// #pragma GCC target("avx2")
// #pragma GCC optimize("O3")
// #pragma GCC optimize("unroll-loops")
#include <bits/stdc++.h>
using namespace std;
// #define INTERACTIVE

namespace templates {
// type
using ll  = long long;
using ull = unsigned long long;
using Pii = pair<int, int>;
using Pil = pair<int, ll>;
using Pli = pair<ll, int>;
using Pll = pair<ll, ll>;
template <class T>
using pq = priority_queue<T>;
template <class T>
using qp = priority_queue<T, vector<T>, greater<T>>;
// clang-format off
#define vec(T, A, ...) vector<T> A(__VA_ARGS__);
#define vvec(T, A, h, ...) vector<vector<T>> A(h, vector<T>(__VA_ARGS__));
#define vvvec(T, A, h1, h2, ...) vector<vector<vector<T>>> A(h1, vector<vector<T>>(h2, vector<T>(__VA_ARGS__)));
// clang-format on

// for loop
#define fori1(a) for (ll _ = 0; _ < (a); _++)
#define fori2(i, a) for (ll i = 0; i < (a); i++)
#define fori3(i, a, b) for (ll i = (a); i < (b); i++)
#define fori4(i, a, b, c) for (ll i = (a); ((c) > 0 || i > (b)) && ((c) < 0 || i < (b)); i += (c))
#define overload4(a, b, c, d, e, ...) e
#define fori(...) overload4(__VA_ARGS__, fori4, fori3, fori2, fori1)(__VA_ARGS__)

// declare and input
// clang-format off
#define INT(...) int __VA_ARGS__; inp(__VA_ARGS__);
#define LL(...) ll __VA_ARGS__; inp(__VA_ARGS__);
#define STRING(...) string __VA_ARGS__; inp(__VA_ARGS__);
#define CHAR(...) char __VA_ARGS__; inp(__VA_ARGS__);
#define DOUBLE(...) double __VA_ARGS__; STRING(str___); __VA_ARGS__ = stod(str___);
#define VEC(T, A, n) vector<T> A(n); inp(A);
#define VVEC(T, A, n, m) vector<vector<T>> A(n, vector<T>(m)); inp(A);
// clang-format on

// const value
const ll MOD1   = 1000000007;
const ll MOD9   = 998244353;
const double PI = acos(-1);

// other macro
#if !defined(RIN__LOCAL) && !defined(INTERACTIVE)
#define endl "\n"
#endif
#define spa ' '
#define len(A) ll(A.size())
#define all(A) begin(A), end(A)

// function
vector<char> stoc(string &S) {
    int n = S.size();
    vector<char> ret(n);
    for (int i = 0; i < n; i++) ret[i] = S[i];
    return ret;
}
string ctos(vector<char> &S) {
    int n      = S.size();
    string ret = "";
    for (int i = 0; i < n; i++) ret += S[i];
    return ret;
}

template <class T>
auto min(const T &a) {
    return *min_element(all(a));
}
template <class T>
auto max(const T &a) {
    return *max_element(all(a));
}
template <class T, class S>
auto clamp(T &a, const S &l, const S &r) {
    return (a > r ? r : a < l ? l : a);
}
template <class T, class S>
inline bool chmax(T &a, const S &b) {
    return (a < b ? a = b, 1 : 0);
}
template <class T, class S>
inline bool chmin(T &a, const S &b) {
    return (a > b ? a = b, 1 : 0);
}
template <class T, class S>
inline bool chclamp(T &a, const S &l, const S &r) {
    auto b = clamp(a, l, r);
    return (a != b ? a = b, 1 : 0);
}

template <typename T>
T sum(vector<T> &A) {
    T tot = 0;
    for (auto a : A) tot += a;
    return tot;
}

template <typename T>
vector<T> compression(vector<T> X) {
    sort(all(X));
    X.erase(unique(all(X)), X.end());
    return X;
}

// input and output
namespace io {

// vector<T>
template <typename T>
istream &operator>>(istream &is, vector<T> &A) {
    for (auto &a : A) is >> a;
    return is;
}
template <typename T>
ostream &operator<<(ostream &os, vector<T> &A) {
    for (size_t i = 0; i < A.size(); i++) {
        os << A[i];
        if (i != A.size() - 1) os << ' ';
    }
    return os;
}

// vector<vector<T>>
template <typename T>
istream &operator>>(istream &is, vector<vector<T>> &A) {
    for (auto &a : A) is >> a;
    return is;
}
template <typename T>
ostream &operator<<(ostream &os, vector<vector<T>> &A) {
    for (size_t i = 0; i < A.size(); i++) {
        os << A[i];
        if (i != A.size() - 1) os << endl;
    }
    return os;
}

// pair<S, T>
template <typename S, typename T>
istream &operator>>(istream &is, pair<S, T> &A) {
    is >> A.first >> A.second;
    return is;
}
template <typename S, typename T>
ostream &operator<<(ostream &os, pair<S, T> &A) {
    os << A.first << ' ' << A.second;
    return os;
}

// vector<pair<S, T>>
template <typename S, typename T>
istream &operator>>(istream &is, vector<pair<S, T>> &A) {
    for (size_t i = 0; i < A.size(); i++) {
        is >> A[i];
    }
    return is;
}
template <typename S, typename T>
ostream &operator<<(ostream &os, vector<pair<S, T>> &A) {
    for (size_t i = 0; i < A.size(); i++) {
        os << A[i];
        if (i != A.size() - 1) os << endl;
    }
    return os;
}

// tuple
template <typename T, size_t N>
struct TuplePrint {
    static ostream &print(ostream &os, const T &t) {
        TuplePrint<T, N - 1>::print(os, t);
        os << ' ' << get<N - 1>(t);
        return os;
    }
};
template <typename T>
struct TuplePrint<T, 1> {
    static ostream &print(ostream &os, const T &t) {
        os << get<0>(t);
        return os;
    }
};
template <typename... Args>
ostream &operator<<(ostream &os, const tuple<Args...> &t) {
    TuplePrint<decltype(t), sizeof...(Args)>::print(os, t);
    return os;
}

// io functions
void FLUSH() {
    cout << flush;
}

void print() {
    cout << endl;
}
template <class Head, class... Tail>
void print(Head &&head, Tail &&...tail) {
    cout << head;
    if (sizeof...(Tail)) cout << spa;
    print(std::forward<Tail>(tail)...);
}

template <typename T, typename S>
void prisep(vector<T> &A, S sep) {
    int n = A.size();
    for (int i = 0; i < n; i++) {
        cout << A[i];
        if (i != n - 1) cout << sep;
    }
    cout << endl;
}
template <typename T, typename S>
void priend(T A, S end) {
    cout << A << end;
}
template <typename T>
void prispa(T A) {
    priend(A, spa);
}
template <typename T, typename S>
bool printif(bool f, T A, S B) {
    if (f)
        print(A);
    else
        print(B);
    return f;
}

template <class... T>
void inp(T &...a) {
    (cin >> ... >> a);
}

} // namespace io
using namespace io;

// read graph
vector<vector<int>> read_edges(int n, int m, bool direct = false, int indexed = 1) {
    vector<vector<int>> edges(n, vector<int>());
    for (int i = 0; i < m; i++) {
        INT(u, v);
        u -= indexed;
        v -= indexed;
        edges[u].push_back(v);
        if (!direct) edges[v].push_back(u);
    }
    return edges;
}
vector<vector<int>> read_tree(int n, int indexed = 1) {
    return read_edges(n, n - 1, false, indexed);
}

template <typename T = long long>
vector<vector<pair<int, T>>> read_wedges(int n, int m, bool direct = false, int indexed = 1) {
    vector<vector<pair<int, T>>> edges(n, vector<pair<int, T>>());
    for (int i = 0; i < m; i++) {
        INT(u, v);
        T w;
        inp(w);
        u -= indexed;
        v -= indexed;
        edges[u].push_back({v, w});
        if (!direct) edges[v].push_back({u, w});
    }
    return edges;
}
template <typename T = long long>
vector<vector<pair<int, T>>> read_wtree(int n, int indexed = 1) {
    return read_wedges<T>(n, n - 1, false, indexed);
}

// yes / no
namespace yesno {

// yes
inline bool yes(bool f = true) {
    cout << (f ? "yes" : "no") << endl;
    return f;
}
inline bool Yes(bool f = true) {
    cout << (f ? "Yes" : "No") << endl;
    return f;
}
inline bool YES(bool f = true) {
    cout << (f ? "YES" : "NO") << endl;
    return f;
}

// no
inline bool no(bool f = true) {
    cout << (!f ? "yes" : "no") << endl;
    return f;
}
inline bool No(bool f = true) {
    cout << (!f ? "Yes" : "No") << endl;
    return f;
}
inline bool NO(bool f = true) {
    cout << (!f ? "YES" : "NO") << endl;
    return f;
}

// possible
inline bool possible(bool f = true) {
    cout << (f ? "possible" : "impossible") << endl;
    return f;
}
inline bool Possible(bool f = true) {
    cout << (f ? "Possible" : "Impossible") << endl;
    return f;
}
inline bool POSSIBLE(bool f = true) {
    cout << (f ? "POSSIBLE" : "IMPOSSIBLE") << endl;
    return f;
}

// impossible
inline bool impossible(bool f = true) {
    cout << (!f ? "possible" : "impossible") << endl;
    return f;
}
inline bool Impossible(bool f = true) {
    cout << (!f ? "Possible" : "Impossible") << endl;
    return f;
}
inline bool IMPOSSIBLE(bool f = true) {
    cout << (!f ? "POSSIBLE" : "IMPOSSIBLE") << endl;
    return f;
}

// Alice Bob
inline bool Alice(bool f = true) {
    cout << (f ? "Alice" : "Bob") << endl;
    return f;
}
inline bool Bob(bool f = true) {
    cout << (f ? "Bob" : "Alice") << endl;
    return f;
}

// Takahashi Aoki
inline bool Takahashi(bool f = true) {
    cout << (f ? "Takahashi" : "Aoki") << endl;
    return f;
}
inline bool Aoki(bool f = true) {
    cout << (f ? "Aoki" : "Takahashi") << endl;
    return f;
}

} // namespace yesno
using namespace yesno;

} // namespace templates
using namespace templates;

template <int MOD>
struct Modint {
    int x;
    Modint() : x(0) {}
    Modint(int64_t y) {
        if (y >= 0)
            x = y % MOD;
        else
            x = (y % MOD + MOD) % MOD;
    }

    Modint &operator+=(const Modint &p) {
        x += p.x;
        if (x >= MOD) x -= MOD;
        return *this;
    }

    Modint &operator-=(const Modint &p) {
        x -= p.x;
        if (x < 0) x += MOD;
        return *this;
    }

    Modint &operator*=(const Modint &p) {
        x = int(1LL * x * p.x % MOD);
        return *this;
    }

    Modint &operator/=(const Modint &p) {
        *this *= p.inverse();
        return *this;
    }

    Modint &operator%=(const Modint &p) {
        assert(p.x == 0);
        return *this;
    }

    Modint operator-() const {
        return Modint(-x);
    }

    Modint &operator++() {
        x++;
        if (x == MOD) x = 0;
        return *this;
    }

    Modint &operator--() {
        if (x == 0) x = MOD;
        x--;
        return *this;
    }

    Modint operator++(int) {
        Modint result = *this;
        ++*this;
        return result;
    }

    Modint operator--(int) {
        Modint result = *this;
        --*this;
        return result;
    }

    friend Modint operator+(const Modint &lhs, const Modint &rhs) {
        return Modint(lhs) += rhs;
    }

    friend Modint operator-(const Modint &lhs, const Modint &rhs) {
        return Modint(lhs) -= rhs;
    }

    friend Modint operator*(const Modint &lhs, const Modint &rhs) {
        return Modint(lhs) *= rhs;
    }

    friend Modint operator/(const Modint &lhs, const Modint &rhs) {
        return Modint(lhs) /= rhs;
    }

    friend Modint operator%(const Modint &lhs, const Modint &rhs) {
        assert(rhs.x == 0);
        return Modint(lhs);
    }

    bool operator==(const Modint &p) const {
        return x == p.x;
    }

    bool operator!=(const Modint &p) const {
        return x != p.x;
    }

    bool operator<(const Modint &rhs) const {
        return x < rhs.x;
    }

    bool operator<=(const Modint &rhs) const {
        return x <= rhs.x;
    }

    bool operator>(const Modint &rhs) const {
        return x > rhs.x;
    }

    bool operator>=(const Modint &rhs) const {
        return x >= rhs.x;
    }

    Modint inverse() const {
        int a = x, b = MOD, u = 1, v = 0, t;
        while (b > 0) {
            t = a / b;
            a -= t * b;
            u -= t * v;
            std::swap(a, b);
            std::swap(u, v);
        }
        return Modint(u);
    }

    Modint pow(int64_t k) const {
        Modint ret(1);
        Modint y(x);
        while (k > 0) {
            if (k & 1) ret *= y;
            y *= y;
            k >>= 1;
        }
        return ret;
    }

    friend std::ostream &operator<<(std::ostream &os, const Modint &p) {
        return os << p.x;
    }

    friend std::istream &operator>>(std::istream &is, Modint &p) {
        int64_t y;
        is >> y;
        p = Modint<MOD>(y);
        return (is);
    }

    static int get_mod() {
        return MOD;
    }
};

struct Arbitrary_Modint {
    int x;
    static int MOD;

    static void set_mod(int mod) {
        MOD = mod;
    }

    Arbitrary_Modint() : x(0) {}
    Arbitrary_Modint(int64_t y) {
        if (y >= 0)
            x = y % MOD;
        else
            x = (y % MOD + MOD) % MOD;
    }

    Arbitrary_Modint &operator+=(const Arbitrary_Modint &p) {
        x += p.x;
        if (x >= MOD) x -= MOD;
        return *this;
    }

    Arbitrary_Modint &operator-=(const Arbitrary_Modint &p) {
        x -= p.x;
        if (x < 0) x += MOD;
        return *this;
    }

    Arbitrary_Modint &operator*=(const Arbitrary_Modint &p) {
        x = int(1LL * x * p.x % MOD);
        return *this;
    }

    Arbitrary_Modint &operator/=(const Arbitrary_Modint &p) {
        *this *= p.inverse();
        return *this;
    }

    Arbitrary_Modint &operator%=(const Arbitrary_Modint &p) {
        assert(p.x == 0);
        return *this;
    }

    Arbitrary_Modint operator-() const {
        return Arbitrary_Modint(-x);
    }

    Arbitrary_Modint &operator++() {
        x++;
        if (x == MOD) x = 0;
        return *this;
    }

    Arbitrary_Modint &operator--() {
        if (x == 0) x = MOD;
        x--;
        return *this;
    }

    Arbitrary_Modint operator++(int) {
        Arbitrary_Modint result = *this;
        ++*this;
        return result;
    }

    Arbitrary_Modint operator--(int) {
        Arbitrary_Modint result = *this;
        --*this;
        return result;
    }

    friend Arbitrary_Modint operator+(const Arbitrary_Modint &lhs, const Arbitrary_Modint &rhs) {
        return Arbitrary_Modint(lhs) += rhs;
    }

    friend Arbitrary_Modint operator-(const Arbitrary_Modint &lhs, const Arbitrary_Modint &rhs) {
        return Arbitrary_Modint(lhs) -= rhs;
    }

    friend Arbitrary_Modint operator*(const Arbitrary_Modint &lhs, const Arbitrary_Modint &rhs) {
        return Arbitrary_Modint(lhs) *= rhs;
    }

    friend Arbitrary_Modint operator/(const Arbitrary_Modint &lhs, const Arbitrary_Modint &rhs) {
        return Arbitrary_Modint(lhs) /= rhs;
    }

    friend Arbitrary_Modint operator%(const Arbitrary_Modint &lhs, const Arbitrary_Modint &rhs) {
        assert(rhs.x == 0);
        return Arbitrary_Modint(lhs);
    }

    bool operator==(const Arbitrary_Modint &p) const {
        return x == p.x;
    }

    bool operator!=(const Arbitrary_Modint &p) const {
        return x != p.x;
    }

    bool operator<(const Arbitrary_Modint &rhs) {
        return x < rhs.x;
    }

    bool operator<=(const Arbitrary_Modint &rhs) {
        return x <= rhs.x;
    }

    bool operator>(const Arbitrary_Modint &rhs) {
        return x > rhs.x;
    }

    bool operator>=(const Arbitrary_Modint &rhs) {
        return x >= rhs.x;
    }

    Arbitrary_Modint inverse() const {
        int a = x, b = MOD, u = 1, v = 0, t;
        while (b > 0) {
            t = a / b;
            a -= t * b;
            u -= t * v;
            std::swap(a, b);
            std::swap(u, v);
        }
        return Arbitrary_Modint(u);
    }

    Arbitrary_Modint pow(int64_t k) const {
        Arbitrary_Modint ret(1);
        Arbitrary_Modint y(x);
        while (k > 0) {
            if (k & 1) ret *= y;
            y *= y;
            k >>= 1;
        }
        return ret;
    }

    friend std::ostream &operator<<(std::ostream &os, const Arbitrary_Modint &p) {
        return os << p.x;
    }

    friend std::istream &operator>>(std::istream &is, Arbitrary_Modint &p) {
        int64_t y;
        is >> y;
        p = Arbitrary_Modint(y);
        return (is);
    }

    static int get_mod() {
        return MOD;
    }
};
int Arbitrary_Modint::MOD = 998244353;

using modint9 = Modint<998244353>;
using modint1 = Modint<1000000007>;
using modint  = Arbitrary_Modint;
using mint    = modint9;

template <typename T>
struct Combination {
    int N;
    std::vector<T> fact, invfact;
    Combination(int N) : N(N) {
        fact.resize(N + 1);
        invfact.resize(N + 1);
        fact[0] = 1;
        for (int i = 1; i <= N; i++) {
            fact[i] = fact[i - 1] * i;
        }
        invfact[N] = T(1) / fact[N];
        for (int i = N - 1; i >= 0; i--) {
            invfact[i] = invfact[i + 1] * (i + 1);
        }
    }

    void extend(int n) {
        int le = fact.size();
        fact.resize(n + 1);
        invfact.resize(n + 1);
        for (int i = le; i <= n; i++) {
            fact[i] = fact[i - 1] * i;
        }
        invfact[n] = T(1) / fact[n];
        for (int i = n - 1; i >= le; i--) {
            invfact[i] = invfact[i + 1] * (i + 1);
        }
    }

    T nCk(int n, int k) {
        if (k > n || k < 0) return T(0);
        if (n >= int(fact.size())) extend(n);
        return fact[n] * invfact[k] * invfact[n - k];
    }

    T nPk(int n, int k) {
        if (k > n || k < 0) return T(0);
        if (n >= int(fact.size())) extend(n);
        return fact[n] * invfact[n - k];
    }

    T nHk(int n, int k) {
        if (n == 0 && k == 0) return T(1);
        return nCk(n + k - 1, k);
    }

    T Catalan(int n) {
        return nCk(2 * n, n) - nCk(2 * n, n + 1);
    }

    // n 個の +1, m 個の -1, 累積和が常にk以下
    T Catalan(int n, int m, int k) {
        if (n > m + k || k < 0)
            return T(0);
        else
            return nCk(n + m, n) - nCk(n + m, m + k + 1);
    }

    T Narayana(int n, int k) {
        return nCk(n, k) * nCk(n, k - 1) / n;
    }

    T inv(int n) {
        assert(n >= 1);
        if (n >= int(fact.size())) extend(n);
        return invfact[n] * fact[n - 1];
    }
};

template <typename mint>
struct NumberTheoreticTransform {
    static std::vector<mint> roots, iroots, rate3, irate3;
    static int max_base;

    NumberTheoreticTransform() = default;

    static void init() {
        if (!roots.empty()) return;
        const unsigned mod = mint::get_mod();
        auto tmp           = mod - 1;
        max_base           = 0;
        while (tmp % 2 == 0) {
            tmp >>= 1;
            max_base++;
        }
        mint root = 2;
        while (root.pow((mod - 1) >> 1) == 1) root++;

        roots.resize(max_base + 1);
        iroots.resize(max_base + 1);
        rate3.resize(max_base + 1);
        irate3.resize(max_base + 1);

        roots[max_base]  = root.pow((mod - 1) >> max_base);
        iroots[max_base] = mint(1) / roots[max_base];
        for (int i = max_base - 1; i >= 0; i--) {
            roots[i]  = roots[i + 1] * roots[i + 1];
            iroots[i] = iroots[i + 1] * iroots[i + 1];
        }

        mint prod = 1, iprod = 1;
        for (int i = 0; i <= max_base - 3; i++) {
            rate3[i]  = roots[i + 3] * prod;
            irate3[i] = iroots[i + 3] * iprod;
            prod *= iroots[i + 3];
            iprod *= roots[i + 3];
        }
    }

    static void ntt(std::vector<mint> &A) {
        init();
        int n     = int(A.size());
        int h     = __builtin_ctz(n);
        int le    = 0;
        mint imag = roots[2];
        if (h & 1) {
            int p = 1 << (h - 1);
            for (int i = 0; i < p; i++) {
                auto r   = A[i + p];
                A[i + p] = A[i] - r;
                A[i] += r;
            }
            le++;
        }
        for (; le + 1 < h; le += 2) {
            int p = 1 << (h - le - 2);

            for (int i = 0; i < p; i++) {
                auto a0        = A[i];
                auto a1        = A[i + p];
                auto a2        = A[i + 2 * p];
                auto a3        = A[i + 3 * p];
                auto a1na3imag = (a1 - a3) * imag;
                A[i]           = a0 + a2 + a1 + a3;
                A[i + p]       = a0 + a2 - (a1 + a3);
                A[i + 2 * p]   = a0 - a2 + a1na3imag;
                A[i + 3 * p]   = a0 - a2 - a1na3imag;
            }

            mint rot = rate3[0];
            for (int s = 1; s < (1 << le); s++) {
                int offset = s << (h - le);
                mint rot2  = rot * rot;
                mint rot3  = rot2 * rot;
                for (int i = 0; i < p; i++) {
                    auto a0               = A[i + offset];
                    auto a1               = A[i + offset + p] * rot;
                    auto a2               = A[i + offset + 2 * p] * rot2;
                    auto a3               = A[i + offset + 3 * p] * rot3;
                    auto a1na3imag        = (a1 - a3) * imag;
                    A[i + offset]         = a0 + a2 + a1 + a3;
                    A[i + offset + p]     = a0 + a2 - (a1 + a3);
                    A[i + offset + 2 * p] = a0 - a2 + a1na3imag;
                    A[i + offset + 3 * p] = a0 - a2 - a1na3imag;
                }
                rot *= rate3[__builtin_ctz(~s)];
            }
        }
    }

    static void intt(std::vector<mint> &A, bool f = true) {
        init();
        int n      = int(A.size());
        int h      = __builtin_ctz(n);
        int le     = h;
        mint iimag = iroots[2];
        for (; le > 1; le -= 2) {
            int p = 1 << (h - le);

            for (int i = 0; i < p; i++) {
                auto a0         = A[i];
                auto a1         = A[i + p];
                auto a2         = A[i + 2 * p];
                auto a3         = A[i + 3 * p];
                auto a2na3iimag = (a2 - a3) * iimag;
                A[i]            = a0 + a1 + a2 + a3;
                A[i + p]        = a0 - a1 + a2na3iimag;
                A[i + 2 * p]    = a0 + a1 - (a2 + a3);
                A[i + 3 * p]    = a0 - a1 - a2na3iimag;
            }

            mint irot = irate3[0];
            for (int s = 1; s < (1 << (le - 2)); s++) {
                int offset = s << (h - le + 2);
                mint irot2 = irot * irot;
                mint irot3 = irot2 * irot;
                for (int i = 0; i < p; i++) {
                    auto a0               = A[i + offset];
                    auto a1               = A[i + offset + p];
                    auto a2               = A[i + offset + 2 * p];
                    auto a3               = A[i + offset + 3 * p];
                    auto a2na3iimag       = (a2 - a3) * iimag;
                    A[i + offset]         = a0 + a1 + a2 + a3;
                    A[i + offset + p]     = (a0 - a1 + a2na3iimag) * irot;
                    A[i + offset + 2 * p] = (a0 + a1 - (a2 + a3)) * irot2;
                    A[i + offset + 3 * p] = (a0 - a1 - a2na3iimag) * irot3;
                }
                irot *= irate3[__builtin_ctz(~s)];
            }
        }
        if (le >= 1) {
            int p = 1 << (h - 1);
            for (int i = 0; i < p; i++) {
                auto ajp = A[i] - A[i + p];
                A[i] += A[i + p];
                A[i + p] = ajp;
            }
        }
        if (f) {
            mint inv = mint(1) / n;
            for (int i = 0; i < n; i++) {
                A[i] *= inv;
            }
        }
    }

    static std::vector<mint> multiply(std::vector<mint> A, std::vector<mint> B) {
        int need = int(A.size() + B.size()) - 1;
        if (std::min(A.size(), B.size()) < 60u) {
            std::vector<mint> C(need, 0);
            for (size_t i = 0; i < A.size(); i++)
                for (size_t j = 0; j < B.size(); j++) {
                    C[i + j] += A[i] * B[j];
                }
            return C;
        }
        int sz = 1;
        while (sz < need) sz <<= 1;
        A.resize(sz, 0);
        B.resize(sz, 0);
        ntt(A);
        ntt(B);
        mint inv = mint(1) / sz;
        for (int i = 0; i < sz; i++) A[i] *= B[i] * inv;
        intt(A, false);
        A.resize(need);
        return A;
    }
};

template <typename mint>
std::vector<mint> NumberTheoreticTransform<mint>::roots = std::vector<mint>();
template <typename mint>
std::vector<mint> NumberTheoreticTransform<mint>::iroots = std::vector<mint>();
template <typename mint>
std::vector<mint> NumberTheoreticTransform<mint>::rate3 = std::vector<mint>();
template <typename mint>
std::vector<mint> NumberTheoreticTransform<mint>::irate3 = std::vector<mint>();
template <typename mint>
int NumberTheoreticTransform<mint>::max_base = 0;

template <typename T, typename S>
T modpow(T a, S b, T MOD) {
    T ret = 1;
    while (b > 0) {
        if (b & 1) {
            ret *= a;
            ret %= MOD;
        }
        a *= a;
        a %= MOD;
        b >>= 1;
    }
    return ret;
}

template <typename T>
T cipolla(T a, T MOD) {
    if (MOD == 2)
        return a;
    else if (a == 0)
        return 0;
    else if (modpow(a, (MOD - 1) / 2, MOD) != 1)
        return -1;
    T b = 1;
    while (modpow((b * b + MOD - a) % MOD, (MOD - 1) / 2, MOD) == 1) {
        b++;
    }

    T base     = (b * b + MOD - a) % MOD;
    auto multi = [&](T a0, T b0, T a1, T b1) -> std::pair<T, T> {
        return {(a0 * a1 + (b0 * b1 % MOD) * base) % MOD, (a0 * b1 + b0 * a1) % MOD};
    };

    auto pow_ = [&](auto self, T a, T b, T n) -> std::pair<T, T> {
        if (n == 0) return {1, 0};
        auto tmp = multi(a, b, a, b);
        auto ret = self(self, tmp.first, tmp.second, n / 2);
        if (n & 1) {
            ret = multi(ret.first, ret.second, a, b);
        }
        return ret;
    };

    return pow_(pow_, b, 1LL, (MOD + 1) / 2).first;
}

template <typename mint>
struct FormalPowerSeries : std::vector<mint> {
    using std::vector<mint>::vector;
    using FPS = FormalPowerSeries;
    static std::vector<mint> inv_x;

    void shrink() {
        while (this->size() && this->back() == mint(0)) {
            this->pop_back();
        }
    }

    FPS &operator+=(const FPS &A) {
        if (A.size() > this->size()) this->resize(A.size());
        for (size_t i = 0; i < A.size(); i++) (*this)[i] += A[i];
        return *this;
    }

    FPS &operator+=(const mint &x) {
        if (this->empty()) this->resize(1);
        (*this)[0] += x;
        return *this;
    }

    FPS &operator-=(const FPS &A) {
        if (A.size() > this->size()) this->resize(A.size());
        for (size_t i = 0; i < A.size(); i++) (*this)[i] -= A[i];
        return *this;
    }

    FPS &operator-=(const mint &x) {
        if (this->empty()) this->resize(1);
        (*this)[0] -= x;
        return *this;
    }

    FPS &operator*=(const FPS &A) {
        if (this->empty() || A.empty()) {
            this->clear();
            return *this;
        }
        auto res     = NumberTheoreticTransform<mint>::multiply(*this, A);
        return *this = {begin(res), end(res)};
    }

    FPS &operator*=(const mint &x) {
        for (size_t i = 0; i < this->size(); i++) (*this)[i] *= x;
        return *this;
    }

    FPS operator+(const FPS &A) const {
        return FPS(*this) += A;
    }
    FPS operator+(const mint &x) const {
        return FPS(*this) += x;
    }
    FPS operator-(const FPS &A) const {
        return FPS(*this) -= A;
    }
    FPS operator-(const mint &x) const {
        return FPS(*this) -= x;
    }
    FPS operator*(const FPS &A) const {
        return FPS(*this) *= A;
    }
    FPS operator*(const mint &x) const {
        return FPS(*this) *= x;
    }
    FPS operator-() const {
        FPS ret(this->size);
        for (size_t i = 0; i < this->size(); i++) ret[i] = -(*this)[i];
        return ret;
    }

    FPS inv(int deg = -1) {
        assert((*this)[0] != mint(0));
        if (deg == -1) deg = this->size();
        FPS g = {mint(1) / (*this)[0]};
        int l = 1;
        while (l < deg) {
            FPS tmp = g * 2;
            l <<= 1;
            FPS tmp2;
            g *= g;
            if (int(this->size()) >= l)
                tmp2 = FPS({this->begin(), this->begin() + l}) * g;
            else
                tmp2 = (*this) * g;
            g = tmp - tmp2;
            g.resize(l);
        }
        g.resize(deg);
        return g;
    }

    void iinv(int deg = -1) {
        *this = inv(deg);
    }

    FPS differential() {
        FPS ret(this->size() - 1);
        for (size_t i = 0; i < this->size() - 1; i++) ret[i] = (*this)[i + 1] * (i + 1);
        return ret;
    }

    void idifferential() {
        *this = this->differential();
    }

    void extend_inv(int n) {
        int bn = inv_x.size();
        if (n >= bn) {
            inv_x.resize(n + 1, 0);
            if (bn == 0) {
                inv_x[0] = 0;
                inv_x[1] = 1;
                bn       = 2;
            }
            long long mod = mint::get_mod();
            for (int i = bn; i <= n; i++) {
                inv_x[i] = mod - inv_x[mod % i].x * (mod / i) % mod;
            }
        }
    }

    FPS integral() {
        extend_inv(this->size());
        FPS ret(this->size() + 1);
        for (size_t i = 0; i < this->size(); i++) ret[i + 1] = (*this)[i] * inv_x[i + 1];
        return ret;
    }

    void iintegral() {
        *this = this->integral();
    }

    FPS log(int deg = -1) {
        assert((*this)[0] == mint(1));
        if (deg == -1) deg = this->size();
        FPS B = (this->differential()) * (this->inv());
        B.resize(deg - 1);
        return B.integral();
    }

    void ilog(int deg = -1) {
        *this = this->log(deg);
    }

    FPS exp(int deg = -1) {
        assert((*this)[0] == mint(0));
        if (deg == -1) deg = this->size();
        FPS g = {1};
        int l = 1;
        while (l < deg * 2) {
            l *= 2;
            FPS tmp = {1};
            tmp -= g.log(l);
            if (int(this->size()) >= l)
                tmp += FPS({this->begin(), this->begin() + l});
            else
                tmp += (*this);
            g *= tmp;
            g.resize(l);
        }
        g.resize(deg);
        return g;
    }

    void iexp(int deg = -1) {
        *this = this->exp(deg);
    }

    FPS pow(long long k, int deg = -1) {
        if (deg == -1) deg = this->size();
        if (k == 0) {
            FPS ret(deg, 0);
            ret[0] = 1;
            return ret;
        }

        int p = -1;
        for (int i = 0; i < deg; i++) {
            if ((*this)[i] != 0) {
                p = i;
                break;
            }
        }

        if (p == -1 || p > deg / k) {
            FPS ret(deg, 0);
            return ret;
        }
        mint inv = mint(1) / (*this)[p];
        FPS A    = FPS({(*this).begin() + p, (*this).end()});
        A *= inv;
        A.ilog(deg);
        A *= k % mint::get_mod();
        A.iexp(deg);
        FPS B(p * k, 0);
        B.insert(B.end(), A.begin(), A.begin() + (deg - p * k));
        B *= (*this)[p].pow(k);
        return B;
    }

    void ipow(long long k, int deg = -1) {
        *this = this->pow(k, deg);
    }

    FPS sqrt(int deg = -1) {
        if (deg == -1) deg = this->size();
        if (this->size() == 0u) {
            FPS ret(deg, 0);
            return ret;
        }
        if ((*this)[0] == mint(0)) {
            for (size_t i = 1; i < this->size(); i++) {
                if ((*this)[i] != 0) {
                    if (i & 1) {
                        FPS ret;
                        return ret;
                    }
                    if (deg <= int(i / 2)) break;
                    FPS ret = FPS({this->begin() + i, this->end()}).sqrt(deg - i / 2);
                    if (ret.size() == 0u) return ret;
                    FPS ret2(i / 2, 0);
                    ret2.insert(ret2.end(), ret.begin(), ret.end());
                    std::swap(ret, ret2);
                    if (int(ret.size()) < deg) ret.resize(deg);
                    return ret;
                }
            }
            FPS ret(deg, 0);
            return ret;
        }
        long long sq = cipolla<long long>((*this)[0].x, mint::get_mod());
        if (sq == -1) {
            FPS ret;
            return ret;
        }
        mint inv2 = mint(1) / 2;
        FPS g     = {sq};
        int l     = 1;
        while (l < deg) {
            l *= 2;
            if (int(this->size()) >= l)
                g += FPS({this->begin(), this->begin() + l}) * g.inv(l);
            else
                g += (*this) * g.inv(l);
            g *= inv2;
        }
        g.resize(deg);
        return g;
    }

    void isqrt(int deg = -1) {
        *this = this->sqrt(deg);
    }

    FPS taylorshift(mint a) {
        auto A  = (*this);
        int deg = A.size();
        extend_inv(deg);
        mint fac = 1;
        for (int i = 0; i < deg; i++) {
            A[i] *= fac;
            fac *= (i + 1);
        }
        reverse(A.begin(), A.end());
        FPS g(deg, 0);
        g[0] = 1;
        for (int i = 1; i < deg; i++) g[i] = g[i - 1] * a * inv_x[i];
        A *= g;
        if (int(A.size()) > deg) A.resize(deg);
        reverse(A.begin(), A.end());
        mint invfac = 1;
        for (int i = 0; i < deg; i++) {
            A[i] *= invfac;
            invfac *= inv_x[i + 1];
        }
        return A;
    }

    void itaylorshift(mint a) {
        int deg = this->size();
        extend_inv(deg);
        mint fac = 1;
        for (int i = 0; i < deg; i++) {
            (*this)[i] *= fac;
            fac *= (i + 1);
        }
        reverse(this->begin(), this->end());
        FPS g(deg, 0);
        g[0] = 1;
        for (int i = 1; i < deg; i++) g[i] = g[i - 1] * a * inv_x[i];
        (*this) *= g;
        if (int(this->size()) > deg) this->resize(deg);
        reverse(this->begin(), this->end());
        mint invfac = 1;
        for (int i = 0; i < deg; i++) {
            (*this)[i] *= invfac;
            invfac *= inv_x[i + 1];
        }
    }

    std::pair<FPS, FPS> division_of_polynomial(FPS G) {
        FPS F = *this;
        if (F.size() < G.size()) {
            return {{}, F};
        }

        reverse(F.begin(), F.end());
        reverse(G.begin(), G.end());
        int deg = F.size() - G.size() + 1u;
        auto Q  = F * G.inv(deg);
        if (int(Q.size()) > deg) Q.resize(deg);
        reverse(Q.begin(), Q.end());
        reverse(F.begin(), F.end());
        reverse(G.begin(), G.end());
        auto R = F - G * Q;
        R.shrink();
        return {Q, R};
    }

    std::vector<mint> multipoint_evaluation(std::vector<mint> &X) {
        int m  = X.size();
        int m2 = 1;
        while (m2 <= m - 1) m2 *= 2;
        std::vector<FPS> G(m2 << 1, FPS(1, 1));
        for (int i = 0; i < m; i++) G[m2 + i] = {-X[i], 1};

        for (int i = m2 - 1; i >= 0; i--) G[i] = G[i << 1] * G[(i << 1) | 1];
        G[1] = this->division_of_polynomial(G[1]).second;
        for (int i = 2; i < m2 + m; i++) G[i] = G[i >> 1].division_of_polynomial(G[i]).second;

        std::vector<mint> Y(m);
        for (int i = 0; i < m; i++) {
            if (G[m2 + i].empty())
                Y[i] = 0;
            else
                Y[i] = G[m2 + i][0];
        }
        return Y;
    }

    std::vector<long long> multipoint_evaluation(std::vector<long long> &X) {
        int m  = X.size();
        int m2 = 1;
        while (m2 <= m - 1) m2 *= 2;
        std::vector<FPS> G(m2 << 1, FPS(1, 1));
        for (int i = 0; i < m; i++) G[m2 + i] = {-X[i], 1};

        for (int i = m2 - 1; i >= 0; i--) G[i] = G[i << 1] * G[(i << 1) | 1];
        G[1] = this->division_of_polynomial(G[1]).second;
        for (int i = 2; i < m2 + m; i++) G[i] = G[i >> 1].division_of_polynomial(G[i]).second;

        std::vector<long long> Y(m);
        for (int i = 0; i < m; i++) {
            if (G[m2 + i].empty())
                Y[i] = 0;
            else
                Y[i] = G[m2 + i][0].x;
        }
        return Y;
    }

    friend std::ostream &operator<<(std::ostream &os, const FPS &A) {
        for (size_t i = 0; i < A.size(); i++) {
            os << A[i];
            if (i != A.size() - 1) os << ' ';
        }
        return os;
    }

    friend std::istream &operator>>(std::istream &is, FPS &A) {
        for (size_t i = 0; i < A.size(); i++) {
            is >> A[i];
        }
        return (is);
    }
};

template <typename mint>
std::vector<mint> FormalPowerSeries<mint>::inv_x = std::vector<mint>();
using FPS                                        = FormalPowerSeries<mint>;

void solve() {
    INT(n);
    VEC(int, A, n);
    int m = A[0];
    Combination<mint> Comb(m + 10);
    vec(ll, C, m + 1, 0);
    for (auto a : A) C[a]++;
    vec(mint, pow2, m + 1, 1);
    fori(i, 1, m + 1) pow2[i] = pow2[i - 1] * 2;

    FPS F(m + 1);
    FPS G(m + 1);
    fori(i, m + 1) G[i] = Comb.invfact[i];
    auto dc             = [&](auto &&self, int l, int r) -> void {
        if (l + 1 == r) {
            if (l == 0) {
                F[0] = 1;
            } else {
                F[l] *= pow2[C[l]] - 1;
            }
            return;
        }
        int m = (l + r) / 2;
        self(self, l, m);

        auto L   = FPS(F.begin() + l, F.begin() + m);
        auto R   = FPS(G.begin(), G.begin() + r - l);
        auto res = L * R;
        fori(i, m, r) {
            F[i] += res[i - l];
        }

        self(self, m, r);
    };

    dc(dc, 0, m + 1);

    mint ans = 0;
    fori(i, 1, m + 1) {
        ans += F[i] * Comb.fact[i];
    }
    print(ans);
}

int main() {
#ifndef INTERACTIVE
    cin.tie(0)->sync_with_stdio(0);
#endif
    // cout << fixed << setprecision(12);
    int t;
    t = 1;
    // cin >> t;
    while (t--) solve();
    return 0;
}