#include <bits/stdc++.h>

#ifdef _MSC_VER
#  include <intrin.h>
#else
#  include <x86intrin.h>
#endif

#include <limits>
#include <type_traits>

namespace suisen {
// ! utility
template <typename ...Types>
using constraints_t = std::enable_if_t<std::conjunction_v<Types...>, std::nullptr_t>;
template <bool cond_v, typename Then, typename OrElse>
constexpr decltype(auto) constexpr_if(Then&& then, OrElse&& or_else) {
    if constexpr (cond_v) {
        return std::forward<Then>(then);
    } else {
        return std::forward<OrElse>(or_else);
    }
}

// ! function
template <typename ReturnType, typename Callable, typename ...Args>
using is_same_as_invoke_result = std::is_same<std::invoke_result_t<Callable, Args...>, ReturnType>;
template <typename F, typename T>
using is_uni_op = is_same_as_invoke_result<T, F, T>;
template <typename F, typename T>
using is_bin_op = is_same_as_invoke_result<T, F, T, T>;

template <typename Comparator, typename T>
using is_comparator = std::is_same<std::invoke_result_t<Comparator, T, T>, bool>;

// ! integral
template <typename T, typename = constraints_t<std::is_integral<T>>>
constexpr int bit_num = std::numeric_limits<std::make_unsigned_t<T>>::digits;
template <typename T, unsigned int n>
struct is_nbit { static constexpr bool value = bit_num<T> == n; };
template <typename T, unsigned int n>
static constexpr bool is_nbit_v = is_nbit<T, n>::value;

// ?
template <typename T>
struct safely_multipliable {};
template <>
struct safely_multipliable<int> { using type = long long; };
template <>
struct safely_multipliable<long long> { using type = __int128_t; };
template <>
struct safely_multipliable<unsigned int> { using type = unsigned long long; };
template <>
struct safely_multipliable<unsigned long int> { using type = __uint128_t; };
template <>
struct safely_multipliable<unsigned long long> { using type = __uint128_t; };
template <>
struct safely_multipliable<float> { using type = float; };
template <>
struct safely_multipliable<double> { using type = double; };
template <>
struct safely_multipliable<long double> { using type = long double; };
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;

} // namespace suisen

// ! type aliases
using i128 = __int128_t;
using u128 = __uint128_t;

template <typename T>
using pq_greater = std::priority_queue<T, std::vector<T>, std::greater<T>>;

// ! macros (internal)
#define DETAIL_OVERLOAD2(_1,_2,name,...) name
#define DETAIL_OVERLOAD3(_1,_2,_3,name,...) name
#define DETAIL_OVERLOAD4(_1,_2,_3,_4,name,...) name

#define DETAIL_REP4(i,l,r,s)  for(std::remove_reference_t<std::remove_const_t<decltype(r)>>i=(l);i<(r);i+=(s))
#define DETAIL_REP3(i,l,r)    DETAIL_REP4(i,l,r,1)
#define DETAIL_REP2(i,n)      DETAIL_REP3(i,0,n)
#define DETAIL_REPINF3(i,l,s) for(std::remove_reference_t<std::remove_const_t<decltype(l)>>i=(l);;i+=(s))
#define DETAIL_REPINF2(i,l)   DETAIL_REPINF3(i,l,1)
#define DETAIL_REPINF1(i)     DETAIL_REPINF2(i,0)
#define DETAIL_RREP4(i,l,r,s) for(std::remove_reference_t<std::remove_const_t<decltype(r)>>i=(l)+fld((r)-(l)-1,s)*(s);i>=(l);i-=(s))
#define DETAIL_RREP3(i,l,r)   DETAIL_RREP4(i,l,r,1)
#define DETAIL_RREP2(i,n)     DETAIL_RREP3(i,0,n)

#define DETAIL_CAT_I(a, b) a##b
#define DETAIL_CAT(a, b) DETAIL_CAT_I(a, b)
#define DETAIL_UNIQVAR(tag) DETAIL_CAT(tag, __LINE__)

// ! macros
#define REP(...)    DETAIL_OVERLOAD4(__VA_ARGS__, DETAIL_REP4   , DETAIL_REP3   , DETAIL_REP2   )(__VA_ARGS__)
#define RREP(...)   DETAIL_OVERLOAD4(__VA_ARGS__, DETAIL_RREP4  , DETAIL_RREP3  , DETAIL_RREP2  )(__VA_ARGS__)
#define REPINF(...) DETAIL_OVERLOAD3(__VA_ARGS__, DETAIL_REPINF3, DETAIL_REPINF2, DETAIL_REPINF1)(__VA_ARGS__)

#define LOOP(n) for (std::remove_reference_t<std::remove_const_t<decltype(n)>> DETAIL_UNIQVAR(loop_variable) = n; DETAIL_UNIQVAR(loop_variable) --> 0;)

#define ALL(iterable) std::begin(iterable), std::end(iterable)
#define INPUT(type, ...) type __VA_ARGS__; read(__VA_ARGS__)

// ! debug

#ifdef LOCAL
#  define debug(...) debug_internal(#__VA_ARGS__, __VA_ARGS__)

template <class T, class... Args>
void debug_internal(const char* s, T&& first, Args&&... args) {
    constexpr const char* prefix = "[\033[32mDEBUG\033[m] ";
    constexpr const char* open_brakets = sizeof...(args) == 0 ? "" : "(";
    constexpr const char* close_brakets = sizeof...(args) == 0 ? "" : ")";
    std::cerr << prefix << open_brakets << s << close_brakets << ": " << open_brakets << std::forward<T>(first);
    ((std::cerr << ", " << std::forward<Args>(args)), ...);
    std::cerr << close_brakets << "\n";
}

#else
#  define debug(...) void(0)
#endif

// ! I/O utilities

// __int128_t
std::ostream& operator<<(std::ostream& dest, __int128_t value) {
    std::ostream::sentry s(dest);
    if (s) {
        __uint128_t tmp = value < 0 ? -value : value;
        char buffer[128];
        char* d = std::end(buffer);
        do {
            --d;
            *d = "0123456789"[tmp % 10];
            tmp /= 10;
        } while (tmp != 0);
        if (value < 0) {
            --d;
            *d = '-';
        }
        int len = std::end(buffer) - d;
        if (dest.rdbuf()->sputn(d, len) != len) {
            dest.setstate(std::ios_base::badbit);
        }
    }
    return dest;
}
// __uint128_t
std::ostream& operator<<(std::ostream& dest, __uint128_t value) {
    std::ostream::sentry s(dest);
    if (s) {
        char buffer[128];
        char* d = std::end(buffer);
        do {
            --d;
            *d = "0123456789"[value % 10];
            value /= 10;
        } while (value != 0);
        int len = std::end(buffer) - d;
        if (dest.rdbuf()->sputn(d, len) != len) {
            dest.setstate(std::ios_base::badbit);
        }
    }
    return dest;
}

// pair
template <typename T, typename U>
std::ostream& operator<<(std::ostream& out, const std::pair<T, U>& a) {
    return out << a.first << ' ' << a.second;
}
// tuple
template <unsigned int N = 0, typename ...Args>
std::ostream& operator<<(std::ostream& out, const std::tuple<Args...>& a) {
    if constexpr (N >= std::tuple_size_v<std::tuple<Args...>>) return out;
    else {
        out << std::get<N>(a);
        if constexpr (N + 1 < std::tuple_size_v<std::tuple<Args...>>) out << ' ';
        return operator<<<N + 1>(out, a);
    }
}
// vector
template <typename T>
std::ostream& operator<<(std::ostream& out, const std::vector<T>& a) {
    for (auto it = a.begin(); it != a.end();) {
        out << *it;
        if (++it != a.end()) out << ' ';
    }
    return out;
}
// array
template <typename T, size_t N>
std::ostream& operator<<(std::ostream& out, const std::array<T, N>& a) {
    for (auto it = a.begin(); it != a.end();) {
        out << *it;
        if (++it != a.end()) out << ' ';
    }
    return out;
}
inline void print() { std::cout << '\n'; }
template <typename Head, typename... Tail>
inline void print(const Head& head, const Tail &...tails) {
    std::cout << head;
    if (sizeof...(tails)) std::cout << ' ';
    print(tails...);
}
template <typename Iterable>
auto print_all(const Iterable& v, std::string sep = " ", std::string end = "\n") -> decltype(std::cout << *v.begin(), void()) {
    for (auto it = v.begin(); it != v.end();) {
        std::cout << *it;
        if (++it != v.end()) std::cout << sep;
    }
    std::cout << end;
}

__int128_t stoi128(const std::string& s) {
    __int128_t ret = 0;
    for (int i = 0; i < int(s.size()); i++) if ('0' <= s[i] and s[i] <= '9') ret = 10 * ret + s[i] - '0';
    if (s[0] == '-') ret = -ret;
    return ret;
}
__uint128_t stou128(const std::string& s) {
    __uint128_t ret = 0;
    for (int i = 0; i < int(s.size()); i++) if ('0' <= s[i] and s[i] <= '9') ret = 10 * ret + s[i] - '0';
    return ret;
}
// __int128_t
std::istream& operator>>(std::istream& in, __int128_t& v) {
    std::string s;
    in >> s;
    v = stoi128(s);
    return in;
}
// __uint128_t
std::istream& operator>>(std::istream& in, __uint128_t& v) {
    std::string s;
    in >> s;
    v = stou128(s);
    return in;
}
// pair
template <typename T, typename U>
std::istream& operator>>(std::istream& in, std::pair<T, U>& a) {
    return in >> a.first >> a.second;
}
// tuple
template <unsigned int N = 0, typename ...Args>
std::istream& operator>>(std::istream& in, std::tuple<Args...>& a) {
    if constexpr (N >= std::tuple_size_v<std::tuple<Args...>>) return in;
    else return operator>><N + 1>(in >> std::get<N>(a), a);
}
// vector
template <typename T>
std::istream& operator>>(std::istream& in, std::vector<T>& a) {
    for (auto it = a.begin(); it != a.end(); ++it) in >> *it;
    return in;
}
// array
template <typename T, size_t N>
std::istream& operator>>(std::istream& in, std::array<T, N>& a) {
    for (auto it = a.begin(); it != a.end(); ++it) in >> *it;
    return in;
}
template <typename ...Args>
void read(Args &...args) {
    (std::cin >> ... >> args);
}

// ! integral utilities

// Returns pow(-1, n)
template <typename T> constexpr inline int pow_m1(T n) {
    return -(n & 1) | 1;
}
// Returns pow(-1, n)
template <> constexpr inline int pow_m1<bool>(bool n) {
    return -int(n) | 1;
}

// Returns floor(x / y)
template <typename T> constexpr inline T fld(const T x, const T y) {
    return (x ^ y) >= 0 ? x / y : (x - (y + pow_m1(y >= 0))) / y;
}
template <typename T> constexpr inline T cld(const T x, const T y) {
    return (x ^ y) <= 0 ? x / y : (x + (y + pow_m1(y >= 0))) / y;
}

template <typename T, std::enable_if_t<std::negation_v<suisen::is_nbit<T, 64>>, std::nullptr_t> = nullptr>
__attribute__((target("popcnt"))) constexpr inline int popcount(const T x) { return _mm_popcnt_u32(x); }
template <typename T, std::enable_if_t<suisen::is_nbit_v<T, 64>, std::nullptr_t> = nullptr>
__attribute__((target("popcnt"))) constexpr inline int popcount(const T x) { return _mm_popcnt_u64(x); }
template <typename T, std::enable_if_t<std::negation_v<suisen::is_nbit<T, 64>>, std::nullptr_t> = nullptr>
constexpr inline int count_lz(const T x) { return x ? __builtin_clz(x) : suisen::bit_num<T>; }
template <typename T, std::enable_if_t<suisen::is_nbit_v<T, 64>, std::nullptr_t> = nullptr>
constexpr inline int count_lz(const T x) { return x ? __builtin_clzll(x) : suisen::bit_num<T>; }
template <typename T, std::enable_if_t<std::negation_v<suisen::is_nbit<T, 64>>, std::nullptr_t> = nullptr>
constexpr inline int count_tz(const T x) { return x ? __builtin_ctz(x) : suisen::bit_num<T>; }
template <typename T, std::enable_if_t<suisen::is_nbit_v<T, 64>, std::nullptr_t> = nullptr>
constexpr inline int count_tz(const T x) { return x ? __builtin_ctzll(x) : suisen::bit_num<T>; }
template <typename T> constexpr inline int floor_log2(const T x) { return suisen::bit_num<T> - 1 - count_lz(x); }
template <typename T> constexpr inline int ceil_log2(const T x) { return floor_log2(x) + ((x & -x) != x); }
template <typename T> constexpr inline int kth_bit(const T x, const unsigned int k) { return (x >> k) & 1; }
template <typename T> constexpr inline int parity(const T x) { return popcount(x) & 1; }

// ! container

template <typename T, typename Comparator>
auto priqueue_comp(const Comparator comparator) {
    return std::priority_queue<T, std::vector<T>, Comparator>(comparator);
}

template <typename Container>
void sort_unique_erase(Container& a) {
    std::sort(a.begin(), a.end());
    a.erase(std::unique(a.begin(), a.end()), a.end());
}

template <typename InputIterator, typename BiConsumer>
auto foreach_adjacent_values(InputIterator first, InputIterator last, BiConsumer f) -> decltype(f(*first++, *last), void()) {
    if (first != last) for (auto itr = first, itl = itr++; itr != last; itl = itr++) f(*itl, *itr);
}
template <typename Container, typename BiConsumer>
auto foreach_adjacent_values(Container &&c, BiConsumer f) -> decltype(c.begin(), c.end(), void()) {
    foreach_adjacent_values(c.begin(), c.end(), f);
}

// ! other utilities

// x <- min(x, y). returns true iff `x` has chenged.
template <typename T>
inline bool chmin(T& x, const T& y) {
    return y >= x ? false : (x = y, true);
}
// x <- max(x, y). returns true iff `x` has chenged.
template <typename T>
inline bool chmax(T& x, const T& y) {
    return y <= x ? false : (x = y, true);
}

template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
std::string bin(T val, int bit_num = -1) {
    std::string res;
    if (bit_num != -1) {
        for (int bit = bit_num; bit-- > 0;) res += '0' + ((val >> bit) & 1);
    } else {
        for (; val; val >>= 1) res += '0' + (val & 1);
        std::reverse(res.begin(), res.end());
    }
    return res;
}

template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
std::vector<T> digits_low_to_high(T val, T base = 10) {
    std::vector<T> res;
    for (; val; val /= base) res.push_back(val % base);
    if (res.empty()) res.push_back(T{ 0 });
    return res;
}
template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
std::vector<T> digits_high_to_low(T val, T base = 10) {
    auto res = digits_low_to_high(val, base);
    std::reverse(res.begin(), res.end());
    return res;
}

template <typename T>
std::string join(const std::vector<T>& v, const std::string& sep, const std::string& end) {
    std::ostringstream ss;
    for (auto it = v.begin(); it != v.end();) {
        ss << *it;
        if (++it != v.end()) ss << sep;
    }
    ss << end;
    return ss.str();
}

template <typename Func, typename Seq>
auto transform_to_vector(const Func &f, const Seq &s) {
    std::vector<std::invoke_result_t<Func, typename Seq::value_type>> v;
    v.reserve(std::size(s)), std::transform(std::begin(s), std::end(s), std::back_inserter(v), f);
    return v;
}
template <typename T, typename Seq>
auto copy_to_vector(const Seq &s) {
    std::vector<T> v;
    v.reserve(std::size(s)), std::copy(std::begin(s), std::end(s), std::back_inserter(v));
    return v;
}
template <typename Seq>
Seq concat(Seq s, const Seq &t) {
    s.reserve(std::size(s) + std::size(t));
    std::copy(std::begin(t), std::end(t), std::back_inserter(s));
    return s;
}
template <typename Seq>
std::vector<Seq> split(const Seq s, typename Seq::value_type delim) {
    std::vector<Seq> res;
    for (auto itl = std::begin(s), itr = itl;; itl = ++itr) {
        while (itr != std::end(s) and *itr != delim) ++itr;
        res.emplace_back(itl, itr);
        if (itr == std::end(s)) return res;
    }
}

int digit_to_int(char c) { return c - '0'; }
int lowercase_to_int(char c) { return c - 'a'; }
int uppercase_to_int(char c) { return c - 'A'; }

std::vector<int> digit_str_to_ints(const std::string &s) {
    return transform_to_vector(digit_to_int, s);
}
std::vector<int> lowercase_str_to_ints(const std::string &s) {
    return transform_to_vector(lowercase_to_int, s);
}
std::vector<int> uppercase_str_to_ints(const std::string &s) {
    return transform_to_vector(uppercase_to_int, s);
}

template <typename T, typename ToKey, typename CompareValue = std::less<>,
    std::enable_if_t<
        std::conjunction_v<
            std::is_invocable<ToKey, T>,
            std::is_invocable_r<bool, CompareValue, std::invoke_result_t<ToKey, T>, std::invoke_result_t<ToKey, T>
        >
    >, std::nullptr_t> = nullptr
>
auto comparator(const ToKey &to_key, const CompareValue &compare_value = std::less<>()) {
    return [to_key, compare_value](const T& x, const T& y) { return compare_value(to_key(x), to_key(y)); };
}

template <typename 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) {
    std::vector<int> p(n);
    std::iota(p.begin(), p.end(), 0);
    std::sort(p.begin(), p.end(), comparator<int>(to_key));
    return p;
}
template <typename 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);
    std::iota(p.begin(), p.end(), 0);
    std::sort(p.begin(), p.end(), compare);
    return p;
}

const std::string Yes = "Yes", No = "No", YES = "YES", NO = "NO";

namespace suisen {}
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_ {};

// ! code from here

#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 <cmath>

namespace suisen {
    template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
    T floor_kth_root(T x, int k) {
        if (k == 1 or x == 0 or x == 1) return x;
        if (k == 2) return ::sqrtl(x);
        if (k >= 64) return 1;
        T res = ::powl(x, ::nextafterl(1 / (long double) k, 0));
        while (::powl(res + 1, k) <= x) ++res;
        return res;
    }
    template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
    T ceil_kth_root(T x, int k) {
        T res = floor_kth_root(x, k);
        res += ::powl(res, k) < x;
        return res;
    }
} // namespace suisen

#include <vector>

namespace suisen {
    template <typename FPSType, typename T>
    std::vector<typename FPSType::value_type> multi_point_eval(const FPSType& f, const std::vector<T>& xs) {
        int n = xs.size();
        if (n == 0) return {};
        std::vector<FPSType> seg(2 * n);
        for (int i = 0; i < n; ++i) seg[n + i] = FPSType{ -xs[i], 1 };
        for (int i = n - 1; i > 0; --i) seg[i] = seg[i * 2] * seg[i * 2 + 1];
        seg[1] = f % seg[1];
        for (int i = 2; i < 2 * n; ++i) seg[i] = seg[i / 2] % seg[i];
        std::vector<typename FPSType::value_type> ys(n);
        for (int i = 0; i < n; ++i) ys[i] = seg[n + i].size() ? seg[n + i][0] : 0;
        return ys;
    }
} // namespace suisen

#include <optional>
#include <queue>

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

#include <cassert>

/**
 * 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() {}
        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();
}

using fps = FormalPowerSeries<mint>;

constexpr int B = (1 << 17);

namespace suisen {
    template <int base_as_int, typename mint>
    struct static_pow_mods {
        static_pow_mods() {}
        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() {}
        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();
    };
}

int main() {
    int t;
    read(t);
    LOOP(t) {
        mint a;
        int n;
        read(a, n);

        const int C = n / B;

        mint res = 0;

        vector<mint> pa(B), pa2(B);
        {
            pa[0] = a;
            REP(r, 1, B) {
                pa[r] = pa[r - 1] * a * a;
            }
            pa2[0] = 1;
            REP(r, 1, B) {
                pa2[r] = pa2[r - 1] * pa[r - 1];
            }
        }
        pow_mods<mint> pow_aB(a.pow(2 * B), B);

        const mint aB2 = a.pow((long long) B * B);
        vector<mint> paB2(B), paB22(B);
        {
            paB2[0] = aB2;
            REP(r, 1, B) {
                paB2[r] = paB2[r - 1] * aB2 * aB2;
            }
            paB22[0] = 1;
            REP(r, 1, B) {
                paB22[r] = paB22[r - 1] * paB2[r - 1];
            }
        }

        fps f = pa2;

        vector<mint> xs(C);
        REP(q, C) {
            xs[q] = pow_aB[q];
        }
        vector<mint> ys = multi_point_eval(f, xs);

        const long long BC = B * C, r = n - B * C;
        const mint aBC = a.pow(BC);

        pow_mods<mint> pow_aBC(a.pow(2 * BC), r);

        REP(q, C) {
            res += paB22[q] * ys[q];
        }
        REP(i, r) {
            res += aBC * pa2[i] * pow_aBC[i];
        }
        print(res);
    }
    return 0;
}