#include<iostream>
#include<string>
#include<vector>
#include<algorithm>
#include<numeric>
#include<cmath>
#include<utility>
#include<tuple>
#include<array>
#include<cstdint>
#include<cstdio>
#include<iomanip>
#include<map>
#include<set>
#include<unordered_map>
#include<unordered_set>
#include<queue>
#include<stack>
#include<deque>
#include<bitset>
#include<cctype>
#include<chrono>
#include<random>
#include<cassert>
#include<cstddef>
#include<iterator>
#include<string_view>
#include<type_traits>
#include<functional>

#ifdef LOCAL
#  include "debug_print.hpp"
#  define debug(...) debug_print::multi_print(#__VA_ARGS__, __VA_ARGS__)
#else
#  define debug(...) (static_cast<void>(0))
#endif

using namespace std;

template<typename T, typename U>
ostream &operator<<(ostream &os, const pair<T, U> &p) {
    os << p.first << " " << p.second;
    return os;
}
template<typename T, typename U>
istream &operator>>(istream &is, pair<T, U> &p) {
    is >> p.first >> p.second;
    return is;
}
template<typename T>
ostream &operator<<(ostream &os, const vector<T> &v) {
    int s = (int)v.size();
    for (int i = 0; i < s; i++) os << (i ? " " : "") << v[i];
    return os;
}
template<typename T>
istream &operator>>(istream &is, vector<T> &v) {
    for (auto &x : v) is >> x;
    return is;
}
void in() {}
template<typename T, class... U>
void in(T &t, U &...u) {
    cin >> t;
    in(u...);
}
void out() { cout << "\n"; }
template<typename T, class... U, char sep = ' '>
void out(const T &t, const U &...u) {
    cout << t;
    if (sizeof...(u)) cout << sep;
    out(u...);
}
void outr() {}
template<typename T, class... U, char sep = ' '>
void outr(const T &t, const U &...u) {
    cout << t;
    outr(u...);
}

using ll = long long;
using D = double;
using LD = long double;
using P = pair<ll, ll>;
using u8 = uint8_t;
using u16 = uint16_t;
using u32 = uint32_t;
using u64 = uint64_t;
using i128 = __int128;
using u128 = unsigned __int128;
using vi = vector<ll>;
template <class T> using vc = vector<T>;
template <class T> using vvc = vector<vc<T>>;
template <class T> using vvvc = vector<vvc<T>>;
template <class T> using vvvvc = vector<vvvc<T>>;
template <class T> using vvvvvc = vector<vvvvc<T>>;
#define vv(type, name, h, ...) \
  vector<vector<type>> name(h, vector<type>(__VA_ARGS__))
#define vvv(type, name, h, w, ...)   \
  vector<vector<vector<type>>> name( \
      h, vector<vector<type>>(w, vector<type>(__VA_ARGS__)))
#define vvvv(type, name, a, b, c, ...)       \
  vector<vector<vector<vector<type>>>> name( \
      a, vector<vector<vector<type>>>(       \
             b, vector<vector<type>>(c, vector<type>(__VA_ARGS__))))
template<typename T> using PQ = priority_queue<T,vector<T>>;
template<typename T> using minPQ = priority_queue<T, vector<T>, greater<T>>;

#define rep1(a)          for(ll i = 0; i < a; i++)
#define rep2(i, a)       for(ll i = 0; i < a; i++)
#define rep3(i, a, b)    for(ll i = a; i < b; i++)
#define rep4(i, a, b, c) for(ll i = a; i < b; i += c)
#define overload4(a, b, c, d, e, ...) e
#define rep(...) overload4(__VA_ARGS__, rep4, rep3, rep2, rep1)(__VA_ARGS__)
#define rrep1(a)          for(ll i = (a)-1; i >= 0; i--)
#define rrep2(i, a)       for(ll i = (a)-1; i >= 0; i--)
#define rrep3(i, a, b)    for(ll i = (b)-1; i >= a; i--)
#define rrep4(i, a, b, c) for(ll i = (b)-1; i >= a; i -= c)
#define rrep(...) overload4(__VA_ARGS__, rrep4, rrep3, rrep2, rrep1)(__VA_ARGS__)
#define for_subset(t, s) for (ll t = (s); t >= 0; t = (t == 0 ? -1 : (t - 1) & (s)))
#define ALL(v) v.begin(), v.end()
#define RALL(v) v.rbegin(), v.rend()
#define UNIQUE(v) v.erase( unique(v.begin(), v.end()), v.end() )
#define SZ(v) ll(v.size())
#define MIN(v) *min_element(ALL(v))
#define MAX(v) *max_element(ALL(v))
#define LB(c, x) distance((c).begin(), lower_bound(ALL(c), (x)))
#define UB(c, x) distance((c).begin(), upper_bound(ALL(c), (x)))
template <typename T, typename U>
T SUM(const vector<U> &v) {
    T res = 0;
    for(auto &&a : v) res += a;
    return res;
}
template <typename T>
vector<pair<T,int>> RLE(const vector<T> &v) {
    if (v.empty()) return {};
    T cur = v.front();
    int cnt = 1;
    vector<pair<T,int>> res;
    for (int i = 1; i < (int)v.size(); i++) {
        if (cur == v[i]) cnt++;
        else {
            res.emplace_back(cur, cnt);
            cnt = 1; cur = v[i];
        }
    }
    res.emplace_back(cur, cnt);
    return res;
}
template<class T, class S>
inline bool chmax(T &a, const S &b) { return (a < b ? a = b, true : false); }
template<class T, class S>
inline bool chmin(T &a, const S &b) { return (a > b ? a = b, true : false); }
void YESNO(bool flag) { out(flag ? "YES" : "NO"); }
void yesno(bool flag) { out(flag ? "Yes" : "No"); }

int popcnt(int x) { return __builtin_popcount(x); }
int popcnt(u32 x) { return __builtin_popcount(x); }
int popcnt(ll x) { return __builtin_popcountll(x); }
int popcnt(u64 x) { return __builtin_popcountll(x); }
int popcnt_sgn(int x) { return (__builtin_parity(x) & 1 ? -1 : 1); }
int popcnt_sgn(u32 x) { return (__builtin_parity(x) & 1 ? -1 : 1); }
int popcnt_sgn(ll x) { return (__builtin_parity(x) & 1 ? -1 : 1); }
int popcnt_sgn(u64 x) { return (__builtin_parity(x) & 1 ? -1 : 1); }
int highbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int highbit(u32 x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int highbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
int highbit(u64 x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
int lowbit(int x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(u32 x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(ll x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }
int lowbit(u64 x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }

template <typename T>
T get_bit(T x, int k) { return x >> k & 1; }
template <typename T>
T set_bit(T x, int k) { return x | T(1) << k; }
template <typename T>
T reset_bit(T x, int k) { return x & ~(T(1) << k); }
template <typename T>
T flip_bit(T x, int k) { return x ^ T(1) << k; }

template <typename T>
T popf(deque<T> &que) { T a = que.front(); que.pop_front(); return a; }
template <typename T>
T popb(deque<T> &que) { T a = que.back(); que.pop_back(); return a; }
template <typename T>
T pop(queue<T> &que) { T a = que.front(); que.pop(); return a; }
template <typename T>
T pop(PQ<T> &que) { T a = que.top(); que.pop(); return a; }
template <typename T>
T pop(minPQ<T> &que) { T a = que.top(); que.pop(); return a; }

template <typename F>
ll binary_search(F check, ll ok, ll ng, bool check_ok = true) {
    if (check_ok) assert(check(ok));
    while (abs(ok -  ng) > 1) {
        ll mid = (ok + ng) / 2;
        (check(mid) ? ok : ng) = mid;
    }
    return ok;
}
template <typename F>
ll binary_search_real(F check, double ok, double ng, int iter = 60) {
    for (int _ = 0; _ < iter; _++) {
        double mid = (ok + ng) / 2;
        (check(mid) ? ok : ng) = mid;
    }
    return (ok + ng) / 2;
}

// max x s.t. b*x <= a
ll div_floor(ll a, ll b) {
    assert(b != 0);
    if (b < 0) a = -a, b = -b;
    return a / b - (a % b < 0);
}
// max x s.t. b*x < a
ll div_under(ll a, ll b) {
    assert(b != 0);
    if (b < 0) a = -a, b = -b;
    return a / b - (a % b <= 0);
}
// min x s.t. b*x >= a
ll div_ceil(ll a, ll b) {
    assert(b != 0);
    if (b < 0) a = -a, b = -b;
    return a / b + (a % b > 0);
}
// min x s.t. b*x > a
ll div_over(ll a, ll b) {
    assert(b != 0);
    if (b < 0) a = -a, b = -b;
    return a / b + (a % b >= 0);
}
// x = a mod b (b > 0), 0 <= x < b
ll modulo(ll a, ll b) {
    assert(b > 0);
    ll c = a % b;
    return c < 0 ? c + b : c;
}
// (q,r) s.t. a = b*q + r, 0 <= r < b (b > 0)
// div_floor(a,b), modulo(a,b)
pair<ll,ll> divmod(ll a, ll b) {
    ll q = div_floor(a,b);
    return {q, a - b*q};
}

template <uint32_t mod>
struct LazyMontgomeryModInt {
    using mint = LazyMontgomeryModInt;
    using i32 = int32_t;
    using u32 = uint32_t;
    using u64 = uint64_t;

    static constexpr u32 get_r() {
        u32 ret = mod;
        for (i32 i = 0; i < 4; ++i) ret *= 2 - mod * ret;
        return ret;
    }

    static constexpr u32 r = get_r();
    static constexpr u32 n2 = -u64(mod) % mod;
    static_assert(mod < (1 << 30), "invalid, mod >= 2 ^ 30");
    static_assert((mod & 1) == 1, "invalid, mod % 2 == 0");
    static_assert(r * mod == 1, "this code has bugs.");

    u32 a;

    constexpr LazyMontgomeryModInt() : a(0) {}
    constexpr LazyMontgomeryModInt(const int64_t &b)
    : a(reduce(u64(b % mod + mod) * n2)){};

    static constexpr u32 reduce(const u64 &b) {
        return (b + u64(u32(b) * u32(-r)) * mod) >> 32;
    }

    constexpr mint &operator+=(const mint &b) {
        if (i32(a += b.a - 2 * mod) < 0) a += 2 * mod;
        return *this;
    }

    constexpr mint &operator-=(const mint &b) {
        if (i32(a -= b.a) < 0) a += 2 * mod;
        return *this;
    }

    constexpr mint &operator*=(const mint &b) {
        a = reduce(u64(a) * b.a);
        return *this;
    }

    constexpr mint &operator/=(const mint &b) {
        *this *= b.inverse();
        return *this;
    }

    constexpr mint operator+(const mint &b) const { return mint(*this) += b; }
    constexpr mint operator-(const mint &b) const { return mint(*this) -= b; }
    constexpr mint operator*(const mint &b) const { return mint(*this) *= b; }
    constexpr mint operator/(const mint &b) const { return mint(*this) /= b; }
    constexpr bool operator==(const mint &b) const {
        return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a);
    }
    constexpr bool operator!=(const mint &b) const {
        return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a);
    }
    constexpr mint operator-() const { return mint() - mint(*this); }
    constexpr mint operator+() const { return mint(*this); }

    constexpr mint pow(u64 n) const {
        mint ret(1), mul(*this);
        while (n > 0) {
            if (n & 1) ret *= mul;
            mul *= mul;
            n >>= 1;
        }
        return ret;
    }

    constexpr mint inverse() const {
        int x = get(), y = mod, u = 1, v = 0, t = 0, tmp = 0;
        while (y > 0) {
            t = x / y;
            x -= t * y, u -= t * v;
            tmp = x, x = y, y = tmp;
            tmp = u, u = v, v = tmp;
        }
        return mint{u};
    }

    friend ostream &operator<<(ostream &os, const mint &b) {
        return os << b.get();
    }

    friend istream &operator>>(istream &is, mint &b) {
        int64_t t;
        is >> t;
        b = LazyMontgomeryModInt<mod>(t);
        return (is);
    }

    constexpr u32 get() const {
        u32 ret = reduce(a);
        return ret >= mod ? ret - mod : ret;
    }

    static constexpr u32 get_mod() { return mod; }
};

template<typename T> struct Binomial {
    vector<T> fact_, inv_, finv_;
    constexpr Binomial() {}
    constexpr Binomial(int n) noexcept : fact_(n, 1), inv_(n, 1), finv_(n, 1) {
        init(n);
    }
    constexpr void init(int n) noexcept {
        constexpr int mod = T::get_mod();
        fact_.assign(n, 1), inv_.assign(n, 1), finv_.assign(n, 1);
        for(int i = 2; i < n; i++){
            fact_[i] = fact_[i-1] * i;
            inv_[i] = -inv_[mod%i] * (mod/i);
            finv_[i] = finv_[i-1] * inv_[i];
        }
    }
    constexpr T com(int n, int k) const noexcept {
        if (n < k || n < 0 || k < 0) return 0;
        return fact_[n] * finv_[k] * finv_[n-k];
    }
    constexpr T perm(int n, int k) const noexcept {
        if (n < k || n < 0 || k < 0) return 0;
        return fact_[n] * finv_[n-k];
    }
    constexpr T fact(int n) const noexcept {
        if (n < 0) return 0;
        return fact_[n];
    }
    constexpr T inv(int n) const noexcept {
        if (n < 0) return 0;
        return inv_[n];
    }
    constexpr T finv(int n) const noexcept {
        if (n < 0) return 0;
        return finv_[n];
    }
    constexpr T com_naive(int n, int k) const noexcept {
        if (n < 0 || k < 0 || n < k) return 0;
        T res = T(1);
        k = min(k, n-k);
        for (int i = 1; i <= k; i++)res *= (n--) * inv(i);
        return res;
    }
    template <typename I>
    constexpr T multi(const vector<I> &v) const noexcept {
        static_assert(is_integral<I>::value);
        I n = 0;
        for (auto& x : v) {
            if (x < 0) return 0;
            n += x;
        }
        T res = fact(n);
        for (auto &x : v) res *= finv(x);
        return res;
    }
    // [x^k] (1-x)^{-n} = com(n+k-1, k)
    constexpr T neg(int n, int k) const noexcept {
        if (n < 0 || k < 0) return 0;
        return k == 0 ? 1 : com(n+k-1, k);
    }
};


template <typename mint>
struct NTT {
    static constexpr uint32_t get_pr() {
        uint32_t _mod = mint::get_mod();
        using u64 = uint64_t;
        u64 ds[32] = {};
        int idx = 0;
        u64 m = _mod - 1;
        for (u64 i = 2; i * i <= m; ++i) {
            if (m % i == 0) {
                ds[idx++] = i;
                while (m % i == 0) m /= i;
            }
        }
        if (m != 1) ds[idx++] = m;

        uint32_t _pr = 2;
        while (1) {
            int flg = 1;
            for (int i = 0; i < idx; ++i) {
                u64 a = _pr, b = (_mod - 1) / ds[i], r = 1;
                while (b) {
                    if (b & 1) r = r * a % _mod;
                    a = a * a % _mod;
                    b >>= 1;
                }
                if (r == 1) {
                    flg = 0;
                    break;
                }
            }
            if (flg == 1) break;
            ++_pr;
        }
        return _pr;
    };

    static constexpr uint32_t mod = mint::get_mod();
    static constexpr uint32_t pr = get_pr();
    static constexpr int level = __builtin_ctzll(mod - 1);
    mint dw[level], dy[level];

    void setwy(int k) {
        mint w[level], y[level];
        w[k - 1] = mint(pr).pow((mod - 1) / (1 << k));
        y[k - 1] = w[k - 1].inverse();
        for (int i = k - 2; i > 0; --i)
            w[i] = w[i + 1] * w[i + 1], y[i] = y[i + 1] * y[i + 1];
        dw[1] = w[1], dy[1] = y[1], dw[2] = w[2], dy[2] = y[2];
        for (int i = 3; i < k; ++i) {
            dw[i] = dw[i - 1] * y[i - 2] * w[i];
            dy[i] = dy[i - 1] * w[i - 2] * y[i];
        }
    }

    NTT() { setwy(level); }

    void fft4(vector<mint> &a, int k) {
        if ((int)a.size() <= 1) return;
        if (k == 1) {
            mint a1 = a[1];
            a[1] = a[0] - a[1];
            a[0] = a[0] + a1;
            return;
        }
        if (k & 1) {
            int v = 1 << (k - 1);
            for (int j = 0; j < v; ++j) {
                mint ajv = a[j + v];
                a[j + v] = a[j] - ajv;
                a[j] += ajv;
            }
        }
        int u = 1 << (2 + (k & 1));
        int v = 1 << (k - 2 - (k & 1));
        mint one = mint(1);
        mint imag = dw[1];
        while (v) {
            // jh = 0
            {
                int j0 = 0;
                int j1 = v;
                int j2 = j1 + v;
                int j3 = j2 + v;
                for (; j0 < v; ++j0, ++j1, ++j2, ++j3) {
                    mint t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3];
                    mint t0p2 = t0 + t2, t1p3 = t1 + t3;
                    mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag;
                    a[j0] = t0p2 + t1p3, a[j1] = t0p2 - t1p3;
                    a[j2] = t0m2 + t1m3, a[j3] = t0m2 - t1m3;
                }
            }
            // jh >= 1
            mint ww = one, xx = one * dw[2], wx = one;
            for (int jh = 4; jh < u;) {
                ww = xx * xx, wx = ww * xx;
                int j0 = jh * v;
                int je = j0 + v;
                int j2 = je + v;
                for (; j0 < je; ++j0, ++j2) {
                    mint t0 = a[j0], t1 = a[j0 + v] * xx, t2 = a[j2] * ww,
                    t3 = a[j2 + v] * wx;
                    mint t0p2 = t0 + t2, t1p3 = t1 + t3;
                    mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag;
                    a[j0] = t0p2 + t1p3, a[j0 + v] = t0p2 - t1p3;
                    a[j2] = t0m2 + t1m3, a[j2 + v] = t0m2 - t1m3;
                }
                xx *= dw[__builtin_ctzll((jh += 4))];
            }
            u <<= 2;
            v >>= 2;
        }
    }

    void ifft4(vector<mint> &a, int k) {
        if ((int)a.size() <= 1) return;
        if (k == 1) {
            mint a1 = a[1];
            a[1] = a[0] - a[1];
            a[0] = a[0] + a1;
            return;
        }
        int u = 1 << (k - 2);
        int v = 1;
        mint one = mint(1);
        mint imag = dy[1];
        while (u) {
            // jh = 0
            {
                int j0 = 0;
                int j1 = v;
                int j2 = v + v;
                int j3 = j2 + v;
                for (; j0 < v; ++j0, ++j1, ++j2, ++j3) {
                    mint t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3];
                    mint t0p1 = t0 + t1, t2p3 = t2 + t3;
                    mint t0m1 = t0 - t1, t2m3 = (t2 - t3) * imag;
                    a[j0] = t0p1 + t2p3, a[j2] = t0p1 - t2p3;
                    a[j1] = t0m1 + t2m3, a[j3] = t0m1 - t2m3;
                }
            }
            // jh >= 1
            mint ww = one, xx = one * dy[2], yy = one;
            u <<= 2;
            for (int jh = 4; jh < u;) {
                ww = xx * xx, yy = xx * imag;
                int j0 = jh * v;
                int je = j0 + v;
                int j2 = je + v;
                for (; j0 < je; ++j0, ++j2) {
                    mint t0 = a[j0], t1 = a[j0 + v], t2 = a[j2], t3 = a[j2 + v];
                    mint t0p1 = t0 + t1, t2p3 = t2 + t3;
                    mint t0m1 = (t0 - t1) * xx, t2m3 = (t2 - t3) * yy;
                    a[j0] = t0p1 + t2p3, a[j2] = (t0p1 - t2p3) * ww;
                    a[j0 + v] = t0m1 + t2m3, a[j2 + v] = (t0m1 - t2m3) * ww;
                }
                xx *= dy[__builtin_ctzll(jh += 4)];
            }
            u >>= 4;
            v <<= 2;
        }
        if (k & 1) {
            u = 1 << (k - 1);
            for (int j = 0; j < u; ++j) {
                mint ajv = a[j] - a[j + u];
                a[j] += a[j + u];
                a[j + u] = ajv;
            }
        }
    }

    void ntt(vector<mint> &a) {
        if ((int)a.size() <= 1) return;
        fft4(a, __builtin_ctz(a.size()));
    }

    void intt(vector<mint> &a) {
        if ((int)a.size() <= 1) return;
        ifft4(a, __builtin_ctz(a.size()));
        mint iv = mint(a.size()).inverse();
        for (auto &x : a) x *= iv;
    }

    vector<mint> multiply(const vector<mint> &a, const vector<mint> &b) {
        int l = a.size() + b.size() - 1;
        if (min<int>(a.size(), b.size()) <= 40) {
            vector<mint> s(l);
            for (int i = 0; i < (int)a.size(); ++i)
                for (int j = 0; j < (int)b.size(); ++j) s[i + j] += a[i] * b[j];
            return s;
        }
        int k = 2, M = 4;
        while (M < l) M <<= 1, ++k;
        setwy(k);
        vector<mint> s(M);
        for (int i = 0; i < (int)a.size(); ++i) s[i] = a[i];
        fft4(s, k);
        if (a.size() == b.size() && a == b) {
            for (int i = 0; i < M; ++i) s[i] *= s[i];
        } else {
            vector<mint> t(M);
            for (int i = 0; i < (int)b.size(); ++i) t[i] = b[i];
            fft4(t, k);
            for (int i = 0; i < M; ++i) s[i] *= t[i];
        }
        ifft4(s, k);
        s.resize(l);
        mint invm = mint(M).inverse();
        for (int i = 0; i < l; ++i) s[i] *= invm;
        return s;
    }

    void ntt_doubling(vector<mint> &a) {
        int M = (int)a.size();
        auto b = a;
        intt(b);
        mint r = 1, zeta = mint(pr).pow((mint::get_mod() - 1) / (M << 1));
        for (int i = 0; i < M; i++) b[i] *= r, r *= zeta;
        ntt(b);
        copy(begin(b), end(b), back_inserter(a));
    }
};

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

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

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

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

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

    FPS &operator*=(const mint &v) {
        for (int k = 0; k < (int)this->size(); k++) (*this)[k] *= v;
        return *this;
    }

    FPS &operator/=(const FPS &r) {
        if (this->size() < r.size()) {
            this->clear();
            return *this;
        }
        int n = this->size() - r.size() + 1;
        if ((int)r.size() <= 64) {
            FPS f(*this), g(r);
            g.shrink();
            mint coeff = g.back().inverse();
            for (auto &x : g) x *= coeff;
            int deg = (int)f.size() - (int)g.size() + 1;
            int gs = g.size();
            FPS quo(deg);
            for (int i = deg - 1; i >= 0; i--) {
                quo[i] = f[i + gs - 1];
                for (int j = 0; j < gs; j++) f[i + j] -= quo[i] * g[j];
            }
            *this = quo * coeff;
            this->resize(n, mint(0));
            return *this;
        }
        return *this = ((*this).rev().pre(n) * r.rev().inv(n)).pre(n).rev();
    }

    FPS &operator%=(const FPS &r) {
        *this -= *this / r * r;
        shrink();
        return *this;
    }

    FPS operator+(const FPS &r) const { return FPS(*this) += r; }
    FPS operator+(const mint &v) const { return FPS(*this) += v; }
    FPS operator-(const FPS &r) const { return FPS(*this) -= r; }
    FPS operator-(const mint &v) const { return FPS(*this) -= v; }
    FPS operator*(const FPS &r) const { return FPS(*this) *= r; }
    FPS operator*(const mint &v) const { return FPS(*this) *= v; }
    FPS operator/(const FPS &r) const { return FPS(*this) /= r; }
    FPS operator%(const FPS &r) const { return FPS(*this) %= r; }
    FPS operator-() const {
        FPS ret(this->size());
        for (int i = 0; i < (int)this->size(); i++) ret[i] = -(*this)[i];
        return ret;
    }

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

    FPS rev() const {
        FPS ret(*this);
        reverse(begin(ret), end(ret));
        return ret;
    }

    FPS dot(FPS r) const {
        FPS ret(min(this->size(), r.size()));
        for (int i = 0; i < (int)ret.size(); i++) ret[i] = (*this)[i] * r[i];
        return ret;
    }

    // 前 sz 項を取ってくる。sz に足りない項は 0 埋めする
    FPS pre(int sz) const {
        FPS ret(begin(*this), begin(*this) + min((int)this->size(), sz));
        if ((int)ret.size() < sz) ret.resize(sz);
        return ret;
    }

    FPS operator>>(int sz) const {
        if ((int)this->size() <= sz) return {};
        FPS ret(*this);
        ret.erase(ret.begin(), ret.begin() + sz);
        return ret;
    }

    FPS operator<<(int sz) const {
        FPS ret(*this);
        ret.insert(ret.begin(), sz, mint(0));
        return ret;
    }

    FPS diff() const {
        const int n = (int)this->size();
        FPS ret(max(0, n - 1));
        mint one(1), coeff(1);
        for (int i = 1; i < n; i++) {
            ret[i - 1] = (*this)[i] * coeff;
            coeff += one;
        }
        return ret;
    }

    FPS integral() const {
        const int n = (int)this->size();
        FPS ret(n + 1);
        ret[0] = mint(0);
        if (n > 0) ret[1] = mint(1);
        auto mod = mint::get_mod();
        for (int i = 2; i <= n; i++) ret[i] = (-ret[mod % i]) * (mod / i);
        for (int i = 0; i < n; i++) ret[i + 1] *= (*this)[i];
        return ret;
    }

    mint eval(mint x) const {
        mint r = 0, w = 1;
        for (auto &v : *this) r += w * v, w *= x;
        return r;
    }

    FPS log(int deg = -1) const {
        assert(!(*this).empty() && (*this)[0] == mint(1));
        if (deg == -1) deg = (int)this->size();
        return (this->diff() * this->inv(deg)).pre(deg - 1).integral();
    }

    FPS pow(int64_t k, int deg = -1) const {
        const int n = (int)this->size();
        if (deg == -1) deg = n;
        if (k == 0) {
            FPS ret(deg);
            if (deg) ret[0] = 1;
            return ret;
        }
        for (int i = 0; i < n; i++) {
            if ((*this)[i] != mint(0)) {
                mint rev = mint(1) / (*this)[i];
                FPS ret = (((*this * rev) >> i).log(deg) * k).exp(deg);
                ret *= (*this)[i].pow(k);
                ret = (ret << (i * k)).pre(deg);
                if ((int)ret.size() < deg) ret.resize(deg, mint(0));
                return ret;
            }
            if (__int128_t(i + 1) * k >= deg) return FPS(deg, mint(0));
        }
        return FPS(deg, mint(0));
    }

    static void *ntt_ptr;
    static void set_fft();
    FPS &operator*=(const FPS &r);
    void ntt();
    void intt();
    void ntt_doubling();
    static int ntt_pr();
    FPS inv(int deg = -1) const;
    FPS exp(int deg = -1) const;
};
template <typename mint>
void *FormalPowerSeries<mint>::ntt_ptr = nullptr;

/**
 * @brief 多項式/形式的冪級数ライブラリ
 * @docs docs/fps/formal-power-series.md
 */

template <typename mint>
void FormalPowerSeries<mint>::set_fft() {
    if (!ntt_ptr) ntt_ptr = new NTT<mint>;
}

template <typename mint>
FormalPowerSeries<mint>& FormalPowerSeries<mint>::operator*=(
    const FormalPowerSeries<mint>& r) {
    if (this->empty() || r.empty()) {
        this->clear();
        return *this;
    }
    set_fft();
    auto ret = static_cast<NTT<mint>*>(ntt_ptr)->multiply(*this, r);
    return *this = FormalPowerSeries<mint>(ret.begin(), ret.end());
}

template <typename mint>
void FormalPowerSeries<mint>::ntt() {
    set_fft();
    static_cast<NTT<mint>*>(ntt_ptr)->ntt(*this);
}

template <typename mint>
void FormalPowerSeries<mint>::intt() {
    set_fft();
    static_cast<NTT<mint>*>(ntt_ptr)->intt(*this);
}

template <typename mint>
void FormalPowerSeries<mint>::ntt_doubling() {
    set_fft();
    static_cast<NTT<mint>*>(ntt_ptr)->ntt_doubling(*this);
}

template <typename mint>
int FormalPowerSeries<mint>::ntt_pr() {
    set_fft();
    return static_cast<NTT<mint>*>(ntt_ptr)->pr;
}

template <typename mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::inv(int deg) const {
    assert((*this)[0] != mint(0));
    if (deg == -1) deg = (int)this->size();
    FormalPowerSeries<mint> res(deg);
    res[0] = {mint(1) / (*this)[0]};
    for (int d = 1; d < deg; d <<= 1) {
        FormalPowerSeries<mint> f(2 * d), g(2 * d);
        for (int j = 0; j < min((int)this->size(), 2 * d); j++) f[j] = (*this)[j];
        for (int j = 0; j < d; j++) g[j] = res[j];
        f.ntt();
        g.ntt();
        for (int j = 0; j < 2 * d; j++) f[j] *= g[j];
        f.intt();
        for (int j = 0; j < d; j++) f[j] = 0;
        f.ntt();
        for (int j = 0; j < 2 * d; j++) f[j] *= g[j];
        f.intt();
        for (int j = d; j < min(2 * d, deg); j++) res[j] = -f[j];
    }
    return res.pre(deg);
}

template <typename mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::exp(int deg) const {
    using fps = FormalPowerSeries<mint>;
    assert((*this).size() == 0 || (*this)[0] == mint(0));
    if (deg == -1) deg = this->size();

    fps inv;
    inv.reserve(deg + 1);
    inv.push_back(mint(0));
    inv.push_back(mint(1));

    auto inplace_integral = [&](fps& F) -> void {
        const int n = (int)F.size();
        auto mod = mint::get_mod();
        while ((int)inv.size() <= n) {
            int i = inv.size();
            inv.push_back((-inv[mod % i]) * (mod / i));
        }
        F.insert(begin(F), mint(0));
        for (int i = 1; i <= n; i++) F[i] *= inv[i];
    };

    auto inplace_diff = [](fps& F) -> void {
        if (F.empty()) return;
        F.erase(begin(F));
        mint coeff = 1, one = 1;
        for (int i = 0; i < (int)F.size(); i++) {
            F[i] *= coeff;
            coeff += one;
        }
    };

    fps b{1, 1 < (int)this->size() ? (*this)[1] : 0}, c{1}, z1, z2{1, 1};
    for (int m = 2; m < deg; m *= 2) {
        auto y = b;
        y.resize(2 * m);
        y.ntt();
        z1 = z2;
        fps z(m);
        for (int i = 0; i < m; ++i) z[i] = y[i] * z1[i];
        z.intt();
        fill(begin(z), begin(z) + m / 2, mint(0));
        z.ntt();
        for (int i = 0; i < m; ++i) z[i] *= -z1[i];
        z.intt();
        c.insert(end(c), begin(z) + m / 2, end(z));
        z2 = c;
        z2.resize(2 * m);
        z2.ntt();
        fps x(begin(*this), begin(*this) + min<int>(this->size(), m));
        x.resize(m);
        inplace_diff(x);
        x.push_back(mint(0));
        x.ntt();
        for (int i = 0; i < m; ++i) x[i] *= y[i];
        x.intt();
        x -= b.diff();
        x.resize(2 * m);
        for (int i = 0; i < m - 1; ++i) x[m + i] = x[i], x[i] = mint(0);
        x.ntt();
        for (int i = 0; i < 2 * m; ++i) x[i] *= z2[i];
        x.intt();
        x.pop_back();
        inplace_integral(x);
        for (int i = m; i < min<int>(this->size(), 2 * m); ++i) x[i] += (*this)[i];
        fill(begin(x), begin(x) + m, mint(0));
        x.ntt();
        for (int i = 0; i < 2 * m; ++i) x[i] *= y[i];
        x.intt();
        b.insert(end(b), begin(x) + m, end(x));
    }
    return fps{begin(b), begin(b) + deg};
}

/**
 * @brief NTT mod用FPSライブラリ
 * @docs docs/fps/ntt-friendly-fps.md
 */

const int mod = 998244353;
using mint = LazyMontgomeryModInt<mod>;
using poly = FormalPowerSeries<mint>;
Binomial<mint> bc(200002);

int main() {
    ios_base::sync_with_stdio(false);
    cin.tie(nullptr);
    ll n,m; in(n,m);
    mint ans = mint(m+3).pow(n) * 2 - mint(m+2).pow(n) * 3 + mint(m+1).pow(n);
    if(m+1 > n) {
        out(ans);
        return 0;
    }

    poly e(n+1), f(n+1);
    rep(i,n+1) e[i] = f[i] = bc.finv(i);
    f[0] = 0;
    poly e2 = e*e;
    poly p = f.pow(m+1, n+1);
    ans += p[n] * bc.fact(n) * (m+1) * (m+1);
    if(m+1 <= n){
        rep(i,n+1) ans -= p[i] * e2[n-i] * (m+1)*(m+1) * bc.fact(n);
    }
    if(m+2 <= n){
        p *= f;
        rep(i,n+1) ans += p[i] * e[n-i] * (m*m*2 + m*2 - 1) * bc.fact(n);
    }
    if(m+3 <= n){
        p *= f;
        ans -= p[n] * m*m * bc.fact(n);
    }
    out(ans);
}