];
        return *this;
    }
    FPS &operator-=(const FPS &r){
        if ((*this).size() < r.size()) (*this).resize(r.size());
        for (int i = 0; i < (int)(r.size()); i++) (*this)[i] -= r[i];
        return *this;
    }
    FPS &operator*=(const FPS &r){
        if ((*this).empty() || r.empty()){
            (*this).clear();
            return *this;
        }
        (*this) = Info::multiply(*this,r);
        return *this;
    }
    FPS operator+(const T &r) const { return FPS(*this) += r; }
    FPS operator-(const T &r) const { return FPS(*this) -= r; }
    FPS operator*(const T &r) const { return FPS(*this) *= r; }
    FPS operator/(const T &r) const { return FPS(*this) /= r; }
    FPS operator<<(const int &d) const { return FPS(*this) <<= d; }
    FPS operator>>(const int &d) const { return FPS(*this) >>= d; }
    FPS operator+(const FPS &r) const { return FPS(*this) += r; }
    FPS operator-(const FPS &r) const { return FPS(*this) -= r; }
    FPS operator*(const FPS &r) const { return FPS(*this) *= r; }
    FPS operator+() const { return *this; }
    FPS operator-() const {
        FPS res(*this);
        for (auto &x : res) x = -x;
        return res;
    }
    T eval(const T &x) const {
        T res = T(0), w = T(1);
        for (auto &e : *this) res += e * w, w *= x;
        return res;
    }
    static FPS dot(const FPS &lhs, const FPS &rhs){
        FPS res(min(lhs.size(),rhs.size()));
        for (int i = 0; i < (int)res.size(); i++) res[i] = lhs[i] * rhs[i];
        return res;
    }
    FPS pre(int siz) const {
        FPS ret((*this).begin(), (*this).begin() + min((int)this->size(), siz));
        if ((int)ret.size() < siz) ret.resize(siz);
        return ret;
    }
    FPS rev() const {
        FPS ret(*this);
        reverse(ret.begin(), ret.end());
        return ret;
    }
    FPS diff() const {
        const int n = (int)this->size();
        FPS ret(max(0, n - 1));
        T one(1), coeff(1);
        for (int i = 1; i < n; i++) {
            ret[i - 1] = (*this)[i] * coeff;
            coeff += one;
        }
        return ret;
    }
    FPS integral() const {
        FPS ret = Info::integral(*this);
        return ret;
    }
    FPS inv(int d = -1) const {
        FPS ret = Info::inv(*this,d);
        return ret;
    }
    FPS exp(int d = -1) const {
        const int n = (*this).size();
        if (d == -1) d = n;
        FPS f = {T(1)+(*this)[0],(*this)[1]}, res = {1,(n > 1 ? (*this)[1] : 0)};
        for (int sz = 2; sz < d; sz <<= 1){
            f.insert(f.end(),(*this).begin()+min(n,sz),(*this).begin()+min(n,sz*2));
            if ((int)f.size() < sz*2) f.resize(sz*2);
            res = res * (f - res.log(2*sz));
            res.resize(sz*2);
        }
        res.resize(d);
        return res;
    }
    FPS log(int d = -1) const {
        assert(!(*this).empty() && (*this)[0] == T(1));
        if (d == -1) d = (*this).size();
        return (this->diff() * this->inv(d)).pre(d - 1).integral();
    }
};

} // namespace noya2
#line 2 "/Users/noya2/Desktop/Noya2_library/fps/ntt.hpp"

#line 5 "/Users/noya2/Desktop/Noya2_library/fps/ntt.hpp"

namespace noya2{

template<Modint mint>
struct NTT {
    static constexpr uint mod = mint::mod();
    static constexpr ull mod2 = (ull)mod * mod;
    static constexpr uint pr  = primitive_root_constexpr(mod);
    static constexpr int level = countr_zero(mod-1);
    mint wp[level+1], wm[level+1];
    void set_ws(){
        mint r = mint(pr).pow((mod-1) >> level);
        wp[level] = r, wm[level] = r.inv();
        for (int i = level-1; i >= 0; i--){
            wp[i] = wp[i+1] * wp[i+1];
            wm[i] = wm[i+1] * wm[i+1];
        }
    }
    NTT () { set_ws(); }
    void fft4(vector<mint> &a, int k, int s = 0){
        uint im = wm[2].val();
        uint n = 1<<k;
        uint len = n;
        int l = k;
        while (len > 1){
            if (l == 1){
                for (int i = 0; i < (1<<(k-1)); i++){
                    int i0 = s + i*2, i1 = i0+1;
                    a[i0] += a[i1];
                    a[i1]  = a[i0] - a[i1] * 2;
                }
                len >>= 1;
                l -= 1;
            }
            else {
                int len4 = len/4;
                int nlen = n/len;
                ull r1 = 1, r2 = 1, r3 = 1, imr1 = im, imr3 = im;
                for (int i = 0; i < len4; i++){
                    int offset = 0;
                    for (int j = 0; j < nlen; j++){
                        int i0 = s + i + offset, i1 = i0 + len4, i2 = i1 + len4, i3 = i2 + len4;
                        uint a0 = a[i0].val();
                        uint a1 = a[i1].val();
                        uint a2 = a[i2].val();
                        uint a3 = a[i3].val();
                        uint a0p2 = a0 + a2;
                        uint a1p3 = a1 + a3;
                        ull b0m2 = (a0 + mod - a2) * r1;
                        ull b1m3 = (a1 + mod - a3) * imr1;
                        ull c0m2 = (a0 + mod - a2) * r3;
                        ull c1m3 = (a1 + mod - a3) * imr3;
                        a[i0] = a0p2 + a1p3;
                        a[i1] = b0m2 + b1m3;
                        a[i2] = (a0p2 + mod*2 - a1p3) * r2;
                        a[i3] = c0m2 + mod2*2 - c1m3;
                        offset += len;
                    }
                    r1 = r1 * wm[l].val() % mod;
                    r2 = r1 * r1 % mod;
                    r3 = r1 * r2 % mod;
                    imr1 = im * r1 % mod;
                    imr3 = im * r3 % mod;
                }
                len >>= 2;
                l -= 2;
            }
        }
    }
    void ifft4(vector<mint> &a, int k, int s = 0){
        uint im = wp[2].val();
        uint n = 1<<k;
        uint len = (k & 1 ? 2 : 4);
        int l = (k & 1 ? 1 : 2);
        while (len <= n){
            if (l == 1){
                for (int i = 0; i < (1<<(k-1)); i++){
                    int i0 = s + i*2, i1 = i0+1;
                    a[i0] += a[i1];
                    a[i1]  = a[i0] - a[i1] * 2;
                }
                len <<= 2;
                l += 2;
            }
            else {
                int len4 = len/4;
                int nlen = n/len;
                ull r1 = 1, r2 = 1, r3 = 1, imr1 = im, imr3 = im;
                for (int i = 0; i < len4; i++){
                    int offset = 0;
                    for (int j = 0; j < nlen; j++){
                        int i0 = s + i + offset, i1 = i0 + len4, i2 = i1 + len4, i3 = i2 + len4;
                        ull a0 = a[i0].val();
                        ull a1 = a[i1].val() * r1;
                        ull a2 = a[i2].val() * r2;
                        ull a3 = a[i3].val() * r3;
                        ull b1 = a[i1].val() * imr1;
                        ull b3 = a[i3].val() * imr3;
                        ull a0p2 = a0 + a2;
                        ull a1p3 = a1 + a3;
                        ull a0m2 = a0 + mod2 - a2;
                        ull b1m3 = b1 + mod2 - b3;
                        a[i0] = a0p2 + a1p3;
                        a[i1] = a0m2 + b1m3;
                        a[i2] = a0p2 + mod2*2 - a1p3;
                        a[i3] = a0m2 + mod2*2 - b1m3;
                        offset += len;
                    }
                    r1 = r1 * wp[l].val() % mod;
                    r2 = r1 * r1 % mod;
                    r3 = r1 * r2 % mod;
                    imr1 = im * r1 % mod;
                    imr3 = im * r3 % mod;
                }
                len <<= 2;
                l += 2;
            }
        }
    }
    void ntt(vector<mint> &a) {
        if ((int)a.size() <= 1) return;
        fft4(a, 63-countl_zero(a.size()));
    }
    void intt(vector<mint> &a, bool stop = false) {
        if ((int)a.size() <= 1) return;
        ifft4(a, 63-countl_zero(a.size()));
        if (stop) return ;
        mint iv = mint(a.size()).inv();
        for (auto &x : a) x *= iv;
    }
    vector<mint> multiply(const vector<mint> &a, const vector<mint> &b) {
        int l = a.size() + b.size() - 1;
        if (min<int>(a.size(), b.size()) <= 40){
            vector<mint> s(l);
            for (int i = 0; i < (int)a.size(); i++) for (int j = 0; j < (int)b.size(); j++) s[i + j] += a[i] * b[j];
            return s;
        }
        int k = 2, M = 4;
        while (M < l) M <<= 1, ++k;
        set_ws();
        vector<mint> s(M);
        for (int i = 0; i < (int)a.size(); ++i) s[i] = a[i];
        fft4(s, k);
        if (a.size() == b.size() && a == b) {
            for (int i = 0; i < M; ++i) s[i] *= s[i];
        }
        else {
            vector<mint> t(M);
            for (int i = 0; i < (int)b.size(); ++i) t[i] = b[i];
            fft4(t, k);
            for (int i = 0; i < M; ++i) s[i] *= t[i];
        }
        ifft4(s, k);
        s.resize(l);
        mint invm = mint(M).inv();
        for (int i = 0; i < l; ++i) s[i] *= invm;
        return s;
    }
};


} // namespace noya2
#line 6 "/Users/noya2/Desktop/Noya2_library/fps/fps_ntt.hpp"

namespace noya2{

template<typename T>
struct fps_ntt{
    using value_type = T;
    static NTT<T> ntt;
    static vector<T> multiply(const vector<T> &a, const vector<T> &b){
        return ntt.multiply(a,b);
    }
    static vector<T> inv(const vector<T> &a, int d = -1){
        const int n = a.size();
        if (d == -1) d = n;
        vector<T> res = {a[0].inv()};
        for (int siz = 1; siz < d; siz <<= 1){
            vector<T> f(a.begin(),a.begin()+min(n,siz*2)), g(res);
            f.resize(siz*2), g.resize(siz*2);
            ntt.ntt(f), ntt.ntt(g);
            for (int i = 0; i < siz*2; i++) f[i] *= g[i];
            ntt.intt(f,true);
            f.erase(f.begin(),f.begin()+siz);
            f.resize(siz*2);
            ntt.ntt(f);
            for (int i = 0; i < siz*2; i++) f[i] *= g[i];
            ntt.intt(f,true);
            T siz2_inv = T(siz*2).inv(); siz2_inv *= -siz2_inv;
            for (int i = 0; i < siz; i++) f[i] *= siz2_inv;
            res.insert(res.end(),f.begin(),f.begin()+siz);
        }
        res.resize(d);
        return res;
    }
    static binomial<T> bnm;
    static vector<T> integral(const vector<T> &a){
        const int n = a.size();
        vector<T> res(n+1);
        for (int i = 1; i <= n; i++) res[i] = a[i-1] * bnm.inv(i);
        return res;
    }
};
template<typename T> NTT<T> fps_ntt<T>::ntt;
template<typename T> using FPS_ntt = FormalPowerSeries<fps_ntt<T>>;

} // namespace noya2

#line 7 "c.cpp"
using fps = FPS_ntt<mint>;

void solve(){
    int n; in(n);
    const int mx = 1<<17;
    vector<bool> exist(mx,false);
    rep(i,n){
        int x; in(x);
        exist[x] = true;
    }
    exist[0] = true;
    binomial<mint> bnm;
    vector<mint> dp(mx);
    dp[0] = 1;
    auto dfs = [&](auto sfs, int l, int r) -> void {
        if (r - l <= 1) return ;
        int m = (l + r) / 2;
        sfs(sfs,l,m);
        fps f(m-l);
        repp(i,l,m){
            if (exist[i]){
                f[i-l] = dp[i];
            }
        }
        fps e(r-l+1);
        rep(i,r-l+1){
            e[i] = bnm.ifact(i);
        }
        f *= e;
        int sz = f.size();
        repp(i,m-l,sz){
            if (i+l < r) dp[i+l] += f[i];
        }
        sfs(sfs,m,r);
    };
    dfs(dfs,0,mx);
    mint ans = 0;
    rep(i,mx){
        if (exist[i]){
            ans += dp[i] * bnm.fact(i);
        }
    }
    out(ans-1);
}

int main(){
    int t = 1; //in(t);
    while (t--) { solve(); }
}