#include <bits/stdc++.h>
namespace suisen {
    template <class T> bool chmin(T& x, const T& y) { return y >= x ? false : (x = y, true); }
    template <class T> bool chmax(T& x, const T& y) { return y <= x ? false : (x = y, true); }
    template <class T> constexpr int pow_m1(T n) { return -(n & 1) | 1; }
    template <class T> constexpr T fld(const T x, const T y) { T q = x / y, r = x % y; return q - ((x ^ y) < 0 and (r != 0)); }
    template <class T> constexpr T cld(const T x, const T y) { T q = x / y, r = x % y; return q + ((x ^ y) > 0 and (r != 0)); }
}
namespace suisen::macro {
#define IMPL_REPITER(cond) auto& begin() { return *this; } auto end() { return nullptr; } auto& operator*() { return _val; } auto& operator++() { return _val += _step, *this; } bool operator!=(std::nullptr_t) { return cond; }
    template <class Int, class IntL = Int, class IntStep = Int, std::enable_if_t<(std::is_signed_v<Int> == std::is_signed_v<IntL>), std::nullptr_t> = nullptr> struct rep_impl {
        Int _val; const Int _end, _step;
        rep_impl(Int n) : rep_impl(0, n) {}
        rep_impl(IntL l, Int r, IntStep step = 1) : _val(l), _end(r), _step(step) {}
        IMPL_REPITER((_val < _end))
    };
    template <class Int, class IntL = Int, class IntStep = Int, std::enable_if_t<(std::is_signed_v<Int> == std::is_signed_v<IntL>), std::nullptr_t> = nullptr> struct rrep_impl {
        Int _val; const Int _end, _step;
        rrep_impl(Int n) : rrep_impl(0, n) {}
        rrep_impl(IntL l, Int r) : _val(r - 1), _end(l), _step(-1) {}
        rrep_impl(IntL l, Int r, IntStep step) : _val(l + fld<Int>(r - l - 1, step) * step), _end(l), _step(-step) {}
        IMPL_REPITER((_val >= _end))
    };
    template <class Int, class IntStep = Int> struct repinf_impl {
        Int _val; const Int _step;
        repinf_impl(Int l, IntStep step = 1) : _val(l), _step(step) {}
        IMPL_REPITER((true))
    };
#undef IMPL_REPITER
}

#include <iostream>

#include <limits>
#include <type_traits>
namespace suisen {
    template <typename ...Constraints> using constraints_t = std::enable_if_t<std::conjunction_v<Constraints...>, std::nullptr_t>;

    template <typename T, typename = std::nullptr_t> struct bitnum { static constexpr int value = 0; };
    template <typename T> struct bitnum<T, constraints_t<std::is_integral<T>>> { static constexpr int value = std::numeric_limits<std::make_unsigned_t<T>>::digits; };
    template <typename T> static constexpr int bitnum_v = bitnum<T>::value;
    template <typename T, size_t n> struct is_nbit { static constexpr bool value = bitnum_v<T> == n; };
    template <typename T, size_t n> static constexpr bool is_nbit_v = is_nbit<T, n>::value;

    template <typename T, typename = std::nullptr_t> struct safely_multipliable { using type = T; };
    template <typename T> struct safely_multipliable<T, constraints_t<std::is_signed<T>, is_nbit<T, 32>>> { using type = long long; };
    template <typename T> struct safely_multipliable<T, constraints_t<std::is_signed<T>, is_nbit<T, 64>>> { using type = __int128_t; };
    template <typename T> struct safely_multipliable<T, constraints_t<std::is_unsigned<T>, is_nbit<T, 32>>> { using type = unsigned long long; };
    template <typename T> struct safely_multipliable<T, constraints_t<std::is_unsigned<T>, is_nbit<T, 64>>> { using type = __uint128_t; };
    template <typename T> using safely_multipliable_t = typename safely_multipliable<T>::type;

    template <typename T, typename = void> struct rec_value_type { using type = T; };
    template <typename T> struct rec_value_type<T, std::void_t<typename T::value_type>> {
        using type = typename rec_value_type<typename T::value_type>::type;
    };
    template <typename T> using rec_value_type_t = typename rec_value_type<T>::type;

    template <typename T> class is_iterable {
        template <typename T_> static auto test(T_ e) -> decltype(e.begin(), e.end(), std::true_type{});
        static std::false_type test(...);
    public:
        static constexpr bool value = decltype(test(std::declval<T>()))::value;
    };
    template <typename T> static constexpr bool is_iterable_v = is_iterable<T>::value;
    template <typename T> class is_writable {
        template <typename T_> static auto test(T_ e) -> decltype(std::declval<std::ostream&>() << e, std::true_type{});
        static std::false_type test(...);
    public:
        static constexpr bool value = decltype(test(std::declval<T>()))::value;
    };
    template <typename T> static constexpr bool is_writable_v = is_writable<T>::value;
    template <typename T> class is_readable {
        template <typename T_> static auto test(T_ e) -> decltype(std::declval<std::istream&>() >> e, std::true_type{});
        static std::false_type test(...);
    public:
        static constexpr bool value = decltype(test(std::declval<T>()))::value;
    };
    template <typename T> static constexpr bool is_readable_v = is_readable<T>::value;
} // namespace suisen
namespace suisen::io {
    template <typename IStream, std::enable_if_t<std::conjunction_v<std::is_base_of<std::istream, std::remove_reference_t<IStream>>, std::negation<std::is_const<std::remove_reference_t<IStream>>>>, std::nullptr_t> = nullptr>
    struct InputStream {
    private:
        using istream_type = std::remove_reference_t<IStream>;
        IStream is;
        struct { InputStream* is; template <typename T> operator T() { T e; *is >> e; return e; } } _reader{ this };
    public:
        template <typename IStream_> InputStream(IStream_ &&is) : is(std::move(is)) {}
        template <typename IStream_> InputStream(IStream_ &is) : is(is) {}
        template <typename T> InputStream& operator>>(T& e) {
            if constexpr (suisen::is_readable_v<T>) is >> e; else _read(e);
            return *this;
        }
        auto read() { return _reader; }
        template <typename Head, typename... Tail>
        void read(Head& head, Tail &...tails) { ((*this >> head) >> ... >> tails); }
        istream_type& get_stream() { return is; }
    private:
        static __uint128_t _stou128(const std::string& s) {
            __uint128_t ret = 0;
            for (char c : s) if ('0' <= c and c <= '9') ret = 10 * ret + c - '0';
            return ret;
        }
        static __int128_t _stoi128(const std::string& s) { return (s[0] == '-' ? -1 : +1) * _stou128(s); }

        void _read(__uint128_t& v) { v = _stou128(std::string(_reader)); }
        void _read(__int128_t& v) { v = _stoi128(std::string(_reader)); }
        template <typename T, typename U>
        void _read(std::pair<T, U>& a) { *this >> a.first >> a.second; }
        template <size_t N = 0, typename ...Args>
        void _read(std::tuple<Args...>& a) { if constexpr (N < sizeof...(Args)) *this >> std::get<N>(a), _read<N + 1>(a); }
        template <typename Iterable, std::enable_if_t<suisen::is_iterable_v<Iterable>, std::nullptr_t> = nullptr>
        void _read(Iterable& a) { for (auto& e : a) *this >> e; }
    };
    template <typename IStream>
    InputStream(IStream &&) -> InputStream<IStream>;
    template <typename IStream>
    InputStream(IStream &) -> InputStream<IStream&>;

    InputStream cin{ std::cin };

    auto read() { return cin.read(); }
    template <typename Head, typename... Tail>
    void read(Head& head, Tail &...tails) { cin.read(head, tails...); }
} // namespace suisen::io
namespace suisen { using io::read; } // namespace suisen

namespace suisen::io {
    template <typename OStream, std::enable_if_t<std::conjunction_v<std::is_base_of<std::ostream, std::remove_reference_t<OStream>>, std::negation<std::is_const<std::remove_reference_t<OStream>>>>, std::nullptr_t> = nullptr>
    struct OutputStream {
    private:
        using ostream_type = std::remove_reference_t<OStream>;
        OStream os;
    public:
        template <typename OStream_> OutputStream(OStream_ &&os) : os(std::move(os)) {}
        template <typename OStream_> OutputStream(OStream_ &os) : os(os) {}
        template <typename T> OutputStream& operator<<(const T& e) {
            if constexpr (suisen::is_writable_v<T>) os << e; else _print(e);
            return *this;
        }
        void print() { *this << '\n'; }
        template <typename Head, typename... Tail>
        void print(const Head& head, const Tail &...tails) { *this << head, ((*this << ' ' << tails), ...), *this << '\n'; }
        template <typename Iterable, std::enable_if_t<suisen::is_iterable_v<Iterable>, std::nullptr_t> = nullptr>
        void print_all(const Iterable& v, std::string sep = " ", std::string end = "\n") {
            for (auto it = v.begin(); it != v.end();) if (*this << *it; ++it != v.end()) *this << sep;
            *this << end;
        }
        ostream_type& get_stream() { return os; }
    private:
        void _print(__uint128_t value) {
            char buffer[41], *d = std::end(buffer);
            do *--d = '0' + (value % 10), value /= 10; while (value);
            os.rdbuf()->sputn(d, std::end(buffer) - d);
        }
        void _print(__int128_t value) {
            if (value < 0) *this << '-';
            _print(__uint128_t(value < 0 ? -value : value));
        }
        template <typename T, typename U>
        void _print(const std::pair<T, U>& a) { *this << a.first << ' ' << a.second; }
        template <size_t N = 0, typename ...Args>
        void _print(const std::tuple<Args...>& a) {
            if constexpr (N < std::tuple_size_v<std::tuple<Args...>>) {
                if constexpr (N) *this << ' ';
                *this << std::get<N>(a), _print<N + 1>(a);
            }
        }
        template <typename Iterable, std::enable_if_t<suisen::is_iterable_v<Iterable>, std::nullptr_t> = nullptr>
        void _print(const Iterable& a) { print_all(a, " ", ""); }
    };
    template <typename OStream_>
    OutputStream(OStream_ &&) -> OutputStream<OStream_>;
    template <typename OStream_>
    OutputStream(OStream_ &) -> OutputStream<OStream_&>;

    OutputStream cout{ std::cout }, cerr{ std::cerr };

    template <typename... Args>
    void print(const Args &... args) { cout.print(args...); }
    template <typename Iterable, std::enable_if_t<suisen::is_iterable_v<Iterable>, std::nullptr_t> = nullptr>
    void print_all(const Iterable& v, const std::string& sep = " ", const std::string& end = "\n") { cout.print_all(v, sep, end); }
} // namespace suisen::io
namespace suisen { using io::print, io::print_all; } // namespace suisen

namespace suisen {
    template <class T, class ToKey, class CompKey = std::less<>, std::enable_if_t<std::conjunction_v<std::is_invocable<ToKey, T>, std::is_invocable_r<bool, CompKey, std::invoke_result_t<ToKey, T>, std::invoke_result_t<ToKey, T>>>, std::nullptr_t> = nullptr>
    auto comparator(const ToKey& to_key, const CompKey& comp_key = std::less<>()) {
        return [=](const T& x, const T& y) { return comp_key(to_key(x), to_key(y)); };
    }
    template <class Compare, std::enable_if_t<std::is_invocable_r_v<bool, Compare, int, int>, std::nullptr_t> = nullptr>
    std::vector<int> sorted_indices(int n, const Compare& compare) {
        std::vector<int> p(n);
        return std::iota(p.begin(), p.end(), 0), std::sort(p.begin(), p.end(), compare), p;
    }
    template <class ToKey, std::enable_if_t<std::is_invocable_v<ToKey, int>, std::nullptr_t> = nullptr>
    std::vector<int> sorted_indices(int n, const ToKey& to_key) { return sorted_indices(n, comparator<int>(to_key)); }
    template <class T, class Comparator>
    auto priority_queue_with_comparator(const Comparator& comparator) { return std::priority_queue<T, std::vector<T>, Comparator>{ comparator }; }
    template <class Iterable, std::enable_if_t<suisen::is_iterable_v<Iterable>, std::nullptr_t> = nullptr>
    void sort_unique_erase(Iterable& a) { std::sort(a.begin(), a.end()), a.erase(std::unique(a.begin(), a.end()), a.end()); }

    template <size_t D> struct Dim : std::array<int, D> {
        template <typename ...Ints> Dim(const Ints& ...ns) : std::array<int, D>::array{ static_cast<int>(ns)... } {}
    };
    template <typename ...Ints> Dim(const Ints& ...) -> Dim<sizeof...(Ints)>;
    template <class T, size_t D, size_t I = 0>
    auto ndvec(const Dim<D> &ns, const T& value = {}) {
        if constexpr (I + 1 < D) {
            return std::vector(ns[I], ndvec<T, D, I + 1>(ns, value));
        } else {
            return std::vector<T>(ns[I], value);
        }
    }
}
namespace suisen {
    using int128 = __int128_t;
    using uint128 = __uint128_t;
    template <class T> using min_priority_queue = std::priority_queue<T, std::vector<T>, std::greater<T>>;
    template <class T> using max_priority_queue = std::priority_queue<T, std::vector<T>, std::less<T>>;
}
namespace suisen { const std::string Yes = "Yes", No = "No", YES = "YES", NO = "NO"; }

#ifdef LOCAL
#  define debug(...) debug_impl(#__VA_ARGS__, __VA_ARGS__)
template <class H, class... Ts> void debug_impl(const char* s, const H& h, const Ts&... t) {
    suisen::io::cerr << "[\033[32mDEBUG\033[m] " << s << ": " << h, ((suisen::io::cerr << ", " << t), ..., (suisen::io::cerr << "\n"));
}
#else
#  define debug(...) void(0)
#endif
#define FOR(e, v) for (auto &&e : v)
#define CFOR(e, v) for (const auto &e : v)
#define REP(i, ...) CFOR(i, suisen::macro::rep_impl(__VA_ARGS__))
#define RREP(i, ...) CFOR(i, suisen::macro::rrep_impl(__VA_ARGS__))
#define REPINF(i, ...) CFOR(i, suisen::macro::repinf_impl(__VA_ARGS__))
#define LOOP(n) for ([[maybe_unused]] const auto& _ : suisen::macro::rep_impl(n))
#define ALL(iterable) std::begin(iterable), std::end(iterable)

using namespace suisen;
using namespace std;
struct io_setup {
    io_setup(int precision = 20) {
        std::ios::sync_with_stdio(false), std::cin.tie(nullptr);
        std::cout << std::fixed << std::setprecision(precision);
    }
} io_setup_ {};

constexpr int iinf = std::numeric_limits<int>::max() / 2;
constexpr long long linf = std::numeric_limits<long long>::max() / 2;

#include <atcoder/modint>

using mint = atcoder::modint998244353;

namespace atcoder {
    std::istream& operator>>(std::istream& in, mint &a) {
        long long e; in >> e; a = e;
        return in;
    }
    
    std::ostream& operator<<(std::ostream& out, const mint &a) {
        out << a.val();
        return out;
    }
} // namespace atcoder

#include <cassert>
#include <vector>

namespace suisen {
    template <typename T, typename U = T>
    struct factorial {
        factorial() = default;
        factorial(int n) { ensure(n); }

        static void ensure(const int n) {
            int sz = _fac.size();
            if (n + 1 <= sz) return;
            int new_size = std::max(n + 1, sz * 2);
            _fac.resize(new_size), _fac_inv.resize(new_size);
            for (int i = sz; i < new_size; ++i) _fac[i] = _fac[i - 1] * i;
            _fac_inv[new_size - 1] = U(1) / _fac[new_size - 1];
            for (int i = new_size - 1; i > sz; --i) _fac_inv[i - 1] = _fac_inv[i] * i;
        }

        T fac(const int i) {
            ensure(i);
            return _fac[i];
        }
        T operator()(int i) {
            return fac(i);
        }
        U fac_inv(const int i) {
            ensure(i);
            return _fac_inv[i];
        }
        U binom(const int n, const int r) {
            if (n < 0 or r < 0 or n < r) return 0;
            ensure(n);
            return _fac[n] * _fac_inv[r] * _fac_inv[n - r];
        }
        U perm(const int n, const int r) {
            if (n < 0 or r < 0 or n < r) return 0;
            ensure(n);
            return _fac[n] * _fac_inv[n - r];
        }
    private:
        static std::vector<T> _fac;
        static std::vector<U> _fac_inv;
    };
    template <typename T, typename U>
    std::vector<T> factorial<T, U>::_fac{ 1 };
    template <typename T, typename U>
    std::vector<U> factorial<T, U>::_fac_inv{ 1 };
} // namespace suisen

namespace suisen {
    template <int base_as_int, typename mint>
    struct static_pow_mods {
        static_pow_mods() = default;
        static_pow_mods(int n) { ensure(n); }
        const mint& operator[](int i) const {
            ensure(i);
            return pows[i];
        }
        static void ensure(int n) {
            int sz = pows.size();
            if (sz > n) return;
            pows.resize(n + 1);
            for (int i = sz; i <= n; ++i) pows[i] = base * pows[i - 1];
        }
    private:
        static inline std::vector<mint> pows { 1 };
        static inline mint base = base_as_int;
        static constexpr int mod = mint::mod();
    };

    template <typename mint>
    struct pow_mods {
        pow_mods() = default;
        pow_mods(mint base, int n) : base(base) { ensure(n); }
        const mint& operator[](int i) const {
            ensure(i);
            return pows[i];
        }
        void ensure(int n) const {
            int sz = pows.size();
            if (sz > n) return;
            pows.resize(n + 1);
            for (int i = sz; i <= n; ++i) pows[i] = base * pows[i - 1];
        }
    private:
        mutable std::vector<mint> pows { 1 };
        mint base;
        static constexpr int mod = mint::mod();
    };
}

namespace suisen {
    template <typename FPSType>
    std::vector<typename FPSType::value_type> bernoulli_number(int n) {
        using mint = typename FPSType::value_type;
        factorial<mint> fac(n);
        FPSType a(n + 1);
        for (int i = 0; i <= n; ++i) a[i] = fac.fac_inv(i + 1);
        a.inv_inplace(n + 1), a.resize(n + 1);
        for (int i = 2; i <= n; ++i) a[i] *= fac(i);
        return a;
    }
} // namespace suisen

namespace suisen {
    // res[p] = Sum[i=1,n] i^p for p=0,...,k
    template <typename FPSType>
    std::vector<typename FPSType::value_type> sum_of_powers(int n, int k) {
        using fps = FPSType;
        using mint = typename FPSType::value_type;

        factorial<mint> fac(k + 1);
        pow_mods<mint> pow_n(n, k + 1);

        fps f(k + 2);
        for (int j = 0; j <= k + 1; ++j) {
            f[j] = pow_n[j] * fac.fac_inv(j);
        }
        std::vector<mint> b = bernoulli_number<fps>(k + 1);
        b[1] *= -1;
        for (int j = 0; j <= k + 1; ++j) {
            b[j] *= fac.fac_inv(j);
        }
        f *= b;

        std::vector<mint> res(k + 1);
        for (int p = 0; p <= k; ++p) {
            res[p] = fac.fac(p) * (f[p + 1] - b[p + 1]);
        }
        return res;
    }
} // namespace suisen

#include <optional>
#include <queue>

#include <atcoder/modint>
#include <atcoder/convolution>

#include <cmath>

/**
 * refernce: https://37zigen.com/tonelli-shanks-algorithm/
 * calculates x s.t. x^2 = a mod p in O((log p)^2).
 */
template <typename mint>
std::optional<mint> safe_sqrt(mint a) {
    static int p = mint::mod();
    if (a == 0) return std::make_optional(0);
    if (p == 2) return std::make_optional(a);
    if (a.pow((p - 1) / 2) != 1) return std::nullopt;
    mint b = 1;
    while (b.pow((p - 1) / 2) == 1) ++b;
    static int tlz = __builtin_ctz(p - 1), q = (p - 1) >> tlz;
    mint x = a.pow((q + 1) / 2);
    b = b.pow(q);
    for (int shift = 2; x * x != a; ++shift) {
        mint e = a.inv() * x * x;
        if (e.pow(1 << (tlz - shift)) != 1) x *= b;
        b *= b;
    }
    return std::make_optional(x);
}

/**
 * calculates x s.t. x^2 = a mod p in O((log p)^2).
 * if not exists, raises runtime error.
 */
template <typename mint>
auto sqrt(mint a) -> decltype(mint::mod(), mint()) {
    return *safe_sqrt(a);
}
template <typename mint>
auto log(mint a) -> decltype(mint::mod(), mint()) {
    assert(a == 1);
    return 0;
}
template <typename mint>
auto exp(mint a) -> decltype(mint::mod(), mint()) {
    assert(a == 0);
    return 1;
}
template <typename mint, typename T>
auto pow(mint a, T b) -> decltype(mint::mod(), mint()) {
    return a.pow(b);
}
template <typename mint>
auto inv(mint a) -> decltype(mint::mod(), mint()) {
    return a.inv();
}

namespace suisen {
    template <typename mint>
    class inv_mods {
    public:
        inv_mods() = default;
        inv_mods(int n) { ensure(n); }
        const mint& operator[](int i) const {
            ensure(i);
            return invs[i];
        }
        static void ensure(int n) {
            int sz = invs.size();
            if (sz < 2) invs = { 0, 1 }, sz = 2;
            if (sz < n + 1) {
                invs.resize(n + 1);
                for (int i = sz; i <= n; ++i) invs[i] = mint(mod - mod / i) * invs[mod % i];
            }
        }
    private:
        static std::vector<mint> invs;
        static constexpr int mod = mint::mod();
    };
    template <typename mint>
    std::vector<mint> inv_mods<mint>::invs{};

    template <typename mint>
    std::vector<mint> get_invs(const std::vector<mint>& vs) {
        const int n = vs.size();

        mint p = 1;
        for (auto& e : vs) {
            p *= e;
            assert(e != 0);
        }
        mint ip = p.inv();

        std::vector<mint> rp(n + 1);
        rp[n] = 1;
        for (int i = n - 1; i >= 0; --i) {
            rp[i] = rp[i + 1] * vs[i];
        }
        std::vector<mint> res(n);
        for (int i = 0; i < n; ++i) {
            res[i] = ip * rp[i + 1];
            ip *= vs[i];
        }
        return res;
    }
}

namespace suisen {
    template <typename T>
    struct FPSNaive : std::vector<T> {
        static inline int MAX_SIZE = std::numeric_limits<int>::max() / 2;

        using value_type = T;
        using element_type = rec_value_type_t<T>;
        using std::vector<value_type>::vector;

        FPSNaive(const std::initializer_list<value_type> l) : std::vector<value_type>::vector(l) {}
        FPSNaive(const std::vector<value_type>& v) : std::vector<value_type>::vector(v) {}

        static void set_max_size(int n) {
            FPSNaive<T>::MAX_SIZE = n;
        }

        const value_type operator[](int n) const {
            return n <= deg() ? unsafe_get(n) : value_type{ 0 };
        }
        value_type& operator[](int n) {
            return ensure_deg(n), unsafe_get(n);
        }

        int size() const {
            return std::vector<value_type>::size();
        }
        int deg() const {
            return size() - 1;
        }
        int normalize() {
            while (size() and this->back() == value_type{ 0 }) this->pop_back();
            return deg();
        }
        FPSNaive& cut_inplace(int n) {
            if (size() > n) this->resize(std::max(0, n));
            return *this;
        }
        FPSNaive cut(int n) const {
            FPSNaive f = FPSNaive(*this).cut_inplace(n);
            return f;
        }

        FPSNaive operator+() const {
            return FPSNaive(*this);
        }
        FPSNaive operator-() const {
            FPSNaive f(*this);
            for (auto& e : f) e = -e;
            return f;
        }
        FPSNaive& operator++() { return ++(*this)[0], * this; }
        FPSNaive& operator--() { return --(*this)[0], * this; }
        FPSNaive& operator+=(const value_type x) { return (*this)[0] += x, *this; }
        FPSNaive& operator-=(const value_type x) { return (*this)[0] -= x, *this; }
        FPSNaive& operator+=(const FPSNaive& g) {
            ensure_deg(g.deg());
            for (int i = 0; i <= g.deg(); ++i) unsafe_get(i) += g.unsafe_get(i);
            return *this;
        }
        FPSNaive& operator-=(const FPSNaive& g) {
            ensure_deg(g.deg());
            for (int i = 0; i <= g.deg(); ++i) unsafe_get(i) -= g.unsafe_get(i);
            return *this;
        }
        FPSNaive& operator*=(const FPSNaive& g) { return *this = *this * g; }
        FPSNaive& operator*=(const value_type x) {
            for (auto& e : *this) e *= x;
            return *this;
        }
        FPSNaive& operator/=(const FPSNaive& g) { return *this = *this / g; }
        FPSNaive& operator%=(const FPSNaive& g) { return *this = *this % g; }
        FPSNaive& operator<<=(const int shamt) {
            this->insert(this->begin(), shamt, value_type{ 0 });
            return *this;
        }
        FPSNaive& operator>>=(const int shamt) {
            if (shamt > size()) this->clear();
            else this->erase(this->begin(), this->begin() + shamt);
            return *this;
        }

        friend FPSNaive operator+(FPSNaive f, const FPSNaive& g) { f += g; return f; }
        friend FPSNaive operator+(FPSNaive f, const value_type& x) { f += x; return f; }
        friend FPSNaive operator-(FPSNaive f, const FPSNaive& g) { f -= g; return f; }
        friend FPSNaive operator-(FPSNaive f, const value_type& x) { f -= x; return f; }
        friend FPSNaive operator*(const FPSNaive& f, const FPSNaive& g) {
            if (f.empty() or g.empty()) return FPSNaive{};
            const int n = f.size(), m = g.size();
            FPSNaive h(std::min(MAX_SIZE, n + m - 1));
            for (int i = 0; i < n; ++i) for (int j = 0; j < m; ++j) {
                if (i + j >= MAX_SIZE) break;
                h.unsafe_get(i + j) += f.unsafe_get(i) * g.unsafe_get(j);
            }
            return h;
        }
        friend FPSNaive operator*(FPSNaive f, const value_type& x) { f *= x; return f; }
        friend FPSNaive operator/(FPSNaive f, const FPSNaive& g) { return std::move(f.div_mod(g).first); }
        friend FPSNaive operator%(FPSNaive f, const FPSNaive& g) { return std::move(f.div_mod(g).second); }
        friend FPSNaive operator*(const value_type x, FPSNaive f) { f *= x; return f; }
        friend FPSNaive operator<<(FPSNaive f, const int shamt) { f <<= shamt; return f; }
        friend FPSNaive operator>>(FPSNaive f, const int shamt) { f >>= shamt; return f; }

        std::pair<FPSNaive, FPSNaive> div_mod(FPSNaive g) const {
            FPSNaive f = *this;
            const int fd = f.normalize(), gd = g.normalize();
            assert(gd >= 0);
            if (fd < gd) return { FPSNaive{}, f };
            if (gd == 0) return { f *= g.unsafe_get(0).inv(), FPSNaive{} };
            const int k = f.deg() - gd;
            value_type head_inv = g.unsafe_get(gd).inv();
            FPSNaive q(k + 1);
            for (int i = k; i >= 0; --i) {
                value_type div = f.unsafe_get(i + gd) * head_inv;
                q.unsafe_get(i) = div;
                for (int j = 0; j <= gd; ++j) f.unsafe_get(i + j) -= div * g.unsafe_get(j);
            }
            f.cut_inplace(gd);
            f.normalize();
            return { q, f };
        }

        friend bool operator==(const FPSNaive& f, const FPSNaive& g) {
            const int n = f.size(), m = g.size();
            if (n < m) return g == f;
            for (int i = 0; i < m; ++i) if (f.unsafe_get(i) != g.unsafe_get(i)) return false;
            for (int i = m; i < n; ++i) if (f.unsafe_get(i) != 0) return false;
            return true;
        }
        friend bool operator!=(const FPSNaive& f, const FPSNaive& g) {
            return not (f == g);
        }

        FPSNaive mul(const FPSNaive& g, int n = -1) const {
            if (n < 0) n = size();
            if (this->empty() or g.empty()) return FPSNaive{};
            const int m = size(), k = g.size();
            FPSNaive h(std::min(n, m + k - 1));
            for (int i = 0; i < m; ++i) {
                for (int j = 0, jr = std::min(k, n - i); j < jr; ++j) {
                    h.unsafe_get(i + j) += unsafe_get(i) * g.unsafe_get(j);
                }
            }
            return h;
        }
        FPSNaive diff() const {
            if (this->empty()) return {};
            FPSNaive g(size() - 1);
            for (int i = 1; i <= deg(); ++i) g.unsafe_get(i - 1) = unsafe_get(i) * i;
            return g;
        }
        FPSNaive intg() const {
            const int n = size();
            FPSNaive g(n + 1);
            for (int i = 0; i < n; ++i) g.unsafe_get(i + 1) = unsafe_get(i) * invs[i + 1];
            if (g.deg() > MAX_SIZE) g.cut_inplace(MAX_SIZE);
            return g;
        }
        FPSNaive inv(int n = -1) const {
            if (n < 0) n = size();
            FPSNaive g(n);
            const value_type inv_f0 = ::inv(unsafe_get(0));
            g.unsafe_get(0) = inv_f0;
            for (int i = 1; i < n; ++i) {
                for (int j = 1; j <= i; ++j) g.unsafe_get(i) -= g.unsafe_get(i - j) * (*this)[j];
                g.unsafe_get(i) *= inv_f0;
            }
            return g;
        }
        FPSNaive exp(int n = -1) const {
            if (n < 0) n = size();
            assert(unsafe_get(0) == value_type{ 0 });
            FPSNaive g(n);
            g.unsafe_get(0) = value_type{ 1 };
            for (int i = 1; i < n; ++i) {
                for (int j = 1; j <= i; ++j) g.unsafe_get(i) += j * g.unsafe_get(i - j) * (*this)[j];
                g.unsafe_get(i) *= invs[i];
            }
            return g;
        }
        FPSNaive log(int n = -1) const {
            if (n < 0) n = size();
            assert(unsafe_get(0) == value_type{ 1 });
            FPSNaive g(n);
            g.unsafe_get(0) = value_type{ 0 };
            for (int i = 1; i < n; ++i) {
                g.unsafe_get(i) = i * (*this)[i];
                for (int j = 1; j < i; ++j) g.unsafe_get(i) -= (i - j) * g.unsafe_get(i - j) * (*this)[j];
                g.unsafe_get(i) *= invs[i];
            }
            return g;
        }
        FPSNaive pow(const long long k, int n = -1) const {
            if (n < 0) n = size();
            if (k == 0) {
                FPSNaive res(n);
                res[0] = 1;
                return res;
            }
            int z = 0;
            while (z < size() and unsafe_get(z) == value_type{ 0 }) ++z;
            if (z == size() or z > (n - 1) / k) return FPSNaive(n, 0);
            const int m = n - z * k;

            FPSNaive g(m);
            const value_type inv_f0 = ::inv(unsafe_get(z));
            g.unsafe_get(0) = unsafe_get(z).pow(k);
            for (int i = 1; i < m; ++i) {
                for (int j = 1; j <= i; ++j) g.unsafe_get(i) += (element_type{ k } *j - (i - j)) * g.unsafe_get(i - j) * (*this)[z + j];
                g.unsafe_get(i) *= inv_f0 * invs[i];
            }
            g <<= z * k;
            return g;
        }

        std::optional<FPSNaive> safe_sqrt(int n = -1) const {
            if (n < 0) n = size();
            int dl = 0;
            while (dl < size() and unsafe_get(dl) == value_type{ 0 }) ++dl;
            if (dl == size()) return FPSNaive(n, 0);
            if (dl & 1) return std::nullopt;

            const int m = n - dl / 2;

            FPSNaive g(m);
            auto opt_g0 = ::safe_sqrt((*this)[dl]);
            if (not opt_g0.has_value()) return std::nullopt;
            g.unsafe_get(0) = *opt_g0;
            value_type inv_2g0 = ::inv(2 * g.unsafe_get(0));
            for (int i = 1; i < m; ++i) {
                g.unsafe_get(i) = (*this)[dl + i];
                for (int j = 1; j < i; ++j) g.unsafe_get(i) -= g.unsafe_get(j) * g.unsafe_get(i - j);
                g.unsafe_get(i) *= inv_2g0;
            }
            g <<= dl / 2;
            return g;
        }
        FPSNaive sqrt(int n = -1) const {
            if (n < 0) n = size();
            return *safe_sqrt(n);
        }

        value_type eval(value_type x) const {
            value_type y = 0;
            for (int i = size() - 1; i >= 0; --i) y = y * x + unsafe_get(i);
            return y;
        }

    private:
        static inline inv_mods<element_type> invs;

        void ensure_deg(int d) {
            if (deg() < d) this->resize(d + 1, value_type{ 0 });
        }
        const value_type& unsafe_get(int i) const {
            return std::vector<value_type>::operator[](i);
        }
        value_type& unsafe_get(int i) {
            return std::vector<value_type>::operator[](i);
        }
    };
} // namespace suisen

template <typename mint>
suisen::FPSNaive<mint> sqrt(suisen::FPSNaive<mint> a) {
    return a.sqrt();
}
template <typename mint>
suisen::FPSNaive<mint> log(suisen::FPSNaive<mint> a) {
    return a.log();
}
template <typename mint>
suisen::FPSNaive<mint> exp(suisen::FPSNaive<mint> a) {
    return a.exp();
}
template <typename mint, typename T>
suisen::FPSNaive<mint> pow(suisen::FPSNaive<mint> a, T b) {
    return a.pow(b);
}
template <typename mint>
suisen::FPSNaive<mint> inv(suisen::FPSNaive<mint> a) {
    return a.inv();
}

namespace suisen {
    template <typename mint, atcoder::internal::is_static_modint_t<mint>* = nullptr>
    struct FormalPowerSeries : std::vector<mint> {
        using base_type = std::vector<mint>;
        using value_type = typename base_type::value_type;
        using base_type::vector;

        FormalPowerSeries(const std::initializer_list<value_type> l) : std::vector<value_type>::vector(l) {}
        FormalPowerSeries(const std::vector<value_type>& v) : std::vector<value_type>::vector(v) {}

        int size() const noexcept {
            return base_type::size();
        }
        int deg() const noexcept {
            return size() - 1;
        }
        void ensure(int n) {
            if (size() < n) this->resize(n);
        }

        value_type safe_get(int d) const {
            return d <= deg() ? (*this)[d] : 0;
        }
        value_type& safe_get(int d) {
            ensure(d + 1);
            return (*this)[d];
        }

        FormalPowerSeries& cut_trailing_zeros() {
            while (size() and this->back() == 0) this->pop_back();
            return *this;
        }
        FormalPowerSeries& cut(int n) {
            if (size() > n) this->resize(std::max(0, n));
            return *this;
        }
        FormalPowerSeries cut_copy(int n) const {
            FormalPowerSeries res(this->begin(), this->begin() + std::min(size(), n));
            res.ensure(n);
            return res;
        }
        FormalPowerSeries cut_copy(int l, int r) const {
            if (l >= size()) return FormalPowerSeries(r - l, 0);
            FormalPowerSeries res(this->begin() + l, this->begin() + std::min(size(), r));
            res.ensure(r - l);
            return res;
        }

        /* Unary Operations */

        FormalPowerSeries operator+() const { return *this; }
        FormalPowerSeries operator-() const {
            FormalPowerSeries res = *this;
            for (auto& e : res) e = -e;
            return res;
        }
        FormalPowerSeries& operator++() { return ++safe_get(0), * this; }
        FormalPowerSeries& operator--() { return --safe_get(0), * this; }
        FormalPowerSeries operator++(int) {
            FormalPowerSeries res = *this;
            ++(*this);
            return res;
        }
        FormalPowerSeries operator--(int) {
            FormalPowerSeries res = *this;
            --(*this);
            return res;
        }

        /* Binary Operations With Constant */

        FormalPowerSeries& operator+=(const value_type& x) { return safe_get(0) += x, *this; }
        FormalPowerSeries& operator-=(const value_type& x) { return safe_get(0) -= x, *this; }
        FormalPowerSeries& operator*=(const value_type& x) {
            for (auto& e : *this) e *= x;
            return *this;
        }
        FormalPowerSeries& operator/=(const value_type& x) { return *this *= x.inv(); }

        friend FormalPowerSeries operator+(FormalPowerSeries f, const value_type& x) { f += x; return f; }
        friend FormalPowerSeries operator+(const value_type& x, FormalPowerSeries f) { f += x; return f; }
        friend FormalPowerSeries operator-(FormalPowerSeries f, const value_type& x) { f -= x; return f; }
        friend FormalPowerSeries operator-(const value_type& x, FormalPowerSeries f) { f -= x; return -f; }
        friend FormalPowerSeries operator*(FormalPowerSeries f, const value_type& x) { f *= x; return f; }
        friend FormalPowerSeries operator*(const value_type& x, FormalPowerSeries f) { f *= x; return f; }
        friend FormalPowerSeries operator/(FormalPowerSeries f, const value_type& x) { f /= x; return f; }

        /* Binary Operations With Formal Power Series */

        FormalPowerSeries& operator+=(const FormalPowerSeries& g) {
            const int n = g.size();
            ensure(n);
            for (int i = 0; i < n; ++i) (*this)[i] += g[i];
            return *this;
        }
        FormalPowerSeries& operator-=(const FormalPowerSeries& g) {
            const int n = g.size();
            ensure(n);
            for (int i = 0; i < n; ++i) (*this)[i] -= g[i];
            return *this;
        }
        FormalPowerSeries& operator*=(const FormalPowerSeries& g) { return *this = *this * g; }
        FormalPowerSeries& operator/=(const FormalPowerSeries& g) { return *this = *this / g; }
        FormalPowerSeries& operator%=(const FormalPowerSeries& g) { return *this = *this % g; }

        friend FormalPowerSeries operator+(FormalPowerSeries f, const FormalPowerSeries& g) { f += g; return f; }
        friend FormalPowerSeries operator-(FormalPowerSeries f, const FormalPowerSeries& g) { f -= g; return f; }
        friend FormalPowerSeries operator*(const FormalPowerSeries& f, const FormalPowerSeries& g) {
            const int siz_f = f.size(), siz_g = g.size();
            if (siz_f < siz_g) return g * f;
            if (std::min(siz_f, siz_g) <= 60) return atcoder::convolution(f, g);
            const int deg = siz_f + siz_g - 2;
            int fpow2 = 1;
            while ((fpow2 << 1) <= deg) fpow2 <<= 1;
            if (const int dif = deg - fpow2 + 1; dif <= 10) {
                FormalPowerSeries h = atcoder::convolution(std::vector<mint>(f.begin(), f.end() - dif), g);
                h.resize(h.size() + dif);
                for (int i = siz_f - dif; i < siz_f; ++i) for (int j = 0; j < siz_g; ++j) {
                    h[i + j] += f[i] * g[j];
                }
                return h;
            }
            return atcoder::convolution(f, g);
        }
        friend FormalPowerSeries operator/(FormalPowerSeries f, FormalPowerSeries g) {
            if (f.size() < 60) return FPSNaive<mint>(f).div_mod(g).first;
            f.cut_trailing_zeros(), g.cut_trailing_zeros();
            const int fd = f.deg(), gd = g.deg();
            assert(gd >= 0);
            if (fd < gd) return {};
            if (gd == 0) {
                f /= g[0];
                return f;
            }
            std::reverse(f.begin(), f.end()), std::reverse(g.begin(), g.end());
            const int qd = fd - gd;
            f.cut(qd + 1);
            FormalPowerSeries q = f * g.inv(qd + 1);
            q.cut(qd + 1);
            std::reverse(q.begin(), q.end());
            return q;
        }
        friend FormalPowerSeries operator%(const FormalPowerSeries& f, const FormalPowerSeries& g) { return f.div_mod(g).second; }
        std::pair<FormalPowerSeries, FormalPowerSeries> div_mod(const FormalPowerSeries& g) const {
            if (size() < 60) {
                auto [q, r] = FPSNaive<mint>(*this).div_mod(g);
                return { q, r };
            }
            FormalPowerSeries q = *this / g, r = *this - g * q;
            r.cut_trailing_zeros();
            return { q, r };
        }

        /* Shift Operations */

        FormalPowerSeries& operator<<=(const int shamt) {
            return this->insert(this->begin(), shamt, 0), * this;
        }
        FormalPowerSeries& operator>>=(const int shamt) {
            return this->erase(this->begin(), this->begin() + std::min(shamt, size())), * this;
        }
        friend FormalPowerSeries operator<<(FormalPowerSeries f, const int shamt) { f <<= shamt; return f; }
        friend FormalPowerSeries operator>>(FormalPowerSeries f, const int shamt) { f >>= shamt; return f; }

        /* Compare */

        friend bool operator==(const FormalPowerSeries& f, const FormalPowerSeries& g) {
            const int n = f.size(), m = g.size();
            if (n < m) return g == f;
            for (int i = 0; i < m; ++i) if (f[i] != g[i]) return false;
            for (int i = m; i < n; ++i) if (f[i] != 0) return false;
            return true;
        }
        friend bool operator!=(const FormalPowerSeries& f, const FormalPowerSeries& g) { return not (f == g); }

        /* Other Operations */

        FormalPowerSeries& diff_inplace() {
            const int n = size();
            for (int i = 1; i < n; ++i) (*this)[i - 1] = (*this)[i] * i;
            return (*this)[n - 1] = 0, *this;
        }
        FormalPowerSeries diff() const {
            FormalPowerSeries res = *this;
            res.diff_inplace();
            return res;
        }
        FormalPowerSeries& intg_inplace() {
            const int n = size();
            inv_mods<value_type> invs(n);
            this->resize(n + 1);
            for (int i = n; i > 0; --i) (*this)[i] = (*this)[i - 1] * invs[i];
            return (*this)[0] = 0, *this;
        }
        FormalPowerSeries intg() const {
            FormalPowerSeries res = *this;
            res.intg_inplace();
            return res;
        }

        FormalPowerSeries& inv_inplace(int n = -1) { return *this = inv(n); }
        // reference: https://opt-cp.com/fps-fast-algorithms/
        FormalPowerSeries inv(int n = -1) const {
            if (n < 0) n = size();
            if (n < 60) return FPSNaive<mint>(cut_copy(n)).inv();
            if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return inv_sparse(std::move(*sp_f), n);
            FormalPowerSeries f_fft, g_fft;
            FormalPowerSeries g{ (*this)[0].inv() };
            for (int k = 1; k < n; k *= 2) {
                f_fft = cut_copy(2 * k), g_fft = g.cut_copy(2 * k);
                atcoder::internal::butterfly(f_fft);
                atcoder::internal::butterfly(g_fft);
                update_inv(k, f_fft, g_fft, g);
            }
            g.resize(n);
            return g;
        }
        FormalPowerSeries& log_inplace(int n = -1) { return *this = log(n); }
        FormalPowerSeries log(int n = -1) const {
            assert(safe_get(0) == 1);
            if (n < 0) n = size();
            if (n < 60) return FPSNaive<mint>(cut_copy(n)).log();
            if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return log_sparse(std::move(*sp_f), n);
            FormalPowerSeries res = inv(n) * diff();
            res.resize(n - 1);
            return res.intg();
        }
        FormalPowerSeries& exp_inplace(int n = -1) { return *this = exp(n); }
        // https://arxiv.org/pdf/1301.5804.pdf
        FormalPowerSeries exp(int n = -1) const {
            assert(safe_get(0) == 0);
            if (n < 0) n = size();
            if (n < 60) return FPSNaive<mint>(cut_copy(n)).exp();
            if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return exp_sparse(std::move(*sp_f), n);
            // h = *this
            // f = exp(h) mod x ^ k
            // g = f^{-1} mod x ^ k
            FormalPowerSeries dh = diff();
            FormalPowerSeries f{ 1 }, f_fft;
            FormalPowerSeries g{ 1 }, g_fft;
            for (int k = 1; k < n; k *= 2) {
                f_fft = f.cut_copy(2 * k), atcoder::internal::butterfly(f_fft);

                if (k > 1) update_inv(k / 2, f_fft, g_fft, g);

                FormalPowerSeries t = f.cut_copy(k);
                t.diff_inplace();
                {
                    FormalPowerSeries r = dh.cut_copy(k);
                    r.back() = 0;
                    atcoder::internal::butterfly(r);
                    for (int i = 0; i < k; ++i) r[i] *= f_fft[i];
                    atcoder::internal::butterfly_inv(r);
                    r /= -k;
                    t += r;
                    t <<= 1, t[0] = t[k], t.pop_back();
                }
                t.resize(2 * k);
                atcoder::internal::butterfly(t);
                g_fft = g.cut_copy(2 * k);
                atcoder::internal::butterfly(g_fft);
                for (int i = 0; i < 2 * k; ++i) t[i] *= g_fft[i];
                atcoder::internal::butterfly_inv(t);
                t.resize(k);
                t /= 2 * k;

                FormalPowerSeries v = cut_copy(2 * k) >>= k;
                t <<= k - 1;
                t.intg_inplace();
                for (int i = 0; i < k; ++i) v[i] -= t[k + i];

                v.resize(2 * k);
                atcoder::internal::butterfly(v);
                for (int i = 0; i < 2 * k; ++i) v[i] *= f_fft[i];
                atcoder::internal::butterfly_inv(v);
                v.resize(k);
                v /= 2 * k;

                f.resize(2 * k);
                for (int i = 0; i < k; ++i) f[k + i] = v[i];
            }
            f.cut(n);
            return f;
        }

        FormalPowerSeries& pow_inplace(long long k, int n = -1) { return *this = pow(k, n); }
        FormalPowerSeries pow(const long long k, int n = -1) const {
            if (n < 0) n = size();
            if (n < 60) return FPSNaive<mint>(cut_copy(n)).pow(k);
            if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return pow_sparse(std::move(*sp_f), k, n);
            if (k == 0) {
                FormalPowerSeries f{ 1 };
                f.resize(n);
                return f;
            }
            int tlz = 0;
            while (tlz < size() and (*this)[tlz] == 0) ++tlz;
            if (tlz == size() or tlz > (n - 1) / k) return FormalPowerSeries(n, 0);
            const int m = n - tlz * k;
            FormalPowerSeries f = *this >> tlz;
            value_type base = f[0];
            return ((((f /= base).log(m) *= k).exp(m) *= base.pow(k)) <<= (tlz * k));
        }

        std::optional<FormalPowerSeries> safe_sqrt(int n = -1) const {
            if (n < 0) n = size();
            if (n < 60) return FPSNaive<mint>(cut_copy(n)).safe_sqrt();
            if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return safe_sqrt_sparse(std::move(*sp_f), n);
            int tlz = 0;
            while (tlz < size() and (*this)[tlz] == 0) ++tlz;
            if (tlz == size()) return FormalPowerSeries(n, 0);
            if (tlz & 1) return std::nullopt;
            const int m = n - tlz / 2;

            FormalPowerSeries h(this->begin() + tlz, this->end());
            auto q0 = ::safe_sqrt(h[0]);
            if (not q0.has_value()) return std::nullopt;

            FormalPowerSeries f{ *q0 }, f_fft, g{ q0->inv() }, g_fft;
            for (int k = 1; k < m; k *= 2) {
                f_fft = f.cut_copy(2 * k), atcoder::internal::butterfly(f_fft);

                if (k > 1) update_inv(k / 2, f_fft, g_fft, g);

                g_fft = g.cut_copy(2 * k);
                atcoder::internal::butterfly(g_fft);
                FormalPowerSeries h_fft = h.cut_copy(2 * k);
                atcoder::internal::butterfly(h_fft);
                for (int i = 0; i < 2 * k; ++i) h_fft[i] = (h_fft[i] - f_fft[i] * f_fft[i]) * g_fft[i];
                atcoder::internal::butterfly_inv(h_fft);
                f.resize(2 * k);
                const value_type iz = value_type(4 * k).inv();
                for (int i = 0; i < k; ++i) f[k + i] = h_fft[k + i] * iz;
            }
            f.resize(m), f <<= (tlz / 2);
            return f;
        }
        FormalPowerSeries& sqrt_inplace(int n = -1) { return *this = sqrt(n); }
        FormalPowerSeries sqrt(int n = -1) const {
            return *safe_sqrt(n);
        }

        value_type eval(value_type x) const {
            value_type y = 0;
            for (int i = size() - 1; i >= 0; --i) y = y * x + (*this)[i];
            return y;
        }

        static FormalPowerSeries prod(const std::vector<FormalPowerSeries>& fs) {
            if (fs.empty()) return { 1 };
            std::deque<FormalPowerSeries> dq(fs.begin(), fs.end());
            std::sort(dq.begin(), dq.end(), [](auto& f, auto& g) { return f.size() < g.size(); });
            while (dq.size() >= 2) {
                dq.push_back(dq[0] * dq[1]);
                dq.pop_front();
                dq.pop_front();
            }
            return dq.front();
        }

        std::optional<std::vector<std::pair<int, value_type>>> sparse_fps_format(int max_size) const {
            std::vector<std::pair<int, value_type>> res;
            for (int i = 0; i <= deg() and int(res.size()) <= max_size; ++i) if (value_type v = (*this)[i]; v != 0) res.emplace_back(i, v);
            if (int(res.size()) > max_size) return std::nullopt;
            return res;
        }

    private:
        static void update_inv(const int k, FormalPowerSeries& f_fft, FormalPowerSeries& g_fft, FormalPowerSeries& g) {
            FormalPowerSeries fg(2 * k);
            for (int i = 0; i < 2 * k; ++i) fg[i] = f_fft[i] * g_fft[i];
            atcoder::internal::butterfly_inv(fg);
            fg >>= k, fg.resize(2 * k);
            atcoder::internal::butterfly(fg);
            for (int i = 0; i < 2 * k; ++i) fg[i] *= g_fft[i];
            atcoder::internal::butterfly_inv(fg);
            const value_type iz = value_type(2 * k).inv(), c = -iz * iz;
            g.resize(2 * k);
            for (int i = 0; i < k; ++i) g[k + i] = fg[i] * c;
        }

        static FormalPowerSeries div_fps_sparse(const FormalPowerSeries& f, const std::vector<std::pair<int, value_type>>& g, int n) {
            const int siz = g.size();
            assert(siz and g[0].first == 0);
            const value_type inv_g0 = g[0].second.inv();
            FormalPowerSeries h(n);
            for (int i = 0; i < n; ++i) {
                value_type v = f.safe_get(i);
                for (int idx = 1; idx < siz; ++idx) {
                    const auto& [j, gj] = g[idx];
                    if (j > i) break;
                    v -= gj * h[i - j];
                }
                h[i] = v * inv_g0;
            }
            return h;
        }
        static FormalPowerSeries inv_sparse(const std::vector<std::pair<int, value_type>>& g, const int n) {
            return div_fps_sparse(FormalPowerSeries{ 1 }, g, n);
        }
        static FormalPowerSeries exp_sparse(const std::vector<std::pair<int, value_type>>& f, const int n) {
            const int siz = f.size();
            assert(not siz or f[0].first != 0);
            FormalPowerSeries g(n);
            g[0] = 1;
            inv_mods<value_type> invs(n);
            for (int i = 1; i < n; ++i) {
                value_type v = 0;
                for (const auto& [j, fj] : f) {
                    if (j > i) break;
                    v += j * fj * g[i - j];
                }
                v *= invs[i];
                g[i] = v;
            }
            return g;
        }
        static FormalPowerSeries log_sparse(const std::vector<std::pair<int, value_type>>& f, const int n) {
            const int siz = f.size();
            assert(siz and f[0].first == 0 and f[0].second == 1);
            FormalPowerSeries g(n);
            for (int idx = 1; idx < siz; ++idx) {
                const auto& [j, fj] = f[idx];
                if (j >= n) break;
                g[j] = j * fj;
            }
            inv_mods<value_type> invs(n);
            for (int i = 1; i < n; ++i) {
                value_type v = g[i];
                for (int idx = 1; idx < siz; ++idx) {
                    const auto& [j, fj] = f[idx];
                    if (j > i) break;
                    v -= fj * g[i - j] * (i - j);
                }
                v *= invs[i];
                g[i] = v;
            }
            return g;
        }
        static FormalPowerSeries pow_sparse(const std::vector<std::pair<int, value_type>>& f, const long long k, const int n) {
            if (k == 0) {
                FormalPowerSeries res(n, 0);
                res[0] = 1;
                return res;
            }
            const int siz = f.size();
            if (not siz) return FormalPowerSeries(n, 0);
            const int p = f[0].first;
            if (p > (n - 1) / k) return FormalPowerSeries(n, 0);
            const value_type inv_f0 = f[0].second.inv();
            const int lz = p * k;
            FormalPowerSeries g(n);
            g[lz] = f[0].second.pow(k);
            inv_mods<value_type> invs(n);
            for (int i = 1; lz + i < n; ++i) {
                value_type v = 0;
                for (int idx = 1; idx < siz; ++idx) {
                    auto [j, fj] = f[idx];
                    j -= p;
                    if (j > i) break;
                    v += fj * g[lz + i - j] * (value_type(k) * j - (i - j));
                }
                v *= invs[i] * inv_f0;
                g[lz + i] = v;
            }
            return g;
        }
        static std::optional<FormalPowerSeries> safe_sqrt_sparse(const std::vector<std::pair<int, value_type>>& f, const int n) {
            const int siz = f.size();
            if (not siz) return FormalPowerSeries(n, 0);
            const int p = f[0].first;
            if (p % 2 == 1) return std::nullopt;
            if (p / 2 >= n) return FormalPowerSeries(n, 0);
            const value_type inv_f0 = f[0].second.inv();
            const int lz = p / 2;
            FormalPowerSeries g(n);
            auto opt_g0 = ::safe_sqrt(f[0].second);
            if (not opt_g0.has_value()) return std::nullopt;
            g[lz] = *opt_g0;
            value_type k = mint(2).inv();
            inv_mods<value_type> invs(n);
            for (int i = 1; lz + i < n; ++i) {
                value_type v = 0;
                for (int idx = 1; idx < siz; ++idx) {
                    auto [j, fj] = f[idx];
                    j -= p;
                    if (j > i) break;
                    v += fj * g[lz + i - j] * (k * j - (i - j));
                }
                v *= invs[i] * inv_f0;
                g[lz + i] = v;
            }
            return g;
        }
        static FormalPowerSeries sqrt_sparse(const std::vector<std::pair<int, value_type>>& f, const int n) {
            return *safe_sqrt(f, n);
        }
    };
} // namespace suisen

template <typename mint>
suisen::FormalPowerSeries<mint> sqrt(suisen::FormalPowerSeries<mint> a) {
    return a.sqrt();
}
template <typename mint>
suisen::FormalPowerSeries<mint> log(suisen::FormalPowerSeries<mint> a) {
    return a.log();
}
template <typename mint>
suisen::FormalPowerSeries<mint> exp(suisen::FormalPowerSeries<mint> a) {
    return a.exp();
}
template <typename mint, typename T>
suisen::FormalPowerSeries<mint> pow(suisen::FormalPowerSeries<mint> a, T b) {
    return a.pow(b);
}
template <typename mint>
suisen::FormalPowerSeries<mint> inv(suisen::FormalPowerSeries<mint> a) {
    return a.inv();
}

void solve() {
    using fps = FormalPowerSeries<mint>;

    long long h, w, n, k;
    read(h, w, n, k);
    const mint m = mint(h - k + 1) * mint(w - k + 1), im = m.inv();

    const long long a = min(h - k, k - 1), c = h - 2 * a;
    const long long b = min(w - k, k - 1), d = w - 2 * b;

    factorial<mint> fac(n);
    std::vector<mint> sum_pow_to_a = sum_of_powers<fps>(a, n);
    std::vector<mint> sum_pow_to_b = sum_of_powers<fps>(b, n);

    pow_mods<mint> pow_am((a + 1) * -im, n + 2), pow_bm((b + 1) * -im, n + 2), pow_im(-im, n + 2);

    mint ans = 0;
    REP(p, n + 1) {
        ans += 4 * fac.binom(n, p) * pow_im[p] * sum_pow_to_a[p] * sum_pow_to_b[p];
    }
    REP(p, n + 1) {
        ans += 2 * c * (fac.binom(n, p) * pow_am[p] * sum_pow_to_b[p]);
    }
    REP(p, n + 1) {
        ans += 2 * d * (fac.binom(n, p) * pow_bm[p] * sum_pow_to_a[p]);
    }
    ans += mint(c) * d * (1 - mint(a + 1) * (b + 1) * im).pow(n);

    print(h * w - ans);
}

int main() {
    int test_case_num = 1;
    // read(test_case_num);
    LOOP(test_case_num) solve();
    return 0;
}