#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 <array>
#include <cassert>
#include <optional>

namespace suisen {
    namespace default_operator {
        template <typename T>
        auto zero() -> decltype(T { 0 }) { return T { 0 }; }
        template <typename T>
        auto one()  -> decltype(T { 1 }) { return T { 1 }; }
        template <typename T>
        auto add(const T &x, const T &y) -> decltype(x + y) { return x + y; }
        template <typename T>
        auto sub(const T &x, const T &y) -> decltype(x - y) { return x - y; }
        template <typename T>
        auto mul(const T &x, const T &y) -> decltype(x * y) { return x * y; }
        template <typename T>
        auto div(const T &x, const T &y) -> decltype(x / y) { return x / y; }
        template <typename T>
        auto mod(const T &x, const T &y) -> decltype(x % y) { return x % y; }
        template <typename T>
        auto neg(const T &x) -> decltype(-x) { return -x; }
        template <typename T>
        auto inv(const T &x) -> decltype(one<T>() / x)  { return one<T>() / x; }
    } // default_operator
    namespace default_operator_noref {
        template <typename T>
        auto zero() -> decltype(T { 0 }) { return T { 0 }; }
        template <typename T>
        auto one()  -> decltype(T { 1 }) { return T { 1 }; }
        template <typename T>
        auto add(T x, T y) -> decltype(x + y) { return x + y; }
        template <typename T>
        auto sub(T x, T y) -> decltype(x - y) { return x - y; }
        template <typename T>
        auto mul(T x, T y) -> decltype(x * y) { return x * y; }
        template <typename T>
        auto div(T x, T y) -> decltype(x / y) { return x / y; }
        template <typename T>
        auto mod(T x, T y) -> decltype(x % y) { return x % y; }
        template <typename T>
        auto neg(T x) -> decltype(-x) { return -x; }
        template <typename T>
        auto inv(T x) -> decltype(one<T>() / x)  { return one<T>() / x; }
    } // default_operator
} // namespace suisen

namespace suisen {
    template <
        typename T, size_t N, size_t M,
        T(*_add)(T, T) = default_operator_noref::add<T>, T(*_neg)(T) = default_operator_noref::neg<T>, T(*_zero)() = default_operator_noref::zero<T>,
        T(*_mul)(T, T) = default_operator_noref::mul<T>, T(*_inv)(T) = default_operator_noref::inv<T>, T(*_one)() = default_operator_noref::one<T>
    >
    struct ArrayMatrix : public std::array<std::array<T, M>, N> {
    private:
        template <typename DummyType = void>
        static constexpr bool is_square_v = N == M;
        template <size_t X, size_t Y>
        using MatrixType = ArrayMatrix<T, X, Y, _add, _neg, _zero, _mul, _inv, _one>;
    public:
        using base_type = std::array<std::array<T, M>, N>;
        using container_type = base_type;
        using row_type = std::array<T, M>;

        using base_type::base_type;
        ArrayMatrix(T diag_val = _zero()) {
            for (size_t i = 0; i < N; ++i) for (size_t j = 0; j < M; ++j) {
                (*this)[i][j] = (i == j ? diag_val : _zero());
            }
        }
        ArrayMatrix(const container_type& c) : base_type{ c } {}
        ArrayMatrix(const std::initializer_list<row_type>& c) {
            assert(c.size() == N);
            size_t i = 0;
            for (const auto& row : c) {
                for (size_t j = 0; j < M; ++j) (*this)[i][j] = row[j];
                ++i;
            }
        }

        static ArrayMatrix e0() { return ArrayMatrix(_zero()); }
        static MatrixType<M, M> e1() { return MatrixType<M, M>(_one()); }

        int size() const {
            static_assert(is_square_v<>);
            return N;
        }
        std::pair<int, int> shape() const { return { N, M }; }
        int row_size() const { return N; }
        int col_size() const { return M; }

        ArrayMatrix operator+() const { return *this; }
        ArrayMatrix operator-() const {
            ArrayMatrix A;
            for (size_t i = 0; i < N; ++i) for (size_t j = 0; j < M; ++j) A[i][j] = _neg((*this)[i][j]);
            return A;
        }
        friend ArrayMatrix& operator+=(ArrayMatrix& A, const ArrayMatrix& B) {
            for (size_t i = 0; i < N; ++i) for (size_t j = 0; j < M; ++j) A[i][j] = _add(A[i][j], B[i][j]);
            return A;
        }
        friend ArrayMatrix& operator-=(ArrayMatrix& A, const ArrayMatrix& B) {
            for (size_t i = 0; i < N; ++i) for (size_t j = 0; j < M; ++j) A[i][j] = _add(A[i][j], _neg(B[i][j]));
            return A;
        }
        template <size_t K>
        friend MatrixType<N, K>& operator*=(ArrayMatrix& A, const MatrixType<M, K>& B) { return A = A * B; }
        friend ArrayMatrix& operator*=(ArrayMatrix& A, const T& val) {
            for (size_t i = 0; i < N; ++i) for (size_t j = 0; j < M; ++j) A[i][j] = _mul(A[i][j], val);
            return A;
        }
        friend ArrayMatrix& operator/=(ArrayMatrix& A, const ArrayMatrix& B) { static_assert(is_square_v<>); return A *= *B.inv(); }
        friend ArrayMatrix& operator/=(ArrayMatrix& A, const T& val) { return A *= _inv(val); }

        friend ArrayMatrix operator+(ArrayMatrix A, const ArrayMatrix& B) { A += B; return A; }
        friend ArrayMatrix operator-(ArrayMatrix A, const ArrayMatrix& B) { A -= B; return A; }
        template <size_t K>
        friend MatrixType<N, K> operator*(const ArrayMatrix& A, const MatrixType<M, K>& B) {
            MatrixType<N, K> C;
            for (size_t i = 0; i < N; ++i) {
                C[i].fill(_zero());
                for (size_t j = 0; j < M; ++j) for (size_t k = 0; k < K; ++k) C[i][k] = _add(C[i][k], _mul(A[i][j], B[j][k]));
            }
            return C;
        }
        friend ArrayMatrix operator*(ArrayMatrix A, const T& val) { A *= val; return A; }
        friend ArrayMatrix operator*(const T& val, ArrayMatrix A) {
            for (size_t i = 0; i < N; ++i) for (size_t j = 0; j < M; ++j) A[i][j] = _mul(val, A[i][j]);
            return A;
        }
        friend std::array<T, N> operator*(const ArrayMatrix& A, const std::array<T, M>& x) {
            std::array<T, N> b;
            b.fill(_zero());
            for (size_t i = 0; i < N; ++i) for (size_t j = 0; j < M; ++j) b[i] = _add(b[i], _mul(A[i][j], x[j]));
            return b;
        }
        friend ArrayMatrix operator/(ArrayMatrix A, const ArrayMatrix& B) { static_assert(is_square_v<>); return A * B.inv(); }
        friend ArrayMatrix operator/(ArrayMatrix A, const T& val) { A /= val; return A; }
        friend ArrayMatrix operator/(const T& val, ArrayMatrix A) { return A.inv() *= val; }

        ArrayMatrix pow(long long b) const {
            static_assert(is_square_v<>);
            assert(b >= 0);
            ArrayMatrix res(e1()), p(*this);
            for (; b; b >>= 1) {
                if (b & 1) res *= p;
                p *= p;
            }
            return res;
        }

        std::optional<ArrayMatrix> safe_inv() const {
            static_assert(is_square_v<>);
            std::array<std::array<T, 2 * N>, N> data;
            for (size_t i = 0; i < N; ++i) {
                for (size_t j = 0; j < N; ++j) {
                    data[i][j] = (*this)[i][j];
                    data[i][N + j] = i == j ? _one() : _zero();
                }
            }
            for (size_t i = 0; i < N; ++i) {
                for (size_t k = i; k < N; ++k) if (data[k][i] != _zero()) {
                    data[i].swap(data[k]);
                    T c = _inv(data[i][i]);
                    for (size_t j = i; j < 2 * N; ++j) data[i][j] = _mul(c, data[i][j]);
                    break;
                }
                if (data[i][i] == _zero()) return std::nullopt;
                for (size_t k = 0; k < N; ++k) if (k != i and data[k][i] != _zero()) {
                    T c = data[k][i];
                    for (size_t j = i; j < 2 * N; ++j) data[k][j] = _add(data[k][j], _neg(_mul(c, data[i][j])));
                }
            }
            ArrayMatrix res;
            for (size_t i = 0; i < N; ++i) std::copy(data[i].begin() + N, data[i].begin() + 2 * N, res[i].begin());
            return res;
        }
        ArrayMatrix inv() const { return *safe_inv(); }
        T det() const {
            static_assert(is_square_v<>);
            ArrayMatrix A = *this;
            bool sgn = false;
            for (size_t j = 0; j < N; ++j) for (size_t i = j + 1; i < N; ++i) if (A[i][j] != _zero()) {
                std::swap(A[j], A[i]);
                T q = _mul(A[i][j], _inv(A[j][j]));
                for (size_t k = j; k < N; ++k) A[i][k] = _add(A[i][k], _neg(_mul(A[j][k], q)));
                sgn = not sgn;
            }
            T res = sgn ? _neg(_one()) : _one();
            for (size_t i = 0; i < N; ++i) res = _mul(res, A[i][i]);
            return res;
        }
        T det_arbitrary_mod() const {
            static_assert(is_square_v<>);
            ArrayMatrix A = *this;
            bool sgn = false;
            for (size_t j = 0; j < N; ++j) for (size_t i = j + 1; i < N; ++i) {
                for (; A[i][j].val(); sgn = not sgn) {
                    std::swap(A[j], A[i]);
                    T q = A[i][j].val() / A[j][j].val();
                    for (size_t k = j; k < N; ++k) A[i][k] -= A[j][k] * q;
                }
            }
            T res = sgn ? -1 : +1;
            for (size_t i = 0; i < N; ++i) res *= A[i][i];
            return res;
        }
    };
    template <
        typename T, size_t N,
        T(*_add)(T, T) = default_operator_noref::add<T>, T(*_neg)(T) = default_operator_noref::neg<T>, T(*_zero)() = default_operator_noref::zero<T>,
        T(*_mul)(T, T) = default_operator_noref::mul<T>, T(*_inv)(T) = default_operator_noref::inv<T>, T(*_one)() = default_operator_noref::one<T>
    >
    using SquareArrayMatrix = ArrayMatrix<T, N, N, _add, _neg, _zero, _mul, _inv, _one>;
} // namespace suisen

void solve() {
    long long n, m;
    read(m, n);

    if (m % 2 == 0) {
        array<mint, 4> pd{};
        pd[0] = 1;
        mint k = m / 2 - 1;

        ArrayMatrix<mint, 4, 4> T;
        if (m >= 4) {
            T = {{
                1, 1, 1, 1,
                m, m - 1, m - 2, m - 2,
                m * (m - 1) / 2 - m - m / 2, (m - 1) * (m - 2) / 2 - (m - 2) - (m / 2 - 1), (k - 1) * (m - 4) + 1, (k - 1) * (m - 4),
                m / 2, m / 2 - 1, m / 2 - 2, m / 2 - 1
            }};
        } else {
            T = {{
                1, 1, 1, 1,
                m, m - 1, m - 2, m - 2,
                0, 0, 0, 0,
                0, 0, 0, 0
            }};
        }
        array y = T.pow(n) * array<mint, 4>{ 1, 0, 0, 0 };
        print(accumulate(ALL(y), mint(0)));
    } else {
        array<mint, 3> pd{};
        pd[0] = 1;
        mint k = m / 2 - 1;

        ArrayMatrix<mint, 3, 3> T;
        if (m >= 4) {
            T = {{
                1, 1, 1,
                m, m - 1, m - 2,
                m * (m - 1) / 2 - m, (m - 1) * (m - 2) / 2 - (m - 2), k * (m - 4) + 1
            }};
        } else {
            T = {{
                1, 1, 1,
                m, m - 1, m - 2,
                0, 0, 0
            }};
        }
        array y = T.pow(n) * array<mint, 3>{ 1, 0, 0 };
        print(accumulate(ALL(y), mint(0)));
    }
}

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