#include <bits/stdc++.h>
// created [2020/03/20] 21:31:31
#pragma GCC diagnostic ignored "-Wsign-compare"
#pragma GCC diagnostic ignored "-Wsign-conversion"

using i32   = int32_t;
using i64   = int64_t;
using u32   = uint32_t;
using u64   = uint64_t;
using uint  = unsigned int;
using usize = std::size_t;
using ll    = long long;
using ull   = unsigned long long;
using ld    = long double;
template<typename T, usize n>
using arr = T (&)[n];
template<typename T, usize n>
using c_arr = const T (&)[n];
template<typename T>
using max_heap = std::priority_queue<T>;
template<typename T>
using min_heap = std::priority_queue<T, std::vector<T>, std::greater<T>>;
template<typename T> constexpr T popcount(const T u) { return u ? static_cast<T>(__builtin_popcountll(static_cast<u64>(u))) : static_cast<T>(0); }
template<typename T> constexpr T log2p1(const T u) { return u ? static_cast<T>(64 - __builtin_clzll(static_cast<u64>(u))) : static_cast<T>(0); }
template<typename T> constexpr T msbp1(const T u) { return log2p1(u); }
template<typename T> constexpr T lsbp1(const T u) { return __builtin_ffsll(u); }
template<typename T> constexpr T clog(const T u) { return u ? log2p1(u - 1) : static_cast<T>(u); }
template<typename T> constexpr bool ispow2(const T u) { return u and (static_cast<u64>(u) & static_cast<u64>(u - 1)) == 0; }
template<typename T> constexpr T ceil2(const T u) { return static_cast<T>(1) << clog(u); }
template<typename T> constexpr T floor2(const T u) { return u == 0 ? static_cast<T>(0) : static_cast<T>(1) << (log2p1(u) - 1); }
template<typename T> constexpr bool btest(const T mask, const usize ind) { return static_cast<bool>((static_cast<u64>(mask) >> ind) & static_cast<u64>(1)); }
template<typename T> void bset(T& mask, const usize ind) { mask |= (static_cast<T>(1) << ind); }
template<typename T> void breset(T& mask, const usize ind) { mask &= ~(static_cast<T>(1) << ind); }
template<typename T> void bflip(T& mask, const usize ind) { mask ^= (static_cast<T>(1) << ind); }
template<typename T> void bset(T& mask, const usize ind, const bool b) { (b ? bset(mask, ind) : breset(mask, ind)); }
template<typename T> constexpr T bcut(const T mask, const usize ind) { return ind == 0 ? static_cast<T>(0) : static_cast<T>((static_cast<u64>(mask) << (64 - ind)) >> (64 - ind)); }
template<typename T> bool chmin(T& a, const T& b) { return (a > b ? a = b, true : false); }
template<typename T> bool chmax(T& a, const T& b) { return (a < b ? a = b, true : false); }
constexpr unsigned int mod                  = 1000000007;
template<typename T> constexpr T inf_v      = std::numeric_limits<T>::max() / 4;
template<typename Real> constexpr Real pi_v = Real{3.141592653589793238462643383279502884};
auto mfp = [](auto&& f) { return [=](auto&&... args) { return f(f, std::forward<decltype(args)>(args)...); }; };

template<typename T>
T in()
{
    T v;
    return std::cin >> v, v;
}
template<typename T, typename Uint, usize n, usize i>
T in_v(typename std::enable_if<(i == n), c_arr<Uint, n>>::type) { return in<T>(); }
template<typename T, typename Uint, usize n, usize i>
auto in_v(typename std::enable_if<(i < n), c_arr<Uint, n>>::type& szs)
{
    const usize s = (usize)szs[i];
    std::vector<decltype(in_v<T, Uint, n, i + 1>(szs))> ans(s);
    for (usize j = 0; j < s; j++) { ans[j] = in_v<T, Uint, n, i + 1>(szs); }
    return ans;
}
template<typename T, typename Uint, usize n>
auto in_v(c_arr<Uint, n> szs) { return in_v<T, Uint, n, 0>(szs); }
template<typename... Types>
auto in_t() { return std::tuple<std::decay_t<Types>...>{in<Types>()...}; }
struct io_init
{
    io_init()
    {
        std::cin.tie(nullptr), std::ios::sync_with_stdio(false);
        std::cout << std::fixed << std::setprecision(20);
    }
    void clear()
    {
        std::cin.tie(), std::ios::sync_with_stdio(true);
    }
} io_setting;

int out() { return 0; }
template<typename T>
int out(const T& v) { return std::cout << v, 0; }
template<typename T>
int out(const std::vector<T>& v)
{
    for (usize i = 0; i < v.size(); i++) {
        if (i > 0) { std::cout << ' '; }
        out(v[i]);
    }
    return 0;
}
template<typename T1, typename T2>
int out(const std::pair<T1, T2>& v) { return out(v.first), std::cout << ' ', out(v.second), 0; }
template<typename T, typename... Args>
int out(const T& v, const Args... args) { return out(v), std::cout << ' ', out(args...), 0; }
template<typename... Args>
int outln(const Args... args) { return out(args...), std::cout << '\n', 0; }
template<typename... Args>
int outel(const Args... args) { return out(args...), std::cout << std::endl, 0; }
#    define SHOW(...) static_cast<void>(0)
constexpr ull TEN(const usize n) { return n == 0 ? 1ULL : TEN(n - 1) * 10ULL; }

template<typename T, typename Uint, usize n, usize i>
auto make_v(typename std::enable_if<(i == n), c_arr<Uint, n>>::type, const T& v = T{}) { return v; }
template<typename T, typename Uint, usize n, usize i>
auto make_v(typename std::enable_if<(i < n), c_arr<Uint, n>>::type szs, const T& v = T{})
{
    const usize s = (usize)szs[i];
    return std::vector<decltype(make_v<T, Uint, n, i + 1>(szs, v))>(s, make_v<T, Uint, n, i + 1>(szs, v));
}
template<typename T, typename Uint, usize n>
auto make_v(c_arr<Uint, n> szs, const T& t = T{}) { return make_v<T, Uint, n, 0>(szs, t); }



class xoshiro
{
public:
    using result_type = uint32_t;
    static constexpr result_type min() { return std::numeric_limits<result_type>::min(); }
    static constexpr result_type max() { return std::numeric_limits<result_type>::max(); }
    xoshiro() : xoshiro(std::random_device{}()) {}
    xoshiro(uint64_t seed)
    {
        uint64_t z = 0;
        for (int i = 0; i < 4; i++) { z = (seed += 0x9e3779b97f4a7c15), z = (z ^ (z >> 33)) * 0x62A9D9ED799705F5, z = (z ^ (z >> 28)) * 0xCB24D0A5C88C35B3, s[i] = static_cast<result_type>(z >> 32); }
    }
    result_type operator()()
    {
        const result_type result = rotl(s[1] * 5, 7) * 9, t = s[1] << 9;
        return s[2] ^= s[0], s[3] ^= s[1], s[1] ^= s[2], s[0] ^= s[3], s[2] ^= t, s[3] = rotl(s[3], 11), result;
    }
    void discard(const usize rep)
    {
        for (usize i = 0; i < rep; i++) { (*this)(); }
    }

private:
    result_type s[4];
    static result_type rotl(const result_type x, const int k) { return (x << k) | (x >> (32 - k)); }
};
class xoshiro_64
{
public:
    using result_type = uint64_t;
    static constexpr result_type min() { return std::numeric_limits<result_type>::min(); }
    static constexpr result_type max() { return std::numeric_limits<result_type>::max(); }
    xoshiro_64() : xoshiro_64(std::random_device{}()) {}
    xoshiro_64(uint64_t seed)
    {
        uint64_t z = 0;
        for (int i = 0; i < 4; i++) { z = (seed += 0x9e3779b97f4a7c15), z = (z ^ (z >> 30)) * 0xbf58476d1ce4e5b9, z = (z ^ (z >> 27)) * 0x94d049bb133111eb, s[i] = static_cast<result_type>(z ^ (z >> 31)); }
    }
    result_type operator()()
    {
        const result_type result = rotl(s[1] * 5, 7) * 9, t = s[1] << 17;
        return s[2] ^= s[0], s[3] ^= s[1], s[1] ^= s[2], s[0] ^= s[3], s[2] ^= t, s[3] = rotl(s[3], 45), result;
    }
    void discard(const usize rep)
    {
        for (usize i = 0; i < rep; i++) { (*this)(); }
    }

private:
    result_type s[4];
    static result_type rotl(const result_type x, const int k) { return (x << k) | (x >> (64 - k)); }
};
template<typename Rng>
class rng_base
{
public:
    using rng_type    = Rng;
    using result_type = typename rng_type::result_type;
    static constexpr result_type min() { return rng_type::min(); }
    static constexpr result_type max() { return rng_type::max(); }
    rng_base() : rng_base(std::random_device{}()) {}
    rng_base(const u64 seed) : rng(seed) {}
    ~rng_base() = default;
    result_type operator()(const result_type max = std::numeric_limits<result_type>::max())
    {
        if (max == std::numeric_limits<result_type>::max()) { return static_cast<result_type>(rng()); }
        if (ispow2(max + 1)) { return static_cast<result_type>(rng() & max); }
        const result_type mask = static_cast<result_type>(ceil2(static_cast<u64>(max + 1))) - 1;
        while (true) {
            const result_type ans = static_cast<result_type>(rng() & mask);
            if (ans <= max) { return ans; }
        }
    }
    template<typename Int = result_type>
    Int operator()(const Int min, const Int max) { return min + (Int)(*this)(max - min); }
    operator bool() { return (bool)(*this)(0, 1); }
    template<typename Int> std::pair<Int, Int> pair(const Int min, const Int max) { return std::pair<Int, Int>{*this(min, max), *this(min, max)}; }
    template<typename Int>
    std::vector<Int> vec(const usize n, const Int min, const Int max)
    {
        std::vector<Int> v(n);
        for (usize i = 0; i < n; i++) { v[i] = (*this)(min, max); }
        return v;
    }
    std::vector<usize> perm(const usize n)
    {
        std::vector<usize> ans(n);
        std::iota(ans.begin(), ans.end(), 0UL);
        std::shuffle(ans.begin(), ans.end(), rng);
        return ans;
    }

private:
    Rng rng;
};
using rng_mt        = rng_base<std::mt19937>;
using rng_mt64      = rng_base<std::mt19937_64>;
using rng_xoshiro   = rng_base<xoshiro>;
using rng_xoshiro64 = rng_base<xoshiro_64>;
rng_mt g_rng_mt;
rng_mt64 g_rng_mt64;
rng_xoshiro g_rng_xo;
rng_xoshiro64 g_rng_xo64;
template<typename mint>
mint modsqrt(const mint& x)
{
    if (x == 0) { return 0; }
    const uint p = mint::mod();
    if (p == 2) { return 1; }
    if ((x ^ ((p - 1) / 2)) != 1) { return 0; }
    uint Q = p - 1, S = 0;
    while (Q % 2 == 0) { Q /= 2, S++; }
    mint z = 1;
    while (true) {
        z = mint(g_rng_xo(2U, p - 1));
        if ((z ^ ((p - 1) / 2)) != 1) { break; }
    }
    uint M = S;
    mint c = z ^ Q, t = x ^ Q, R = x ^ ((Q + 1) / 2);
    while (true) {
        if (t == 0) { return 0; }
        if (t == 1) { return R; }
        mint s = t * t;
        uint i = 1;
        for (;; i++) {
            if (s == 1) { break; }
            s *= s;
        }
        mint b = c;
        for (uint j = 0; j < M - i - 1; j++) { b *= b; }
        M = i, c = b * b, t *= b * b, R *= b;
    }
    return 0;
}

template<typename Real>
struct complex
{
    using value_type = Real;
    complex() : real{Real{0}}, imag{Real{0}} {}
    complex(const complex&) = default;
    complex(const Real& theta) : real(std::cos(theta)), imag(std::sin(theta)) {}
    complex(const Real& r, const Real& i) : real{r}, imag{i} {}
    ~complex() = default;
    friend complex operator+(const complex& c) { return c; }
    friend complex operator-(const complex& c) { return complex{-c.real, -c.imag}; }
    friend complex operator+(const complex& c1, const complex& c2) { return complex{c1.real + c2.real, c1.imag + c2.imag}; }
    friend complex operator-(const complex& c1, const complex& c2) { return complex{c1.real - c2.real, c1.imag - c2.imag}; }
    friend complex operator*(const complex& c1, const complex& c2) { return complex{c1.real * c2.real - c1.imag * c2.imag, c1.real * c2.imag + c1.imag * c2.real}; }
    friend complex operator*(const complex& c, const Real& r) { return complex{c.real * r, c.imag * r}; }
    friend complex operator/(complex& c1, complex& c2) { c1* c2.conj() / c2.norm(); }
    friend bool operator==(const complex& c1, const complex& c2) { return c1.real == c2.real and c1.imag == c2.imag; }
    friend bool operator!=(const complex& c1, const complex& c2) { return not(c1 == c2); }
    friend complex& operator+=(complex& c1, const complex& c2) { return c1.real += c2.real, c1.imag += c2.imag, c1; }
    friend complex& operator-=(complex& c1, const complex& c2) { return c1.real += c2.real, c1.imag += c2.imag, c1; }
    friend complex& operator*=(complex& c1, const complex& c2) { return c1 = c1 * c2; }
    friend complex& operator*=(complex& c, const Real& r) { return c = c * r; }
    friend complex& operator/=(complex& c1, const complex& c2) { return c1 = c1 / c2; }
    complex conj() const { return complex{real, -imag}; }
    Real norm() const { return real * real + imag * imag; }
    Real abs() const { return std::sqrt(norm()); }
    Real arg() const { return std::atan2(imag, real); }
    friend std::ostream& operator<<(std::ostream& os, const complex& c) { return os << c.real << "+" << c.imag << "i"; }
    Real real, imag;
};


template<typename T> T gcd(const T& a, const T& b) { return a < 0 ? gcd(-a, b) : b < 0 ? gcd(a, -b) : (a > b ? gcd(b, a) : a == 0 ? b : gcd(b % a, a)); }
template<typename T> T lcm(const T& a, const T& b) { return a / gcd(a, b) * b; }
template<typename T>
constexpr std::pair<T, T> extgcd(const T a, const T b)
{
    if (b == 0) { return std::pair<T, T>{1, 0}; }
    const auto g = gcd(a, b), da = std::abs(b) / g;
    const auto p = extgcd(b, a % b);
    const auto x = (da + p.second % da) % da, y = (g - a * x) / b;
    return {x, y};
}
template<typename T>
constexpr T inverse(const T a, const T mod) { return extgcd(a, mod).first; }
template<uint mod_value, bool dynamic = false>
class modint_base
{
public:
    template<typename UInt = uint>
    static std::enable_if_t<dynamic, const UInt> mod() { return mod_ref(); }
    template<typename UInt = uint>
    static constexpr std::enable_if_t<not dynamic, const UInt> mod() { return mod_value; }
    template<typename UInt = uint>
    static void set_mod(const std::enable_if_t<dynamic, const UInt> mod) { mod_ref() = mod, inv_ref() = {1, 1}; }
    modint_base() : v{0} {}
    modint_base(const ll val) : v{norm(static_cast<uint>(val % static_cast<ll>(mod()) + static_cast<ll>(mod())))} {}
    modint_base(const modint_base& n) : v{n()} {}
    explicit operator bool() const { return v != 0; }
    bool operator!() const { return not static_cast<bool>(*this); }
    modint_base& operator=(const modint_base& m) { return v = m(), (*this); }
    modint_base& operator=(const ll val) { return v = norm(uint(val % static_cast<ll>(mod()) + static_cast<ll>(mod()))), (*this); }
    friend modint_base operator+(const modint_base& m) { return m; }
    friend modint_base operator-(const modint_base& m) { return make(norm(mod() - m.v)); }
    friend modint_base operator+(const modint_base& m1, const modint_base& m2) { return make(norm(m1.v + m2.v)); }
    friend modint_base operator-(const modint_base& m1, const modint_base& m2) { return make(norm(m1.v + mod() - m2.v)); }
    friend modint_base operator*(const modint_base& m1, const modint_base& m2) { return make(static_cast<uint>(static_cast<ll>(m1.v) * static_cast<ll>(m2.v) % static_cast<ll>(mod()))); }
    friend modint_base operator/(const modint_base& m1, const modint_base& m2) { return m1 * inv(m2.v); }
    friend modint_base operator+(const modint_base& m, const ll val) { return modint_base{static_cast<ll>(m.v) + val}; }
    friend modint_base operator-(const modint_base& m, const ll val) { return modint_base{static_cast<ll>(m.v) - val}; }
    friend modint_base operator*(const modint_base& m, const ll val) { return modint_base{static_cast<ll>(m.v) * (val % static_cast<ll>(mod()))}; }
    friend modint_base operator/(const modint_base& m, const ll val) { return modint_base{static_cast<ll>(m.v) * inv(val)}; }
    friend modint_base operator+(const ll val, const modint_base& m) { return modint_base{static_cast<ll>(m.v) + val}; }
    friend modint_base operator-(const ll val, const modint_base& m) { return modint_base{-static_cast<ll>(m.v) + val}; }
    friend modint_base operator*(const ll val, const modint_base& m) { return modint_base{static_cast<ll>(m.v) * (val % static_cast<ll>(mod()))}; }
    friend modint_base operator/(const ll val, const modint_base& m) { return modint_base{val * inv(static_cast<ll>(m.v))}; }
    friend modint_base& operator+=(modint_base& m1, const modint_base& m2) { return m1 = m1 + m2; }
    friend modint_base& operator-=(modint_base& m1, const modint_base& m2) { return m1 = m1 - m2; }
    friend modint_base& operator*=(modint_base& m1, const modint_base& m2) { return m1 = m1 * m2; }
    friend modint_base& operator/=(modint_base& m1, const modint_base& m2) { return m1 = m1 / m2; }
    friend modint_base& operator+=(modint_base& m, const ll val) { return m = m + val; }
    friend modint_base& operator-=(modint_base& m, const ll val) { return m = m - val; }
    friend modint_base& operator*=(modint_base& m, const ll val) { return m = m * val; }
    friend modint_base& operator/=(modint_base& m, const ll val) { return m = m / val; }
    friend modint_base operator^(const modint_base& m, const ll n) { return power(m.v, n); }
    friend modint_base& operator^=(modint_base& m, const ll n) { return m = m ^ n; }
    friend bool operator==(const modint_base& m1, const modint_base& m2) { return m1.v == m2.v; }
    friend bool operator!=(const modint_base& m1, const modint_base& m2) { return not(m1 == m2); }
    friend bool operator==(const modint_base& m, const ll val) { return m.v == norm(static_cast<uint>(static_cast<ll>(mod()) + val % static_cast<ll>(mod()))); }
    friend bool operator!=(const modint_base& m, const ll val) { return not(m == val); }
    friend bool operator==(const ll val, const modint_base& m) { return m.v == norm(static_cast<uint>(static_cast<ll>(mod()) + val % static_cast<ll>(mod()))); }
    friend bool operator!=(const ll val, const modint_base& m) { return not(m == val); }
    friend std::istream& operator>>(std::istream& is, modint_base& m)
    {
        ll v;
        return is >> v, m = v, is;
    }
    friend std::ostream& operator<<(std::ostream& os, const modint_base& m) { return os << m(); }
    uint operator()() const { return v; }
    static modint_base small_inv(const usize n)
    {
        auto& in = inv_ref();
        if (n < in.size()) { return in[n]; }
        for (usize i = in.size(); i <= n; i++) { in.push_back(-in[modint_base::mod() % i] * (modint_base::mod() / i)); }
        return in.back();
    }
    std::pair<ll, ll> quad() const
    {
        const auto ans = quad_r(v, mod());
        ll x = std::get<0>(ans), y = std::get<1>(ans);
        if (y < 0) { x = -x, y = -y; }
        return {x, y};
    }

private:
    static std::tuple<ll, ll, ll> quad_r(const ll r, const ll p)  // r = x/y (mod p), (x,y,z) s.t. x=yr+pz
    {
        if (std::abs(r) <= 1000) { return {r, 1, 0}; }
        ll nr = p % r, q = p / r;
        if (nr * 2LL >= r) { nr -= r, q++; }
        if (nr * 2LL <= -r) { nr += r, q--; }
        const auto sub = quad_r(nr, r);
        const ll x = std::get<0>(sub), z = std::get<1>(sub), y = std::get<2>(sub);
        return {x, y - q * z, z};
    }

    template<typename UInt = uint>
    static std::enable_if_t<dynamic, UInt&> mod_ref()
    {
        static UInt mod = 0;
        return mod;
    }
    static uint norm(const uint x) { return x < mod() ? x : x - mod(); }
    static modint_base make(const uint x)
    {
        modint_base m;
        return m.v = x, m;
    }
    static modint_base power(modint_base x, ull n)
    {
        modint_base ans = 1;
        for (; n; n >>= 1, x *= x) {
            if (n & 1) { ans *= x; }
        }
        return ans;
    }
    static modint_base inv(const ll v) { return v <= 2000000 ? small_inv(static_cast<usize>(v)) : modint_base{inverse(v, static_cast<ll>(mod()))}; }
    static std::vector<modint_base>& inv_ref()
    {
        static std::vector<modint_base> in{1, 1};
        return in;
    }
    uint v;
};
template<uint mod>
using modint = modint_base<mod, false>;
template<uint id>
using dynamic_modint = modint_base<id, true>;
template<typename Real = double>
class fft
{
private:
    static constexpr usize depth = 30;
    static constexpr Real pi     = pi_v<Real>;
    static void transform(std::vector<complex<Real>>& a, const usize lg, const bool rev)
    {
        static std::vector<complex<Real>> root[depth];
        const usize sz = a.size();
        assert((1UL << lg) == sz);
        if (root[lg].empty()) {
            root[lg].reserve(sz), root[lg].resize(sz);
            for (usize i = 0; i < sz; i++) { root[lg][i] = complex<Real>(pi * Real(2 * i) / Real(sz)); }
        }
        std::vector<complex<Real>> tmp(sz);
        for (usize w = (sz >> 1); w > 0; w >>= 1) {
            for (usize y = 0; y < (sz >> 1); y += w) {
                const complex<Real> r = rev ? root[lg][y].conj() : root[lg][y];
                for (usize x = 0; x < w; x++) {
                    const auto u = a[y << 1 | x], v = a[y << 1 | x | w] * r;
                    tmp[y | x] = u + v, tmp[y | x | (sz >> 1)] = u - v;
                }
            }
            std::swap(tmp, a);
        }
    }

public:
    using value_type = Real;
    fft()            = delete;
    template<typename T = ll, typename I = int>
    static std::vector<T> simple_convolute(const std::vector<I>& a, const std::vector<I>& b)
    {
        const usize need = a.size() + b.size() - 1, lg = clog(need), sz = 1UL << lg;
        std::vector<complex<Real>> x(sz), y(sz);
        for (usize i = 0; i < a.size(); i++) { x[i] = {(Real)a[i], (Real)0}; }
        for (usize i = 0; i < b.size(); i++) { y[i] = {(Real)b[i], (Real)0}; }
        transform(x, lg, false), transform(y, lg, false);
        for (usize i = 0; i < sz; i++) { x[i] *= y[i]; }
        transform(x, lg, true);
        std::vector<T> ans(need);
        for (usize i = 0; i < need; i++) { ans[i] = (T)std::round(x[i].real / (Real)sz); }
        return ans;
    }
    template<typename T = ll, usize division = 2, typename I = int>
    static std::vector<T> convolute(const std::vector<I>& a, const std::vector<I>& b)
    {
        constexpr usize bitnum = (depth + division - 1) / division;
        const usize need = a.size() + b.size() - 1, lg = clog(need), sz = 1UL << lg;
        std::vector<complex<value_type>> x[division], y[division], tmp(sz);
        for (usize i = 0; i < division; i++) {
            x[i].reserve(sz), x[i].resize(sz), y[i].reserve(sz), y[i].resize(sz);
            std::fill(tmp.begin() + std::min(a.size(), b.size()), tmp.end(), complex<value_type>{});
            for (usize j = 0; j < a.size(); j++) { tmp[j].real = value_type((a[j] >> (bitnum * i)) & ((1 << bitnum) - 1)); }
            for (usize j = 0; j < b.size(); j++) { tmp[j].imag = value_type((b[j] >> (bitnum * i)) & ((1 << bitnum) - 1)); }
            transform(tmp, lg, false);
            for (usize j = 0; j < sz; j++) { tmp[j] *= value_type(0.5); }
            for (usize j = 0; j < sz; j++) {
                const usize k = j == 0 ? 0UL : sz - j;
                x[i][j] = complex<value_type>{tmp[j].real + tmp[k].real, tmp[j].imag - tmp[k].imag}, y[i][j] = complex<value_type>{tmp[j].imag + tmp[k].imag, -tmp[j].real + tmp[k].real};
            }
        }
        std::vector<complex<value_type>> z[division];
        for (usize i = 0; i < division; i++) { z[i].reserve(sz), z[i].resize(sz); }
        for (usize a = 0; a < division; a++) {
            for (usize b = 0; b < division; b++) {
                for (usize i = 0; i < sz; i++) {
                    if (a + b < division) {
                        z[a + b][i] += x[a][i] * y[b][i];
                    } else {
                        z[a + b - division][i] += x[a][i] * y[b][i] * complex<value_type>(0, 1);
                    }
                }
            }
        }
        for (usize i = 0; i < division; i++) { transform(z[i], lg, true); }
        std::vector<T> ans(need);
        T base = 1;
        for (usize k = 0; k < 2 * division - 1; k++, base *= (1LL << bitnum)) {
            for (usize i = 0; i < need; i++) {
                if (k < division) {
                    ans[i] += base * T(std::round(z[k][i].real / value_type(sz)));
                } else {
                    ans[i] += base * T(std::round(z[k - division][i].imag / value_type(sz)));
                }
            }
        }
        return ans;
    }
    template<uint mod, bool dynamic = false, usize division = 2>
    static std::vector<modint_base<mod, dynamic>> convolute(const std::vector<modint_base<mod, dynamic>>& a, const std::vector<modint_base<mod, dynamic>>& b)
    {
        using mint             = modint_base<mod, dynamic>;
        constexpr usize bitnum = (depth + division - 1) / division;
        const usize need = a.size() + b.size() - 1, lg = clog(need), sz = 1UL << lg;
        std::vector<complex<value_type>> x[division], y[division], tmp(sz);
        for (usize i = 0; i < division; i++) {
            x[i].reserve(sz), x[i].resize(sz), y[i].reserve(sz), y[i].resize(sz);
            std::fill(tmp.begin() + std::min(a.size(), b.size()), tmp.end(), complex<value_type>{});
            for (usize j = 0; j < a.size(); j++) { tmp[j].real = value_type((a[j]() >> (bitnum * i)) & ((1 << bitnum) - 1)); }
            for (usize j = 0; j < b.size(); j++) { tmp[j].imag = value_type((b[j]() >> (bitnum * i)) & ((1 << bitnum) - 1)); }
            transform(tmp, lg, false);
            for (usize j = 0; j < sz; j++) { tmp[j] *= value_type(0.5); }
            for (usize j = 0; j < sz; j++) {
                const usize k = j == 0 ? 0UL : sz - j;
                x[i][j] = complex<value_type>{tmp[j].real + tmp[k].real, tmp[j].imag - tmp[k].imag}, y[i][j] = complex<value_type>{tmp[j].imag + tmp[k].imag, -tmp[j].real + tmp[k].real};
            }
        }
        std::vector<complex<value_type>> z[division];
        for (usize i = 0; i < division; i++) { z[i].reserve(sz), z[i].resize(sz); }
        for (usize a = 0; a < division; a++) {
            for (usize b = 0; b < division; b++) {
                for (usize i = 0; i < sz; i++) {
                    if (a + b < division) {
                        z[a + b][i] += x[a][i] * y[b][i];
                    } else {
                        z[a + b - division][i] += x[a][i] * y[b][i] * complex<value_type>(0, 1);
                    }
                }
            }
        }
        for (usize i = 0; i < division; i++) { transform(z[i], lg, true); }
        std::vector<mint> ans(need);
        mint base = 1;
        for (usize k = 0; k < 2 * division - 1; k++, base *= (1LL << bitnum)) {
            for (usize i = 0; i < need; i++) {
                if (k < division) {
                    ans[i] += int((base * ll(std::round(z[k][i].real / value_type(sz))))());
                } else {
                    ans[i] += int((base * ll(std::round(z[k - division][i].imag / value_type(sz))))());
                }
            }
        }
        return ans;
    }
};

template<uint mod = 998244353, uint root = 3>
class ntt
{
private:
    using value_type             = modint<mod>;
    static constexpr usize depth = 30;
    static void transform(std::vector<value_type>& a, const usize lg, const bool rev)
    {
        const usize N = a.size();
        assert(1UL << lg == N);
        static std::vector<value_type> R[depth];
        if (R[lg].empty()) {
            R[lg].reserve(N), R[lg].resize(N, value_type(1));
            const value_type r = value_type(root) ^ ((mod - 1) / N);
            for (usize i = 1; i < N; i++) { R[lg][i] = R[lg][i - 1] * r; }
        }
        std::vector<value_type> tmp(N);
        for (usize w = (N >> 1); w > 0; w >>= 1) {
            for (usize y = 0; y < (N >> 1); y += w) {
                const value_type r = rev ? R[lg][y == 0 ? 0 : N - y] : R[lg][y];
                for (usize x = 0; x < w; x++) {
                    const auto u = a[y << 1 | x], v = a[y << 1 | x | w]() * r;
                    tmp[y | x] = u + v, tmp[y | x | (N >> 1)] = u - v;
                }
            }
            std::swap(tmp, a);
        }
        if (rev) {
            for (usize i = 0; i < N; i++) { a[i] /= value_type(N); }
        }
    }

public:
    ntt() = delete;
    static std::vector<value_type> convolute(const std::vector<value_type>& a, const std::vector<value_type>& b)
    {
        const usize need = a.size() + b.size() - 1, lg = clog(need), sz = 1UL << lg;
        std::vector<value_type> A(sz, 0), B(sz, 0);
        for (usize i = 0; i < a.size(); i++) { A[i] = a[i](); }
        for (usize i = 0; i < b.size(); i++) { B[i] = b[i](); }
        transform(A, lg, false), transform(B, lg, false);
        for (usize i = 0; i < sz; i++) { A[i] *= B[i]; }
        transform(A, lg, true);
        std::vector<value_type> ans(need);
        for (usize i = 0; i < need; i++) { ans[i] = int(A[i]()); }
        return ans;
    }
};
template<uint mod, uint root, bool dynamic, uint fft_division>
class polynomial_base
{
public:
    using value_type = modint_base<mod, dynamic>;
    polynomial_base() : v(0) {}
    polynomial_base(const value_type& r) : v{r} { shrink(); }
    polynomial_base(const polynomial_base& p) : v{p()} {}
    template<typename InIt>
    polynomial_base(const InIt first, const InIt last) : v{first, last} { shrink(); }
    polynomial_base(const std::initializer_list<value_type>&& list) : v{list} { shrink(); }
    const std::vector<value_type>& operator()() const { return v; }
    value_type& operator[](const usize i) { return v[i]; }
    const value_type operator[](const usize i) const { return (i < size() ? v[i] : value_type(0)); }
    void set(const usize k, const value_type& c)
    {
        if (k >= size()) { v.resize(k + 1); }
        v[k] = c;
    }
    polynomial_base& operator=(const polynomial_base& m) { return v = m(), *this; }
    polynomial_base& operator=(const value_type& val) { return (*this) = polynomial_base(val); }
    friend polynomial_base operator+(const polynomial_base& p) { return p; }
    friend polynomial_base operator-(const polynomial_base& p)
    {
        polynomial_base ans;
        for (usize i = 0; i < p.size(); i++) { ans.set(i, -p[i]); }
        return ans;
    }
    friend polynomial_base operator+(const polynomial_base& p, const polynomial_base& q)
    {
        const usize sz = std::max(p.size(), q.size());
        polynomial_base ans;
        for (usize i = 0; i < sz; i++) { ans.set(i, p[i] + q[i]); }
        return ans;
    }
    friend polynomial_base operator-(const polynomial_base& p, const polynomial_base& q)
    {
        const usize sz = std::max(p.size(), q.size());
        polynomial_base ans;
        for (usize i = 0; i < sz; i++) { ans.set(i, p[i] - q[i]); }
        return ans;
    }
    friend polynomial_base operator*(const polynomial_base& p, const polynomial_base& q)
    {
        constexpr int L = 300;
        return p.size() <= L or q.size() <= L ? naive_multiply(p, q) : fft_multiply(p, q);
    }
    friend polynomial_base operator+(const polynomial_base& p, const value_type& r)
    {
        polynomial_base ans = p;
        ans.set(0, p[0] + r);
        return ans;
    }
    friend polynomial_base operator-(const polynomial_base& p, const value_type& r)
    {
        polynomial_base ans = p;
        ans.set(0, p[0] - r);
        return ans;
    }
    friend polynomial_base operator*(const polynomial_base& p, const value_type& r)
    {
        polynomial_base ans;
        for (usize i = 0; i < p.size(); i++) { ans.set(i, p[i] * r); }
        return ans;
    }
    friend polynomial_base operator/(const polynomial_base& p, const value_type& r)
    {
        polynomial_base ans;
        for (usize i = 0; i < p.size(); i++) { ans.set(i, p[i] / r); }
        return ans;
    }
    friend polynomial_base operator+(const value_type& r, const polynomial_base& q) { return q + r; }
    friend polynomial_base operator-(const value_type& r, const polynomial_base& q) { return -q + r; }
    friend polynomial_base operator*(const value_type& r, const polynomial_base& q) { return q * r; }
    friend polynomial_base operator>>(const polynomial_base& p, const usize s) { return quot_by_pow(p, s); }
    friend polynomial_base operator<<(const polynomial_base& p, const usize s) { return prod_by_pow(p, s); }
    friend polynomial_base operator/(const polynomial_base& p, const polynomial_base& q) { return div(p, q); }
    friend polynomial_base operator%(const polynomial_base& p, const polynomial_base& q) { return rem(p, q); }
    friend bool operator==(const polynomial_base& p, const polynomial_base& q) { return p() == q(); }
    friend bool operator!=(const polynomial_base& p, const polynomial_base& q) { return not(p == q); }
    friend polynomial_base& operator+=(polynomial_base& p, const polynomial_base& q) { return p = p + q; }
    friend polynomial_base& operator-=(polynomial_base& p, const polynomial_base& q) { return p = p - q; }
    friend polynomial_base& operator*=(polynomial_base& p, const polynomial_base& q) { return p = p * q; }
    friend polynomial_base& operator+=(polynomial_base& p, const value_type& r) { return p = p + r; }
    friend polynomial_base& operator-=(polynomial_base& p, const value_type& r) { return p = p - r; }
    friend polynomial_base& operator*=(polynomial_base& p, const value_type& r) { return p = p * r; }
    friend polynomial_base& operator/=(polynomial_base& p, const value_type& r) { return p = p / r; }
    friend polynomial_base& operator>>=(polynomial_base& p, const usize s) { return p = (p >> s); }
    friend polynomial_base& operator<<=(polynomial_base& p, const usize s) { return p = (p << s); }
    friend polynomial_base& operator/=(polynomial_base& p, const polynomial_base& q) { return p = p / q; }
    friend polynomial_base& operator%=(polynomial_base& p, const polynomial_base& q) { return p = p % q; }
    static polynomial_base prod_by_pow(const polynomial_base p, const usize s)
    {
        const usize sz = p.size();
        if (sz == 0) { return polynomial_base(); }
        polynomial_base ans;
        for (usize i = 0; i < sz; i++) { ans.set(i + s, p[i]); }
        return ans;
    }
    static polynomial_base quot_by_pow(const polynomial_base& p, const usize s)
    {
        const usize N = p.size();
        if (N <= s) { return polynomial_base(); }
        polynomial_base ans;
        for (usize i = 0; i < N - s; i++) { ans.set(i, p[i + s]); }
        return ans;
    }
    static polynomial_base rem_by_pow(const polynomial_base& p, const usize k) { return p.size() <= k ? p : polynomial_base(p().begin(), p().begin() + k); }
    static polynomial_base derivative_of(const polynomial_base& p)
    {
        if (p.size() <= 1) { return polynomial_base{0}; }
        polynomial_base ans;
        for (usize i = 1; i < p.size(); i++) { ans.set(i - 1, p[i] * i); }
        return ans;
    }
    static polynomial_base primitive_of(const polynomial_base& p)
    {
        polynomial_base ans;
        for (usize i = 1; i <= p.size(); i++) { ans.set(i, p[i - 1] / i); }
        return ans;
    }
    static polynomial_base inv(const polynomial_base& p, const usize k)
    {
        polynomial_base q{value_type(1) / p[0]};
        for (usize i = 1; i < k; i *= 2) { q = rem_by_pow(q * (2 - rem_by_pow(p, 2 * i) * q), 2 * i); }
        return rem_by_pow(q, k);
    }
    static polynomial_base log(const polynomial_base& p, const usize k) { return k == 0 ? polynomial_base{} : primitive_of(rem_by_pow(derivative_of(rem_by_pow(p, k)) * inv(p, k - 1), k - 1)); }
    static polynomial_base exp(const polynomial_base& p, const usize k)
    {
        polynomial_base q{value_type(1)};
        for (usize i = 1; i < k; i *= 2) { q = rem_by_pow(q * (rem_by_pow(p, 2 * i) + 1 - log(q, 2 * i)), 2 * i); }
        return rem_by_pow(q, k);
    }
    static polynomial_base sqrt(const polynomial_base& p_, usize k)
    {
        if (p_.size() == 0) { return polynomial_base{}; }
        usize z = 0;
        for (; z < p_.size(); z++) {
            if (p_[z] != 0) { break; }
        }
        if (z % 2 == 1 or z >= 2 * k) { return polynomial_base{}; }
        const auto p = quot_by_pow(p_, z);
        k -= z / 2;
        const value_type rt = modsqrt(p[0]);
        if (rt * rt != p[0]) { return polynomial_base{}; }
        polynomial_base q{rt};
        for (usize i = 1; i < k; i *= 2) { q = rem_by_pow(q + rem_by_pow(p, 2 * i) * inv(q, 2 * i), 2 * i) / 2; }
        return prod_by_pow(rem_by_pow(q, k), z / 2);
    }
    template<typename Int>
    static polynomial_base pow(const polynomial_base& p, const Int k, const usize s)
    {
        if (k == 0) { return polynomial_base(1); }
        if (k % 2 == 1) {
            return rem_by_pow(pow(p, k - 1, s) * p, s);
        } else {
            const auto q = pow(p, k / 2, s);
            return rem_by_pow(q * q, s);
        }
    }
    template<typename Int>
    static polynomial_base rem_of_pow(const polynomial_base& p, const Int k)
    {
        const usize B = p.size() * 2 - 1;
        const auto q  = p.pseudo_inv(B);
        polynomial_base ans{1};
        const usize D = log2p1<usize>(k);
        for (usize i = 0; i < D; i++) {
            if (k & (static_cast<Int>(1) << (D - i - 1))) { ans = (ans.multiply_power(1)).rem(p, q, B); }
            if (D - i - 1) { ans = (ans * ans).rem(p, q, B); }
        }
        return ans;
    }
    usize size() const { return v.size(); }
    friend std::ostream& operator<<(std::ostream& os, const polynomial_base& p)
    {
        if (p.size() == 0) { return os << "0"; }
        for (usize i = 0; i < p.size(); i++) { os << (i != 0 ? "+" : "") << p[i] << (i != 0 ? i == 1 ? "X" : "X^" + std::to_string(i) : ""); }
        return os;
    }

private:
    static std::vector<value_type> naive_convolute(const std::vector<value_type>& a, const std::vector<value_type>& b)
    {
        std::vector<value_type> ans(a.size() + b.size() - 1, 0);
        for (usize i = 0; i < a.size(); i++) {
            for (usize j = 0; j < b.size(); j++) { ans[i + j] += a[i] * b[j]; }
        }
        return ans;
    }
    static polynomial_base naive_multiply(const polynomial_base& p, const polynomial_base& q)
    {
        if (p.size() == 0 or q.size() == 0) { return polynomial_base{}; }
        const auto v = naive_convolute(p(), q());
        return polynomial_base(v.begin(), v.end());
    }
    template<typename Poly = polynomial_base>
    static std::enable_if_t<root == 0, Poly> fft_multiply(const polynomial_base& p, const polynomial_base& q)
    {
        if (p.size() == 0 or q.size() == 0) { return polynomial_base{}; }
        const auto v = fft<double>::convolute<mod, dynamic, fft_division>(p(), q());
        return polynomial_base(v.begin(), v.end());
    }
    template<typename Poly = polynomial_base>
    static std::enable_if_t<root != 0, Poly> fft_multiply(const polynomial_base& p, const polynomial_base& q)
    {
        if (p.size() == 0 or q.size() == 0) { return polynomial_base{}; }
        const auto v = ntt<mod, root>::convolute(p(), q());
        return polynomial_base(v.begin(), v.end());
    }
    static polynomial_base rev(const polynomial_base& p, const usize l)
    {
        std::vector<value_type> ans = p();
        ans.resize(l), std::reverse(ans.begin(), ans.end());
        return polynomial_base(ans.begin(), ans.end());
    }
    static polynomial_base div(const polynomial_base& p, const polynomial_base& q)
    {
        assert(q.size() > 0);
        if (p.size() < q.size()) { return polynomial_base(); }
        const usize N = p.size();
        const auto iq = pseudoInv(q, N);
        return div_by_pow(p * iq, N - 1);
    }
    static polynomial_base rem(const polynomial_base& p, const polynomial_base& q) { return p - div(p, q) * q; }
    static polynomial_base rem(const polynomial_base& p, const polynomial_base& q, const polynomial_base& iq, const usize B) { return p - q * (quot_by_pow(p * iq, B - 1)); }
    void shrink()
    {
        for (; not v.empty() and v.back() == 0; v.pop_back()) {}
    }
    static polynomial_base pseudo_inv(const polynomial_base& p, const usize B)
    {
        const usize N = p.size();
        return rev(inv(rev(p, N), B + 2 > N ? clog(B - N + 2) : 0), B + 1 - N);
    }
    std::vector<value_type> v;
};
template<uint mod, uint fft_division = 2>
using polynomial = polynomial_base<mod, 0, false, fft_division>;
template<uint mod, uint fft_division = 2>
using dynamic_polynomial = polynomial_base<mod, 0, true, fft_division>;
template<uint mod = 998244353, uint root = 3>
using ntt_polynomial = polynomial_base<mod, root, false, 0>;
int main()
{
    const auto [N, d, K] = in_t<int, int, int>();
    using poly           = polynomial<mod>;
    std::vector<modint<mod>> v(d + 1, 0);
    for (int i = 1; i <= d; i++) { v[i] = 1; }
    poly f(v.begin(), v.end());
    SHOW(f);
    f = poly::pow(f, N, K + 1);
    outln(f[K]);

    return 0;
}