#pragma region competitive_programming
#ifdef __LOCAL
#define _GLIBCXX_DEBUG
#endif
#pragma GCC optimize("Ofast")
#include<bits/stdc++.h>

//#include<atcoder/dsu>
//#include "Rollback_dsu.hpp"
//#include "Partial_Persistent_DSU.hpp"

//#include "Number_Theory.hpp"


#include <cassert>
#include <numeric>
#include <type_traits>

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


#include <utility>

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

namespace atcoder {

namespace internal {

constexpr long long safe_mod(long long x, long long m) {
    x %= m;
    if (x < 0) x += m;
    return x;
}

struct barrett {
    unsigned int _m;
    unsigned long long im;

    explicit barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}

    unsigned int umod() const { return _m; }

    unsigned int mul(unsigned int a, unsigned int b) const {

        unsigned long long z = a;
        z *= b;
#ifdef _MSC_VER
        unsigned long long x;
        _umul128(z, im, &x);
#else
        unsigned long long x =
            (unsigned long long)(((unsigned __int128)(z)*im) >> 64);
#endif
        unsigned int v = (unsigned int)(z - x * _m);
        if (_m <= v) v += _m;
        return v;
    }
};

constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
    if (m == 1) return 0;
    unsigned int _m = (unsigned int)(m);
    unsigned long long r = 1;
    unsigned long long y = safe_mod(x, m);
    while (n) {
        if (n & 1) r = (r * y) % _m;
        y = (y * y) % _m;
        n >>= 1;
    }
    return r;
}

constexpr bool is_prime_constexpr(int n) {
    if (n <= 1) return false;
    if (n == 2 || n == 7 || n == 61) return true;
    if (n % 2 == 0) return false;
    long long d = n - 1;
    while (d % 2 == 0) d /= 2;
    constexpr long long bases[3] = {2, 7, 61};
    for (long long a : bases) {
        long long t = d;
        long long y = pow_mod_constexpr(a, t, n);
        while (t != n - 1 && y != 1 && y != n - 1) {
            y = y * y % n;
            t <<= 1;
        }
        if (y != n - 1 && t % 2 == 0) {
            return false;
        }
    }
    return true;
}
template <int n> constexpr bool is_prime = is_prime_constexpr(n);

constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) {
    a = safe_mod(a, b);
    if (a == 0) return {b, 0};

    long long s = b, t = a;
    long long m0 = 0, m1 = 1;

    while (t) {
        long long u = s / t;
        s -= t * u;
        m0 -= m1 * u;  // |m1 * u| <= |m1| * s <= b


        auto tmp = s;
        s = t;
        t = tmp;
        tmp = m0;
        m0 = m1;
        m1 = tmp;
    }
    if (m0 < 0) m0 += b / s;
    return {s, m0};
}

constexpr int primitive_root_constexpr(int m) {
    if (m == 2) return 1;
    if (m == 167772161) return 3;
    if (m == 469762049) return 3;
    if (m == 754974721) return 11;
    if (m == 998244353) return 3;
    int divs[20] = {};
    divs[0] = 2;
    int cnt = 1;
    int x = (m - 1) / 2;
    while (x % 2 == 0) x /= 2;
    for (int i = 3; (long long)(i)*i <= x; i += 2) {
        if (x % i == 0) {
            divs[cnt++] = i;
            while (x % i == 0) {
                x /= i;
            }
        }
    }
    if (x > 1) {
        divs[cnt++] = x;
    }
    for (int g = 2;; g++) {
        bool ok = true;
        for (int i = 0; i < cnt; i++) {
            if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) {
                ok = false;
                break;
            }
        }
        if (ok) return g;
    }
}
template <int m> constexpr int primitive_root = primitive_root_constexpr(m);

unsigned long long floor_sum_unsigned(unsigned long long n,
                                      unsigned long long m,
                                      unsigned long long a,
                                      unsigned long long b) {
    unsigned long long ans = 0;
    while (true) {
        if (a >= m) {
            ans += n * (n - 1) / 2 * (a / m);
            a %= m;
        }
        if (b >= m) {
            ans += n * (b / m);
            b %= m;
        }

        unsigned long long y_max = a * n + b;
        if (y_max < m) break;
        n = (unsigned long long)(y_max / m);
        b = (unsigned long long)(y_max % m);
        std::swap(m, a);
    }
    return ans;
}

}  // namespace internal

}  // namespace atcoder


#include <cassert>
#include <numeric>
#include <type_traits>

namespace atcoder {

namespace internal {

#ifndef _MSC_VER
template <class T>
using is_signed_int128 =
    typename std::conditional<std::is_same<T, __int128_t>::value ||
                                  std::is_same<T, __int128>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using is_unsigned_int128 =
    typename std::conditional<std::is_same<T, __uint128_t>::value ||
                                  std::is_same<T, unsigned __int128>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using make_unsigned_int128 =
    typename std::conditional<std::is_same<T, __int128_t>::value,
                              __uint128_t,
                              unsigned __int128>;

template <class T>
using is_integral = typename std::conditional<std::is_integral<T>::value ||
                                                  is_signed_int128<T>::value ||
                                                  is_unsigned_int128<T>::value,
                                              std::true_type,
                                              std::false_type>::type;

template <class T>
using is_signed_int = typename std::conditional<(is_integral<T>::value &&
                                                 std::is_signed<T>::value) ||
                                                    is_signed_int128<T>::value,
                                                std::true_type,
                                                std::false_type>::type;

template <class T>
using is_unsigned_int =
    typename std::conditional<(is_integral<T>::value &&
                               std::is_unsigned<T>::value) ||
                                  is_unsigned_int128<T>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using to_unsigned = typename std::conditional<
    is_signed_int128<T>::value,
    make_unsigned_int128<T>,
    typename std::conditional<std::is_signed<T>::value,
                              std::make_unsigned<T>,
                              std::common_type<T>>::type>::type;

#else

template <class T> using is_integral = typename std::is_integral<T>;

template <class T>
using is_signed_int =
    typename std::conditional<is_integral<T>::value && std::is_signed<T>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using is_unsigned_int =
    typename std::conditional<is_integral<T>::value &&
                                  std::is_unsigned<T>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using to_unsigned = typename std::conditional<is_signed_int<T>::value,
                                              std::make_unsigned<T>,
                                              std::common_type<T>>::type;

#endif

template <class T>
using is_signed_int_t = std::enable_if_t<is_signed_int<T>::value>;

template <class T>
using is_unsigned_int_t = std::enable_if_t<is_unsigned_int<T>::value>;

template <class T> using to_unsigned_t = typename to_unsigned<T>::type;

}  // namespace internal

}  // namespace atcoder


namespace atcoder {

namespace internal {

struct modint_base {};
struct static_modint_base : modint_base {};

template <class T> using is_modint = std::is_base_of<modint_base, T>;
template <class T> using is_modint_t = std::enable_if_t<is_modint<T>::value>;

}  // namespace internal

template <int m, std::enable_if_t<(1 <= m)>* = nullptr>
struct static_modint : internal::static_modint_base {
    using mint = static_modint;

  public:
    static constexpr int mod() { return m; }
    static mint raw(int v) {
        mint x;
        x._v = v;
        return x;
    }

    static_modint() : _v(0) {}
    template <class T, internal::is_signed_int_t<T>* = nullptr>
    static_modint(T v) {
        long long x = (long long)(v % (long long)(umod()));
        if (x < 0) x += umod();
        _v = (unsigned int)(x);
    }
    template <class T, internal::is_unsigned_int_t<T>* = nullptr>
    static_modint(T v) {
        _v = (unsigned int)(v % umod());
    }

    unsigned int val() const { return _v; }

    mint& operator++() {
        _v++;
        if (_v == umod()) _v = 0;
        return *this;
    }
    mint& operator--() {
        if (_v == 0) _v = umod();
        _v--;
        return *this;
    }
    mint operator++(int) {
        mint result = *this;
        ++*this;
        return result;
    }
    mint operator--(int) {
        mint result = *this;
        --*this;
        return result;
    }

    mint& operator+=(const mint& rhs) {
        _v += rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    mint& operator-=(const mint& rhs) {
        _v -= rhs._v;
        if (_v >= umod()) _v += umod();
        return *this;
    }
    mint& operator*=(const mint& rhs) {
        unsigned long long z = _v;
        z *= rhs._v;
        _v = (unsigned int)(z % umod());
        return *this;
    }
    mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }

    mint operator+() const { return *this; }
    mint operator-() const { return mint() - *this; }

    mint pow(long long n) const {
        assert(0 <= n);
        mint x = *this, r = 1;
        while (n) {
            if (n & 1) r *= x;
            x *= x;
            n >>= 1;
        }
        return r;
    }
    mint inv() const {
        if (prime) {
            assert(_v);
            return pow(umod() - 2);
        } else {
            auto eg = internal::inv_gcd(_v, m);
            assert(eg.first == 1);
            return eg.second;
        }
    }

    friend mint operator+(const mint& lhs, const mint& rhs) {
        return mint(lhs) += rhs;
    }
    friend mint operator-(const mint& lhs, const mint& rhs) {
        return mint(lhs) -= rhs;
    }
    friend mint operator*(const mint& lhs, const mint& rhs) {
        return mint(lhs) *= rhs;
    }
    friend mint operator/(const mint& lhs, const mint& rhs) {
        return mint(lhs) /= rhs;
    }
    friend bool operator==(const mint& lhs, const mint& rhs) {
        return lhs._v == rhs._v;
    }
    friend bool operator!=(const mint& lhs, const mint& rhs) {
        return lhs._v != rhs._v;
    }

  private:
    unsigned int _v;
    static constexpr unsigned int umod() { return m; }
    static constexpr bool prime = internal::is_prime<m>;
};

template <int id> struct dynamic_modint : internal::modint_base {
    using mint = dynamic_modint;

  public:
    static int mod() { return (int)(bt.umod()); }
    static void set_mod(int m) {
        assert(1 <= m);
        bt = internal::barrett(m);
    }
    static mint raw(int v) {
        mint x;
        x._v = v;
        return x;
    }

    dynamic_modint() : _v(0) {}
    template <class T, internal::is_signed_int_t<T>* = nullptr>
    dynamic_modint(T v) {
        long long x = (long long)(v % (long long)(mod()));
        if (x < 0) x += mod();
        _v = (unsigned int)(x);
    }
    template <class T, internal::is_unsigned_int_t<T>* = nullptr>
    dynamic_modint(T v) {
        _v = (unsigned int)(v % mod());
    }

    unsigned int val() const { return _v; }

    mint& operator++() {
        _v++;
        if (_v == umod()) _v = 0;
        return *this;
    }
    mint& operator--() {
        if (_v == 0) _v = umod();
        _v--;
        return *this;
    }
    mint operator++(int) {
        mint result = *this;
        ++*this;
        return result;
    }
    mint operator--(int) {
        mint result = *this;
        --*this;
        return result;
    }

    mint& operator+=(const mint& rhs) {
        _v += rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    mint& operator-=(const mint& rhs) {
        _v += mod() - rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    mint& operator*=(const mint& rhs) {
        _v = bt.mul(_v, rhs._v);
        return *this;
    }
    mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }

    mint operator+() const { return *this; }
    mint operator-() const { return mint() - *this; }

    mint pow(long long n) const {
        assert(0 <= n);
        mint x = *this, r = 1;
        while (n) {
            if (n & 1) r *= x;
            x *= x;
            n >>= 1;
        }
        return r;
    }
    mint inv() const {
        auto eg = internal::inv_gcd(_v, mod());
        assert(eg.first == 1);
        return eg.second;
    }

    friend mint operator+(const mint& lhs, const mint& rhs) {
        return mint(lhs) += rhs;
    }
    friend mint operator-(const mint& lhs, const mint& rhs) {
        return mint(lhs) -= rhs;
    }
    friend mint operator*(const mint& lhs, const mint& rhs) {
        return mint(lhs) *= rhs;
    }
    friend mint operator/(const mint& lhs, const mint& rhs) {
        return mint(lhs) /= rhs;
    }
    friend bool operator==(const mint& lhs, const mint& rhs) {
        return lhs._v == rhs._v;
    }
    friend bool operator!=(const mint& lhs, const mint& rhs) {
        return lhs._v != rhs._v;
    }

  private:
    unsigned int _v;
    static internal::barrett bt;
    static unsigned int umod() { return bt.umod(); }
};
template <int id> internal::barrett dynamic_modint<id>::bt(998244353);

using modint998244353 = static_modint<998244353>;
using modint1000000007 = static_modint<1000000007>;
using modint = dynamic_modint<-1>;

namespace internal {

template <class T>
using is_static_modint = std::is_base_of<internal::static_modint_base, T>;

template <class T>
using is_static_modint_t = std::enable_if_t<is_static_modint<T>::value>;

template <class> struct is_dynamic_modint : public std::false_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint<id>> : public std::true_type {};

template <class T>
using is_dynamic_modint_t = std::enable_if_t<is_dynamic_modint<T>::value>;

}  // namespace internal

}  // namespace atcoder


namespace tomo0608 {
    std::istream& operator>>(std::istream& is, atcoder::modint998244353& a) { long long v; is >> v; a = v; return is; }
    std::ostream& operator<<(std::ostream& os, const atcoder::modint998244353& a) { return os << a.val(); }
    std::istream& operator>>(std::istream& is, atcoder::modint1000000007& a) { long long v; is >> v; a = v; return is; }
    std::ostream& operator<<(std::ostream& os, const atcoder::modint1000000007& a) { return os << a.val(); }
    template<int m> std::istream& operator>>(std::istream& is, atcoder::static_modint<m>& a) { long long v; is >> v; a = v; return is; }
    template<int m> std::ostream& operator<<(std::ostream& os, const atcoder::static_modint<m>& a) { return os << a.val(); }
    template<int m> std::istream& operator>>(std::istream& is, atcoder::dynamic_modint<m>& a) { long long v; is >> v; a = v; return is; }
    template<int m> std::ostream& operator<<(std::ostream& os, const atcoder::dynamic_modint<m>& a) { return os << a.val(); }

    // Binomial Coefficient of modint
    template<class mint> struct BiCoef {
        std::vector<mint> fact_, inv_, finv_;
        constexpr BiCoef() {}
        constexpr BiCoef(int n) noexcept : fact_(n, 1), inv_(n, 1), finv_(n, 1) {
            init(n);
        }
        constexpr void init(int n) noexcept {
            fact_.assign(n, 1), inv_.assign(n, 1), finv_.assign(n, 1);
            int MOD = fact_[0].mod();
            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 mint 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 mint perm(int n, int k) const noexcept {
            if (n < k || n < 0 || k < 0)return 0;
            return fact_[n] * finv_[n - k];
        }
        constexpr mint homo(int n, int r) { // The number of cases where k indistinguishable balls are put into n distinct boxes
            if (n < 0 || r < 0)return 0;
            return r == 0 ? 1 : com(n + r - 1, r);
        }
        constexpr mint second_stirling_number(int n, int r) { // The number of cases where n distinct balls are put into k indistinguishable boxes, with at least one ball in each box
            mint ret = 0;
            for (int i = 0; i <= r; i++) {
                mint tmp = com(r, i) * mint(i).pow(n);
                ret += ((r - i) & 1) ? -tmp : tmp;
            }
            return ret * finv_[r];
        }
        constexpr mint bell_number(int n, int r) { // The number of cases where n distinct balls are put into k indistinguishable boxes
            if (n == 0) return 1;
            r = std::min(r, n);
            std::vector<mint> pref(r + 1);
            pref[0] = 1;
            for (int i = 1; i <= r; i++) {
                if (i & 1) {
                    pref[i] = pref[i - 1] - finv_[i];
                }
                else {
                    pref[i] = pref[i - 1] + finv_[i];
                }
            }
            mint ret = 0;
            for (int i = 1; i <= r; i++) ret += mint(i).pow(n) * fact_[i] * pref[r - i];
            return ret;
        }

        constexpr mint fact(int n) const noexcept {
            if (n < 0)return 0;
            return fact_[n];
        }
        constexpr mint inv(int n) const noexcept {
            if (n < 0)return 0;
            return inv_[n];
        }
        constexpr mint finv(int n) const noexcept {
            if (n < 0)return 0;
            return finv_[n];
        }
        inline mint operator()(int n, int k) { return com(n, k); }

        constexpr mint com_naive(long long n, long long k) {
            if (n < k || n < 0 || k < 0)return 0;
            mint res = 1;
            k = std::min(k, n - k);
            for (int i = 1; i <= k; i++)res *= inv(i) * (n--);
            return res;
        }
    };
} // namespace tomo0608
//typedef atcoder::modint1000000007 mint;
typedef atcoder::modint998244353 mint;
//#include "Matrix.hpp"

//#include<atcoder/convolution>

#include <algorithm>
#include <array>
#include <cassert>
#include <type_traits>
#include <vector>


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

namespace atcoder {

namespace internal {

int ceil_pow2(int n) {
    int x = 0;
    while ((1U << x) < (unsigned int)(n)) x++;
    return x;
}

constexpr int bsf_constexpr(unsigned int n) {
    int x = 0;
    while (!(n & (1 << x))) x++;
    return x;
}

int bsf(unsigned int n) {
#ifdef _MSC_VER
    unsigned long index;
    _BitScanForward(&index, n);
    return index;
#else
    return __builtin_ctz(n);
#endif
}

}  // namespace internal

}  // namespace atcoder


namespace atcoder {

namespace internal {

template <class mint,
          int g = internal::primitive_root<mint::mod()>,
          internal::is_static_modint_t<mint>* = nullptr>
struct fft_info {
    static constexpr int rank2 = bsf_constexpr(mint::mod() - 1);
    std::array<mint, rank2 + 1> root;   // root[i]^(2^i) == 1
    std::array<mint, rank2 + 1> iroot;  // root[i] * iroot[i] == 1

    std::array<mint, std::max(0, rank2 - 2 + 1)> rate2;
    std::array<mint, std::max(0, rank2 - 2 + 1)> irate2;

    std::array<mint, std::max(0, rank2 - 3 + 1)> rate3;
    std::array<mint, std::max(0, rank2 - 3 + 1)> irate3;

    fft_info() {
        root[rank2] = mint(g).pow((mint::mod() - 1) >> rank2);
        iroot[rank2] = root[rank2].inv();
        for (int i = rank2 - 1; i >= 0; i--) {
            root[i] = root[i + 1] * root[i + 1];
            iroot[i] = iroot[i + 1] * iroot[i + 1];
        }

        {
            mint prod = 1, iprod = 1;
            for (int i = 0; i <= rank2 - 2; i++) {
                rate2[i] = root[i + 2] * prod;
                irate2[i] = iroot[i + 2] * iprod;
                prod *= iroot[i + 2];
                iprod *= root[i + 2];
            }
        }
        {
            mint prod = 1, iprod = 1;
            for (int i = 0; i <= rank2 - 3; i++) {
                rate3[i] = root[i + 3] * prod;
                irate3[i] = iroot[i + 3] * iprod;
                prod *= iroot[i + 3];
                iprod *= root[i + 3];
            }
        }
    }
};

template <class mint, internal::is_static_modint_t<mint>* = nullptr>
void butterfly(std::vector<mint>& a) {
    int n = int(a.size());
    int h = internal::ceil_pow2(n);

    static const fft_info<mint> info;

    int len = 0;  // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
    while (len < h) {
        if (h - len == 1) {
            int p = 1 << (h - len - 1);
            mint rot = 1;
            for (int s = 0; s < (1 << len); s++) {
                int offset = s << (h - len);
                for (int i = 0; i < p; i++) {
                    auto l = a[i + offset];
                    auto r = a[i + offset + p] * rot;
                    a[i + offset] = l + r;
                    a[i + offset + p] = l - r;
                }
                if (s + 1 != (1 << len))
                    rot *= info.rate2[bsf(~(unsigned int)(s))];
            }
            len++;
        } else {
            int p = 1 << (h - len - 2);
            mint rot = 1, imag = info.root[2];
            for (int s = 0; s < (1 << len); s++) {
                mint rot2 = rot * rot;
                mint rot3 = rot2 * rot;
                int offset = s << (h - len);
                for (int i = 0; i < p; i++) {
                    auto mod2 = 1ULL * mint::mod() * mint::mod();
                    auto a0 = 1ULL * a[i + offset].val();
                    auto a1 = 1ULL * a[i + offset + p].val() * rot.val();
                    auto a2 = 1ULL * a[i + offset + 2 * p].val() * rot2.val();
                    auto a3 = 1ULL * a[i + offset + 3 * p].val() * rot3.val();
                    auto a1na3imag =
                        1ULL * mint(a1 + mod2 - a3).val() * imag.val();
                    auto na2 = mod2 - a2;
                    a[i + offset] = a0 + a2 + a1 + a3;
                    a[i + offset + 1 * p] = a0 + a2 + (2 * mod2 - (a1 + a3));
                    a[i + offset + 2 * p] = a0 + na2 + a1na3imag;
                    a[i + offset + 3 * p] = a0 + na2 + (mod2 - a1na3imag);
                }
                if (s + 1 != (1 << len))
                    rot *= info.rate3[bsf(~(unsigned int)(s))];
            }
            len += 2;
        }
    }
}

template <class mint, internal::is_static_modint_t<mint>* = nullptr>
void butterfly_inv(std::vector<mint>& a) {
    int n = int(a.size());
    int h = internal::ceil_pow2(n);

    static const fft_info<mint> info;

    int len = h;  // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
    while (len) {
        if (len == 1) {
            int p = 1 << (h - len);
            mint irot = 1;
            for (int s = 0; s < (1 << (len - 1)); s++) {
                int offset = s << (h - len + 1);
                for (int i = 0; i < p; i++) {
                    auto l = a[i + offset];
                    auto r = a[i + offset + p];
                    a[i + offset] = l + r;
                    a[i + offset + p] =
                        (unsigned long long)(mint::mod() + l.val() - r.val()) *
                        irot.val();
                    ;
                }
                if (s + 1 != (1 << (len - 1)))
                    irot *= info.irate2[bsf(~(unsigned int)(s))];
            }
            len--;
        } else {
            int p = 1 << (h - len);
            mint irot = 1, iimag = info.iroot[2];
            for (int s = 0; s < (1 << (len - 2)); s++) {
                mint irot2 = irot * irot;
                mint irot3 = irot2 * irot;
                int offset = s << (h - len + 2);
                for (int i = 0; i < p; i++) {
                    auto a0 = 1ULL * a[i + offset + 0 * p].val();
                    auto a1 = 1ULL * a[i + offset + 1 * p].val();
                    auto a2 = 1ULL * a[i + offset + 2 * p].val();
                    auto a3 = 1ULL * a[i + offset + 3 * p].val();

                    auto a2na3iimag =
                        1ULL *
                        mint((mint::mod() + a2 - a3) * iimag.val()).val();

                    a[i + offset] = a0 + a1 + a2 + a3;
                    a[i + offset + 1 * p] =
                        (a0 + (mint::mod() - a1) + a2na3iimag) * irot.val();
                    a[i + offset + 2 * p] =
                        (a0 + a1 + (mint::mod() - a2) + (mint::mod() - a3)) *
                        irot2.val();
                    a[i + offset + 3 * p] =
                        (a0 + (mint::mod() - a1) + (mint::mod() - a2na3iimag)) *
                        irot3.val();
                }
                if (s + 1 != (1 << (len - 2)))
                    irot *= info.irate3[bsf(~(unsigned int)(s))];
            }
            len -= 2;
        }
    }
}

template <class mint, internal::is_static_modint_t<mint>* = nullptr>
std::vector<mint> convolution_naive(const std::vector<mint>& a,
                                    const std::vector<mint>& b) {
    int n = int(a.size()), m = int(b.size());
    std::vector<mint> ans(n + m - 1);
    if (n < m) {
        for (int j = 0; j < m; j++) {
            for (int i = 0; i < n; i++) {
                ans[i + j] += a[i] * b[j];
            }
        }
    } else {
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) {
                ans[i + j] += a[i] * b[j];
            }
        }
    }
    return ans;
}

template <class mint, internal::is_static_modint_t<mint>* = nullptr>
std::vector<mint> convolution_fft(std::vector<mint> a, std::vector<mint> b) {
    int n = int(a.size()), m = int(b.size());
    int z = 1 << internal::ceil_pow2(n + m - 1);
    a.resize(z);
    internal::butterfly(a);
    b.resize(z);
    internal::butterfly(b);
    for (int i = 0; i < z; i++) {
        a[i] *= b[i];
    }
    internal::butterfly_inv(a);
    a.resize(n + m - 1);
    mint iz = mint(z).inv();
    for (int i = 0; i < n + m - 1; i++) a[i] *= iz;
    return a;
}

}  // namespace internal

template <class mint, internal::is_static_modint_t<mint>* = nullptr>
std::vector<mint> convolution(std::vector<mint>&& a, std::vector<mint>&& b) {
    int n = int(a.size()), m = int(b.size());
    if (!n || !m) return {};
    if (std::min(n, m) <= 60) return convolution_naive(a, b);
    return internal::convolution_fft(a, b);
}

template <class mint, internal::is_static_modint_t<mint>* = nullptr>
std::vector<mint> convolution(const std::vector<mint>& a,
                              const std::vector<mint>& b) {
    int n = int(a.size()), m = int(b.size());
    if (!n || !m) return {};
    if (std::min(n, m) <= 60) return convolution_naive(a, b);
    return internal::convolution_fft(a, b);
}

template <unsigned int mod = 998244353,
          class T,
          std::enable_if_t<internal::is_integral<T>::value>* = nullptr>
std::vector<T> convolution(const std::vector<T>& a, const std::vector<T>& b) {
    int n = int(a.size()), m = int(b.size());
    if (!n || !m) return {};

    using mint = static_modint<mod>;
    std::vector<mint> a2(n), b2(m);
    for (int i = 0; i < n; i++) {
        a2[i] = mint(a[i]);
    }
    for (int i = 0; i < m; i++) {
        b2[i] = mint(b[i]);
    }
    auto c2 = convolution(move(a2), move(b2));
    std::vector<T> c(n + m - 1);
    for (int i = 0; i < n + m - 1; i++) {
        c[i] = c2[i].val();
    }
    return c;
}

std::vector<long long> convolution_ll(const std::vector<long long>& a,
                                      const std::vector<long long>& b) {
    int n = int(a.size()), m = int(b.size());
    if (!n || !m) return {};

    static constexpr unsigned long long MOD1 = 754974721;  // 2^24
    static constexpr unsigned long long MOD2 = 167772161;  // 2^25
    static constexpr unsigned long long MOD3 = 469762049;  // 2^26
    static constexpr unsigned long long M2M3 = MOD2 * MOD3;
    static constexpr unsigned long long M1M3 = MOD1 * MOD3;
    static constexpr unsigned long long M1M2 = MOD1 * MOD2;
    static constexpr unsigned long long M1M2M3 = MOD1 * MOD2 * MOD3;

    static constexpr unsigned long long i1 =
        internal::inv_gcd(MOD2 * MOD3, MOD1).second;
    static constexpr unsigned long long i2 =
        internal::inv_gcd(MOD1 * MOD3, MOD2).second;
    static constexpr unsigned long long i3 =
        internal::inv_gcd(MOD1 * MOD2, MOD3).second;

    auto c1 = convolution<MOD1>(a, b);
    auto c2 = convolution<MOD2>(a, b);
    auto c3 = convolution<MOD3>(a, b);

    std::vector<long long> c(n + m - 1);
    for (int i = 0; i < n + m - 1; i++) {
        unsigned long long x = 0;
        x += (c1[i] * i1) % MOD1 * M2M3;
        x += (c2[i] * i2) % MOD2 * M1M3;
        x += (c3[i] * i3) % MOD3 * M1M2;
        long long diff =
            c1[i] - internal::safe_mod((long long)(x), (long long)(MOD1));
        if (diff < 0) diff += MOD1;
        static constexpr unsigned long long offset[5] = {
            0, 0, M1M2M3, 2 * M1M2M3, 3 * M1M2M3};
        x -= offset[diff % 5];
        c[i] = x;
    }

    return c;
}

}  // namespace atcoder


namespace tomo0608 {
    template<typename mint>
    struct Formal_Power_Series : public std::vector<mint> {
        using FPS = Formal_Power_Series;
        using std::vector<mint>::vector;
        using std::vector<mint>::operator=;

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

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

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

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

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

        FPS& operator*=(const mint& rhs) {
            for (auto& e : *this)e *= rhs;
            return *this;
        }

        FPS& operator*=(const FPS& rhs) {
            *this = atcoder::convolution((*this), rhs);
            return *this;
        }

        FPS& operator/=(const mint& c) {
            assert(c != mint(0));
            *this *= c.inv();
            return *this;
        }

        FPS& operator/=(const FPS& rhs) {
            *this *= rhs.inv();
            this->resize(rhs.size());
            return *this;
        }

        friend FPS operator+(const FPS& lhs, const FPS& rhs) {
            return FPS(lhs) += rhs;
        }
        friend FPS operator+(const FPS& lhs, const mint& rhs) {
            return FPS(lhs) += rhs;
        }

        friend FPS operator-(const FPS& lhs, const FPS& rhs) {
            return FPS(lhs) -= rhs;
        }

        friend FPS operator-(const FPS& lhs, const mint& rhs) {
            return FPS(lhs) -= rhs;
        }

        friend FPS operator*(const FPS& lhs, const FPS& rhs) {
            return FPS(lhs) *= rhs;
        }

        friend FPS operator*(const FPS& lhs, const mint& rhs) {
            return FPS(lhs) *= rhs;
        }

        friend FPS operator/(const FPS& lhs, const mint& rhs) {
            return FPS(lhs) /= rhs;
        }
        // friend std::ostream& operator<<(std::ostream& os, FPS& f) {
        //     for (int i = 0; i < f.size(); i++) {
        //         os << f[i] << (i + 1 == f.size() ? "\n" : " ");
        //     }
        //     return (os);
        // }

        FPS inv(int deg = -1) const {
            assert(this->front() != mint(0));
            if (deg == -1)deg = this->size();
            FPS res = { this->front().inv() };
            for (int d = 1; d < deg; d <<= 1) {
                FPS f(2 * d), g(2 * d);
                for (int j = 0; j < std::min((int)this->size(), 2 * d); j++)f[j] = (*this)[j];
                for (int j = 0; j < d; j++)g[j] = res[j];

                atcoder::internal::butterfly(f);
                atcoder::internal::butterfly(g);

                for (int j = 0; j < 2 * d; j++)f[j] *= g[j];

                atcoder::internal::butterfly_inv(f);
                f /= 2 * d;

                for (int j = 0; j < d; j++) {
                    f[j] = f[j + d];
                    f[j + d] = 0;
                }

                atcoder::internal::butterfly(f);

                for (int j = 0; j < 2 * d; j++)f[j] *= -g[j];

                atcoder::internal::butterfly_inv(f);
                f /= 2 * d;

                for (int j = 0; j < d; j++)res.emplace_back(f[j]);
            }
            res.resize(deg);
            return res;
        }

        FPS operator-() const {
            FPS res(*this);
            for (auto& e : res) e = -e;
            return res;
        }

        FPS derivative() const {
            FPS res(this->begin() + 1, this->end());
            res.emplace_back(0);
            for (int j = 0; j < res.size(); j++)res[j] *= (j + 1);
            return res;
        }

        FPS integral(bool truncate = true) const {
            FPS res(this->size() + 1 - truncate, 0);
            for (int j = 0; j < this->size() - truncate; j++)res[j + 1] = (*this)[j] / (j + 1);
            return res;
        }

        FPS log() const {
            FPS f = this->derivative();
            f /= (*this);
            return f.integral();
        }
        // https://arxiv.org/pdf/1301.5804.pdf
        FPS exp(int deg = -1) const {
            assert(this->front() == mint(0));
            if (deg == -1)deg = this->size();
            FPS f = { 1 }, g = { 1 };
            for (int d = 1; d < deg; d <<= 1) {
                FPS f_next(f);
                f_next.resize(2 * d, 0);
                atcoder::internal::butterfly(f_next);

                g.resize(2 * d, 0);
                atcoder::internal::butterfly(g);
                for (int j = 0; j < 2 * d; j++)g[j] = 2 * g[j] - f_next[j] * g[j] * g[j];
                atcoder::internal::butterfly_inv(g);
                g /= 2 * d;
                g.resize(d);

                FPS q(2 * d);
                for (int j = 0; j < d && j < this->size() - 1; j++)q[j] = (*this)[j + 1] * (j + 1);

                FPS w(2 * d);
                FPS G(g);
                G.resize(2 * d, 0);
                atcoder::internal::butterfly(G);
                for (int j = 0; j < 2 * d; j++)w[j] = f_next[j] * G[j];
                atcoder::internal::butterfly_inv(w);
                w /= 2 * d;

                for (int j = 0; j < d; j++) {
                    w[j] = w[j + d];
                    w[j + d] = 0;
                }

                atcoder::internal::butterfly(w);
                atcoder::internal::butterfly(q);
                for (int j = 0; j < 2 * d; j++)w[j] *= q[j];
                atcoder::internal::butterfly_inv(w);
                w /= 2 * d;

                FPS df(f.derivative());
                df.resize(2 * d, 0);
                atcoder::internal::butterfly(df);
                for (int j = 0;j < 2 * d;j++)df[j] *= G[j];
                atcoder::internal::butterfly_inv(df);
                df /= 2 * d;

                for (int j = 0; j < d;j++) {
                    w[j + d] = w[j];
                    w[j] = 0;
                }
                w -= df;

                w = w.integral();

                for (int j = 0; j < 2 * d && j < this->size(); j++)w[j] += (*this)[j];
                for (int j = 0; j < d;j++) {
                    w[j] = w[j + d];
                    w[j + d] = 0;
                }
                atcoder::internal::butterfly(w);
                for (int j = 0; j < 2 * d; j++)f_next[j] *= w[j];
                atcoder::internal::butterfly_inv(f_next);
                f_next /= 2 * d;
                f.resize(2 * d, 0);
                for (int j = 0; j < d; j++)f[j + d] = f_next[j];
            }
            f.resize(deg);
            return f;
        }

        FPS pow(long long m) const {
            int n = this->size();
            if (m == 0) {
                auto res = FPS(n, 0);
                res[0] = 1;
                return res;
            }
            int l = std::find_if(this->begin(), this->end(), [](mint x) {return x != mint(0);}) - this->begin();
            if (l == this->size() || (l && m >= (n + l - 1) / l))return FPS(n, 0);

            FPS res(this->begin() + l, this->end());
            mint c = (*this)[l];
            res /= c;
            res.resize(n, 0);
            res = (res.log() * mint(m)).exp();
            res.erase(res.begin() + (n - m * l), res.end());
            res *= c.pow(m);
            std::reverse(res.begin(), res.end());
            res.resize(n, 0);
            std::reverse(res.begin(), res.end());
            return res;
        }

    };


    // for sparse fps
    template <typename mint>
    Formal_Power_Series<mint> positive_unit_fractions(int n) { // res[i] = 1 / i, res[0] = 0 length: n+1
        static const int mod = mint::mod();
        static Formal_Power_Series<mint> res = { 0, 1 };
        assert(0 < n);
        if (n >= mod) n -= mod;
        while (int(res.size()) <= n) {
            int num = res.size();
            int q = (mod + num - 1) / num;
            res.emplace_back(res[num * q - mod] * mint(q));
        }
        return res;
    }

    template<typename mint>
    std::vector<std::pair<int, mint>> compress_fps(const Formal_Power_Series<mint>& f) {
        int n = f.size();
        std::vector<std::pair<int, mint>> cf;
        for (int i = 0; i < n; i++) {
            if (f[i] != 0)cf.emplace_back(i, f[i]);
        }
        return cf;
    }

    template<typename mint>
    Formal_Power_Series<mint> mul_sparse(const Formal_Power_Series<mint>& f, const Formal_Power_Series<mint>& g) {
        int n = f.size(), m = g.size();
        auto cf = compress_fps<mint>(f), cg = compress_fps<mint>(g);
        Formal_Power_Series<mint> h(n + m - 1);
        for (auto [i, p] : cf)for (auto [j, q] : cg)h[i + j] += p * q;
        return h;
    }

    template<typename mint>
    Formal_Power_Series<mint> inv_sparse(const Formal_Power_Series<mint>& f, int deg = -1) {
        assert(f[0] != 0);
        if (deg == -1)deg = f.size();
        auto cf = compress_fps<mint>(f);
        Formal_Power_Series<mint> f_inv(deg);
        f_inv[0] = f[0].inv();
        for (int i = 1; i < deg; i++) {
            for (auto [k, p] : cf) {
                if (i - k < 0)break;
                f_inv[i] -= f_inv[i - k] * p;
            }
            f_inv[i] *= f_inv[0];
        }
        return f_inv;
    }

    template<typename mint>
    Formal_Power_Series<mint> exp_sparse(const Formal_Power_Series<mint>& f, int deg = -1) {
        assert(f[0] == 0);
        if (deg == -1)deg = f.size();
        auto cf = compress_fps<mint>(f);
        Formal_Power_Series<mint> f_exp(deg);
        Formal_Power_Series<mint> inv_num = positive_unit_fractions<mint>(deg);
        f_exp[0] = 1;
        for (int i = 1; i < deg; i++) {
            for (auto [k, p] : cf) {
                if (i - k < 0)break;
                f_exp[i] += f_exp[i - k] * p * k;
            }
            f_exp[i] *= inv_num[i];
        }
        return f_exp;
    }

    template<typename mint>
    Formal_Power_Series<mint> log_sparse(const Formal_Power_Series<mint>& f, int deg = -1) {
        assert(f[0] == 1);
        if (deg == -1)deg = f.size();
        Formal_Power_Series<mint> df = f.derivative();
        Formal_Power_Series<mint> df_log = mul_sparse<mint>(df, inv_sparse<mint>(f));
        Formal_Power_Series<mint> f_log = df_log.integral();
        f_log.resize(deg);
        return f_log;
    }

    template<typename mint>
    Formal_Power_Series<mint> __pow_sparse_const_1(const Formal_Power_Series<mint>& f, long long k, int deg) {
        int n = f.size();
        assert(n > 0 && f[0] == 1);
        auto cf = compress_fps<mint>(f);
        Formal_Power_Series<mint> f_pow_k(deg);
        Formal_Power_Series<mint> inv_num = positive_unit_fractions<mint>(deg);
        f_pow_k[0] = 1;
        for (int i = 1; i < deg; i++) {
            for (const auto& [j, coef] : cf) {
                if (i - j < 0) break;
                f_pow_k[i] += (mint(k) * mint(j) - mint(i - j)) * coef * f_pow_k[i - j];
            }
            f_pow_k[i] *= inv_num[i];
        }
        return f_pow_k;
    }

    template<typename mint>
    Formal_Power_Series<mint> pow_sparse(const Formal_Power_Series<mint>& f, long long k, int deg = -1) {
        int n = f.size();
        if (deg < 0)deg = n;
        assert(k >= 0);
        if (k == 0) {
            Formal_Power_Series<mint> res(deg, 0);
            if (deg)res[0] = 1;
            return res;
        }
        int l = std::find_if(f.begin(), f.end(), [](mint x) {return x != mint(0);}) - f.begin();
        if (l == f.size() || (l && k >= (deg + l - 1) / l))return Formal_Power_Series<mint>(deg, 0);

        Formal_Power_Series<mint> res(f.begin() + l, f.end());
        mint c = f[l];
        res /= c;
        res.resize(deg, 0);
        res = __pow_sparse_const_1<mint>(res, k, deg);
        res.erase(res.begin() + (deg - k * l), res.end());
        res *= c.pow(k);
        std::reverse(res.begin(), res.end());
        res.resize(deg, 0);
        std::reverse(res.begin(), res.end());
        return res;
    }
} // namespace tomo0608
//#include "Bit_Convolution.hpp"

//#include<atcoder/maxflow>
//#include<atcoder/mincostflow>
//#include "Primal_Dual.hpp"
//#include "maxflow_mincap.hpp"

//#include<atcoder/fenwicktree>
//#include<atcoder/segtree>
//#include<atcoder/lazysegtree>
//#include "2D_Segment_Tree.hpp"
//#include "DisjointSparseTable.hpp"
//#include "SWAG.hpp"
//#include "Mo_algorithm.hpp"

//#include "Heavy_Light_Decomposition.hpp"
//#include "Binary_Trie.hpp"
//#include "LCT.hpp"
//#include "Slope_Trick.hpp"
//#include<atcoder/string>

//#include<atcoder/scc>
//#include "TwoEdgeCC.hpp"



namespace tomo0608 {
    typedef long long ll;
    typedef long double ld;
    template <class T> using V = std::vector<T>;
    template <class T> using VV = V<V<T>>;
    template <class T> using VVV = V<VV<T>>;
    typedef std::pair<int, int> pii;
    typedef std::pair<long long, long long> pll;
    template<class... T>void input(T&... a) { (std::cin >> ... >> a); };

#define INT(...) int __VA_ARGS__; IN(__VA_ARGS__)
#define LL(...) long long __VA_ARGS__; IN(__VA_ARGS__)
#define STR(...) string __VA_ARGS__; IN(__VA_ARGS__)
#define DBL(...) double __VA_ARGS__; IN(__VA_ARGS__)
#define VEC(type, name, size) std::vector<type> name(size);IN(name)
#define VECVEC(type, name, h, w) std::vector<std::vector<type>> name(h, std::vector<type>(w));IN(name)
    template<class T1, class T2> std::istream& operator>>(std::istream& is, std::pair<T1, T2>& p) { is >> p.first >> p.second; return is; }
    template<class T1, class T2> std::ostream& operator<<(std::ostream& os, const std::pair<T1, T2>& p) { os << '(' << p.first << ", " << p.second << ')'; return os; }
    template<class T> std::istream& operator>>(std::istream& is, std::vector<T>& v) { for (auto& e : v) is >> e; return is; }
    template<class T> std::ostream& operator<<(std::ostream& os, const std::vector<T>& v) { for (auto& e : v) os << e << ' '; return os; }
    template<typename T> std::ostream& operator << (std::ostream& os, std::set<T>& set_var) { os << "{"; for (auto itr = set_var.begin(); itr != set_var.end(); itr++) { os << *itr;++itr;if (itr != set_var.end()) os << ", ";itr--; }os << "}";return os; }
    template <typename T, typename U> std::ostream& operator<<(std::ostream& os, std::map<T, U>& map_var) { os << "{";for (auto itr = map_var.begin(); itr != map_var.end(); itr++) { os << *itr;itr++;if (itr != map_var.end()) os << ", ";itr--; }os << "}";return os; }

    void IN() {}
    template <class Head, class... Tail> void IN(Head& head, Tail &...tail) {
        std::cin >> head;
        IN(tail...);
    }
    void print() { std::cout << '\n'; }
    template<class T, class... Ts>void print(const T& a, const Ts&... b) { std::cout << a; (std::cout << ... << (std::cout << ' ', b)); std::cout << '\n'; }
    void drop() { std::cout << '\n';exit(0); }
    template<class T, class... Ts>void drop(const T& a, const Ts&... b) { std::cout << a; (std::cout << ... << (std::cout << ' ', b)); std::cout << '\n';exit(0); }
#ifdef __LOCAL
    void debug_out() { std::cerr << std::endl; }
    template < class Head, class... Tail> void debug_out(Head H, Tail... T) { std::cerr << ' ' << H; debug_out(T...); }
#define debug(...) std::cerr << 'L' << __LINE__ << " [" << #__VA_ARGS__ << "]:", debug_out(__VA_ARGS__)
#define dump(x) std::cerr << 'L' << __LINE__ << " " << #x << " = " << (x) << std::endl
#else
#define debug(...) (void(0))
#define dump(x) (void(0))
#endif


#define rep1(a)          for(long long i = 0; i < a; i++)
#define rep2(i, a)       for(long long i = 0; i < a; i++)
#define rep3(i, a, b)    for(long long i = a; i < b; i++)
#define rep4(i, a, b, c) for(long long i = a; i < b; i += c)
#define drep1(a)          for(long long i = a-1; i >= 0; i--)
#define drep2(i, a)       for(long long i = a-1; i >= 0; i--)
#define drep3(i, a, b)    for(long long i = a-1; i >= b; i--)
#define drep4(i, a, b, c) for(long long i = a-1; i >= b; i -= c)
#define overload4(a, b, c, d, e, ...) e
#define rep(...) overload4(__VA_ARGS__, rep4, rep3, rep2, rep1)(__VA_ARGS__)
#define drep(...) overload4(__VA_ARGS__, drep4, drep3, drep2, drep1)(__VA_ARGS__)
#define endl '\n'
} // namespace tomo0608

namespace tomo0608 {
#define ALL(x) x.begin(),x.end()
    template <class T = long long, class S> T SUM(const S& v) { return accumulate(ALL(v), T(0)); }
#define MIN(v) *min_element(ALL(v))
#define MAX(v) *max_element(ALL(v))
#define SORT(v) sort(ALL(v))
#define REVERSE(v) reverse(ALL(v))
#define RSORT(v) sort(ALL(v)); reverse(ALL(v))
#define UNIQUE(x) SORT(x), x.erase(unique(ALL(x)), x.end())
#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> void zip(std::vector<T>& x) { std::vector<T> y = x;UNIQUE(y);for (int i = 0; i < x.size(); ++i) { x[i] = lb(y, x[i]); } }
    template<class T> using priority_queue_rev = std::priority_queue<T, std::vector<T>, std::greater<T>>;
    template<class T, class U> inline bool chmax(T& a, const U& b) { if (a < b) { a = b; return 1; } return 0; }
    template<class T, class U> inline bool chmin(T& a, const U& b) { if (a > b) { a = b; return 1; } return 0; }
    template<class T> inline int count_between(std::vector<T>& a, T l, T r) { return lower_bound(ALL(a), r) - lower_bound(ALL(a), l); } // [l, r)
#define bittest(n, k) (((n) >> (k)) & 1)
    int topbit(signed t) { return t == 0 ? -1 : 31 - __builtin_clz(t); }
    int topbit(long long t) { return t == 0 ? -1 : 63 - __builtin_clzll(t); }
    int lowbit(signed a) { return a == 0 ? 32 : __builtin_ctz(a); }
    int lowbit(long long a) { return a == 0 ? 64 : __builtin_ctzll(a); }
#define perm(v) for(bool permrepflag = true; (permrepflag ? exchange(permrepflag, false) : next_permutation(ALL(v)));)
    template <typename T, typename S> T ceil(T x, S y) {
        assert(y);
        return (y < 0 ? ceil(-x, -y) : (x > 0 ? (x + y - 1) / y : x / y));
    }

    template <typename T, typename S> T floor(T x, S y) {
        assert(y);
        return (y < 0 ? floor(-x, -y) : (x > 0 ? x / y : x / y - (x % y == 0 ? 0 : 1)));
    }
}
using namespace atcoder;
using namespace std;
using namespace tomo0608;
int dx[8] = { 1, 0, -1, 0, 1, 1, -1, -1 };
int dy[8] = { 0, 1, 0, -1, 1, -1, -1, 1 };



// インタラクティブ問題のときは出力するたびにcout.flush();を忘れない!!!!!

void solve();
int main() {
    std::cin.tie(0);
    std::ios_base::sync_with_stdio(false);
    std::cout << std::setprecision(20);
    int codeforces = 1;
    //cin >> codeforces;
    while (codeforces--) {
        solve();
    }
    return 0;
}
#pragma endregion
const int m2 = 999630629;
typedef Formal_Power_Series<mint> fps;
void solve() {
    INT(n);
    VEC(ll, a, n);
    ll sum_a = SUM(a); // sum_a < 2 * m2
    mint ans = sum_a * mint(2).pow(n-1);
    if(sum_a < m2)drop(ans);
    // \sum_{i \in S}a_i >= m2となるようなSの個数を求める
    // \sum_{i \in T}a_i <= sum_a - m2となるようなTの個数を求めればよい
    ll sz = sum_a - m2 + 1;
    fps sum_log(sz);
    BiCoef<mint> bc(sz);
    rep(i, n){
        // log(1 + x^a_i)を求める
        for(int j = a[i], num = 1, c = 1; j < sz; j += a[i], num++, c *= -1){
            sum_log[j] += bc.inv(num) * c;
        }
    }
    fps prd = sum_log.exp();
    mint cnt = mint(2).pow(n);
    rep(i, sz)cnt -= prd[i];
    print(ans - cnt * m2);
}