#define NDEBUG

using namespace std;


#include <immintrin.h>

#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <cctype>
#include <cfenv>
#include <cfloat>
#include <chrono>
#include <cinttypes>
#include <climits>
#include <cmath>
#include <complex>
#include <cstdarg>
#include <cstddef>
#include <cstdint>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <deque>
#include <fstream>
#include <functional>
#include <initializer_list>
#include <iomanip>
#include <ios>
#include <iostream>
#include <istream>
#include <iterator>
#include <limits>
#include <list>
#include <map>
#include <memory>
#include <new>
#include <numeric>
#include <ostream>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <stack>
#include <streambuf>
#include <string>
#include <tuple>
#include <type_traits>
#include <typeinfo>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>



namespace Nyaan {
using ll = long long;
using i64 = long long;
using u64 = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;

template <typename T>
using V = vector<T>;
template <typename T>
using VV = vector<vector<T>>;
using vi = vector<int>;
using vl = vector<long long>;
using vd = V<double>;
using vs = V<string>;
using vvi = vector<vector<int>>;
using vvl = vector<vector<long long>>;
template <typename T>
using minpq = priority_queue<T, vector<T>, greater<T>>;

constexpr int inf = 1001001001;
constexpr long long infLL = 4004004004004004004LL;

template <typename T>
int sz(const T &t) {
 return t.size();
}


constexpr long long TEN(int n) {
 long long ret = 1, x = 10;
 for (; n; x *= x, n >>= 1) ret *= (n & 1 ? x : 1);
 return ret;
}

template <typename T, typename U>
pair<T, U> mkp(const T &t, const U &u) {
 return make_pair(t, u);
}

template <typename T>
vector<T> mkrui(const vector<T> &v, bool rev = false) {
 vector<T> ret(v.size() + 1);
 if (rev) {
 for (int i = int(v.size()) - 1; i >= 0; i--) ret[i] = v[i] + ret[i + 1];
 } else {
 for (int i = 0; i < int(v.size()); i++) ret[i + 1] = ret[i] + v[i];
 }
 return ret;
};


} 




namespace Nyaan {
__attribute__((target("popcnt"))) inline int popcnt(const u64 &a) {
 return _mm_popcnt_u64(a);
}
inline int lsb(const u64 &a) { return a ? __builtin_ctzll(a) : 64; }
inline int ctz(const u64 &a) { return a ? __builtin_ctzll(a) : 64; }
inline int msb(const u64 &a) { return a ? 63 - __builtin_clzll(a) : -1; }
template <typename T>
inline int gbit(const T &a, int i) {
 return (a >> i) & 1;
}
template <typename T>
inline void sbit(T &a, int i, bool b) {
 if (gbit(a, i) != b) a ^= T(1) << i;
}
constexpr long long PW(int n) { return 1LL << n; }
constexpr long long MSK(int n) { return (1LL << n) - 1; }
} 




namespace Nyaan {

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;
}

istream &operator>>(istream &is, __int128_t &x) {
 string S;
 is >> S;
 x = 0;
 int flag = 0;
 for (auto &c : S) {
 if (c == '-') {
 flag = true;
 continue;
 }
 x *= 10;
 x += c - '0';
 }
 if (flag) x = -x;
 return is;
}

istream &operator>>(istream &is, __uint128_t &x) {
 string S;
 is >> S;
 x = 0;
 for (auto &c : S) {
 x *= 10;
 x += c - '0';
 }
 return is;
}

ostream &operator<<(ostream &os, __int128_t x) {
 if (x == 0) return os << 0;
 if (x < 0) os << '-', x = -x;
 string S;
 while (x) S.push_back('0' + x % 10), x /= 10;
 reverse(begin(S), end(S));
 return os << S;
}
ostream &operator<<(ostream &os, __uint128_t x) {
 if (x == 0) return os << 0;
 string S;
 while (x) S.push_back('0' + x % 10), x /= 10;
 reverse(begin(S), end(S));
 return os << S;
}

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...);
}

struct IoSetupNya {
 IoSetupNya() {
 cin.tie(nullptr);
 ios::sync_with_stdio(false);
 cout << fixed << setprecision(15);
 cerr << fixed << setprecision(7);
 }
} iosetupnya;

} 





#ifdef NyaanDebug
#define trc(...) (void(0))
#else
#define trc(...) (void(0))
#endif

#ifdef NyaanLocal
#define trc2(...) (void(0))
#else
#define trc2(...) (void(0))
#endif




#define each(x, v) for (auto&& x : v)
#define each2(x, y, v) for (auto&& [x, y] : v)
#define all(v) (v).begin(), (v).end()
#define rep(i, N) for (long long i = 0; i < (long long)(N); i++)
#define repr(i, N) for (long long i = (long long)(N)-1; i >= 0; i--)
#define rep1(i, N) for (long long i = 1; i <= (long long)(N); i++)
#define repr1(i, N) for (long long i = (N); (long long)(i) > 0; i--)
#define reg(i, a, b) for (long long i = (a); i < (b); i++)
#define regr(i, a, b) for (long long i = (b)-1; i >= (a); i--)
#define fi first
#define se second
#define ini(...) \
 int __VA_ARGS__; \
 in(__VA_ARGS__)
#define inl(...) \
 long long __VA_ARGS__; \
 in(__VA_ARGS__)
#define ins(...) \
 string __VA_ARGS__; \
 in(__VA_ARGS__)
#define in2(s, t) \
 for (int i = 0; i < (int)s.size(); i++) { \
 in(s[i], t[i]); \
 }
#define in3(s, t, u) \
 for (int i = 0; i < (int)s.size(); i++) { \
 in(s[i], t[i], u[i]); \
 }
#define in4(s, t, u, v) \
 for (int i = 0; i < (int)s.size(); i++) { \
 in(s[i], t[i], u[i], v[i]); \
 }
#define die(...) \
 do { \
 Nyaan::out(__VA_ARGS__); \
 return; \
 } while (0)


namespace Nyaan {
void solve();
}
int main() { Nyaan::solve(); }






using namespace std;




using namespace std;

namespace internal {
template <typename T>
using is_broadly_integral =
 typename conditional_t<is_integral_v<T> || is_same_v<T, __int128_t> ||
 is_same_v<T, __uint128_t>,
 true_type, false_type>::type;

template <typename T>
using is_broadly_signed =
 typename conditional_t<is_signed_v<T> || is_same_v<T, __int128_t>,
 true_type, false_type>::type;

template <typename T>
using is_broadly_unsigned =
 typename conditional_t<is_unsigned_v<T> || is_same_v<T, __uint128_t>,
 true_type, false_type>::type;

#define ENABLE_VALUE(x) \
 template <typename T> \
 constexpr bool x##_v = x<T>::value;

ENABLE_VALUE(is_broadly_integral);
ENABLE_VALUE(is_broadly_signed);
ENABLE_VALUE(is_broadly_unsigned);
#undef ENABLE_VALUE

#define ENABLE_HAS_TYPE(var) \
 template <class, class = void> \
 struct has_##var : false_type {}; \
 template <class T> \
 struct has_##var<T, void_t<typename T::var>> : true_type {}; \
 template <class T> \
 constexpr auto has_##var##_v = has_##var<T>::value;

#define ENABLE_HAS_VAR(var) \
 template <class, class = void> \
 struct has_##var : false_type {}; \
 template <class T> \
 struct has_##var<T, void_t<decltype(T::var)>> : true_type {}; \
 template <class T> \
 constexpr auto has_##var##_v = has_##var<T>::value;

} 




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; }
};






__attribute__((target("sse4.2"))) inline __m128i my128_mullo_epu32(
 const __m128i &a, const __m128i &b) {
 return _mm_mullo_epi32(a, b);
}

__attribute__((target("sse4.2"))) inline __m128i my128_mulhi_epu32(
 const __m128i &a, const __m128i &b) {
 __m128i a13 = _mm_shuffle_epi32(a, 0xF5);
 __m128i b13 = _mm_shuffle_epi32(b, 0xF5);
 __m128i prod02 = _mm_mul_epu32(a, b);
 __m128i prod13 = _mm_mul_epu32(a13, b13);
 __m128i prod = _mm_unpackhi_epi64(_mm_unpacklo_epi32(prod02, prod13),
 _mm_unpackhi_epi32(prod02, prod13));
 return prod;
}

__attribute__((target("sse4.2"))) inline __m128i montgomery_mul_128(
 const __m128i &a, const __m128i &b, const __m128i &r, const __m128i &m1) {
 return _mm_sub_epi32(
 _mm_add_epi32(my128_mulhi_epu32(a, b), m1),
 my128_mulhi_epu32(my128_mullo_epu32(my128_mullo_epu32(a, b), r), m1));
}

__attribute__((target("sse4.2"))) inline __m128i montgomery_add_128(
 const __m128i &a, const __m128i &b, const __m128i &m2, const __m128i &m0) {
 __m128i ret = _mm_sub_epi32(_mm_add_epi32(a, b), m2);
 return _mm_add_epi32(_mm_and_si128(_mm_cmpgt_epi32(m0, ret), m2), ret);
}

__attribute__((target("sse4.2"))) inline __m128i montgomery_sub_128(
 const __m128i &a, const __m128i &b, const __m128i &m2, const __m128i &m0) {
 __m128i ret = _mm_sub_epi32(a, b);
 return _mm_add_epi32(_mm_and_si128(_mm_cmpgt_epi32(m0, ret), m2), ret);
}

__attribute__((target("avx2"))) inline __m256i my256_mullo_epu32(
 const __m256i &a, const __m256i &b) {
 return _mm256_mullo_epi32(a, b);
}

__attribute__((target("avx2"))) inline __m256i my256_mulhi_epu32(
 const __m256i &a, const __m256i &b) {
 __m256i a13 = _mm256_shuffle_epi32(a, 0xF5);
 __m256i b13 = _mm256_shuffle_epi32(b, 0xF5);
 __m256i prod02 = _mm256_mul_epu32(a, b);
 __m256i prod13 = _mm256_mul_epu32(a13, b13);
 __m256i prod = _mm256_unpackhi_epi64(_mm256_unpacklo_epi32(prod02, prod13),
 _mm256_unpackhi_epi32(prod02, prod13));
 return prod;
}

__attribute__((target("avx2"))) inline __m256i montgomery_mul_256(
 const __m256i &a, const __m256i &b, const __m256i &r, const __m256i &m1) {
 return _mm256_sub_epi32(
 _mm256_add_epi32(my256_mulhi_epu32(a, b), m1),
 my256_mulhi_epu32(my256_mullo_epu32(my256_mullo_epu32(a, b), r), m1));
}

__attribute__((target("avx2"))) inline __m256i montgomery_add_256(
 const __m256i &a, const __m256i &b, const __m256i &m2, const __m256i &m0) {
 __m256i ret = _mm256_sub_epi32(_mm256_add_epi32(a, b), m2);
 return _mm256_add_epi32(_mm256_and_si256(_mm256_cmpgt_epi32(m0, ret), m2),
 ret);
}

__attribute__((target("avx2"))) inline __m256i montgomery_sub_256(
 const __m256i &a, const __m256i &b, const __m256i &m2, const __m256i &m0) {
 __m256i ret = _mm256_sub_epi32(a, b);
 return _mm256_add_epi32(_mm256_and_si256(_mm256_cmpgt_epi32(m0, ret), m2),
 ret);
}

namespace ntt_inner {
using u64 = uint64_t;
constexpr uint32_t get_pr(uint32_t mod) {
 if (mod == 2) return 1;
 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;
}

constexpr int SZ_FFT_BUF = 1 << 23;
uint32_t _buf1[SZ_FFT_BUF] __attribute__((aligned(64)));
uint32_t _buf2[SZ_FFT_BUF] __attribute__((aligned(64)));
} 

template <typename mint>
struct NTT {
 static constexpr uint32_t mod = mint::get_mod();
 static constexpr uint32_t pr = ntt_inner::get_pr(mint::get_mod());
 static constexpr int level = __builtin_ctzll(mod - 1);
 mint dw[level], dy[level];
 mint *buf1, *buf2;

 constexpr NTT() {
 setwy(level);
 union raw_cast {
 mint dat;
 uint32_t _;
 };
 buf1 = &(((raw_cast *)(ntt_inner::_buf1))->dat);
 buf2 = &(((raw_cast *)(ntt_inner::_buf2))->dat);
 }

 constexpr 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[0] = dy[0] = w[1] * w[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];
 }
 }

 __attribute__((target("avx2"))) void ntt(mint *a, int n) {
 int k = n ? __builtin_ctz(n) : 0;
 if (k == 0) 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);
 if (v < 8) {
 for (int j = 0; j < v; ++j) {
 mint ajv = a[j + v];
 a[j + v] = a[j] - ajv;
 a[j] += ajv;
 }
 } else {
 const __m256i m0 = _mm256_set1_epi32(0);
 const __m256i m2 = _mm256_set1_epi32(mod + mod);
 int j0 = 0;
 int j1 = v;
 for (; j0 < v; j0 += 8, j1 += 8) {
 __m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
 __m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
 __m256i naj = montgomery_add_256(T0, T1, m2, m0);
 __m256i najv = montgomery_sub_256(T0, T1, m2, m0);
 _mm256_storeu_si256((__m256i *)(a + j0), naj);
 _mm256_storeu_si256((__m256i *)(a + j1), najv);
 }
 }
 }
 int u = 1 << (2 + (k & 1));
 int v = 1 << (k - 2 - (k & 1));
 mint one = mint(1);
 mint imag = dw[1];
 while (v) {
 if (v == 1) {
 mint ww = one, xx = one, wx = one;
 for (int jh = 0; jh < u;) {
 ww = xx * xx, wx = ww * xx;
 mint t0 = a[jh + 0], t1 = a[jh + 1] * xx;
 mint t2 = a[jh + 2] * ww, t3 = a[jh + 3] * wx;
 mint t0p2 = t0 + t2, t1p3 = t1 + t3;
 mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag;
 a[jh + 0] = t0p2 + t1p3, a[jh + 1] = t0p2 - t1p3;
 a[jh + 2] = t0m2 + t1m3, a[jh + 3] = t0m2 - t1m3;
 xx *= dw[__builtin_ctz((jh += 4))];
 }
 } else if (v == 4) {
 const __m128i m0 = _mm_set1_epi32(0);
 const __m128i m1 = _mm_set1_epi32(mod);
 const __m128i m2 = _mm_set1_epi32(mod + mod);
 const __m128i r = _mm_set1_epi32(mint::r);
 const __m128i Imag = _mm_set1_epi32(imag.a);
 mint ww = one, xx = one, wx = one;
 for (int jh = 0; jh < u;) {
 if (jh == 0) {
 int j0 = 0;
 int j1 = v;
 int j2 = j1 + v;
 int j3 = j2 + v;
 int je = v;
 for (; j0 < je; j0 += 4, j1 += 4, j2 += 4, j3 += 4) {
 const __m128i T0 = _mm_loadu_si128((__m128i *)(a + j0));
 const __m128i T1 = _mm_loadu_si128((__m128i *)(a + j1));
 const __m128i T2 = _mm_loadu_si128((__m128i *)(a + j2));
 const __m128i T3 = _mm_loadu_si128((__m128i *)(a + j3));
 const __m128i T0P2 = montgomery_add_128(T0, T2, m2, m0);
 const __m128i T1P3 = montgomery_add_128(T1, T3, m2, m0);
 const __m128i T0M2 = montgomery_sub_128(T0, T2, m2, m0);
 const __m128i T1M3 = montgomery_mul_128(
 montgomery_sub_128(T1, T3, m2, m0), Imag, r, m1);
 _mm_storeu_si128((__m128i *)(a + j0),
 montgomery_add_128(T0P2, T1P3, m2, m0));
 _mm_storeu_si128((__m128i *)(a + j1),
 montgomery_sub_128(T0P2, T1P3, m2, m0));
 _mm_storeu_si128((__m128i *)(a + j2),
 montgomery_add_128(T0M2, T1M3, m2, m0));
 _mm_storeu_si128((__m128i *)(a + j3),
 montgomery_sub_128(T0M2, T1M3, m2, m0));
 }
 } else {
 ww = xx * xx, wx = ww * xx;
 const __m128i WW = _mm_set1_epi32(ww.a);
 const __m128i WX = _mm_set1_epi32(wx.a);
 const __m128i XX = _mm_set1_epi32(xx.a);
 int j0 = jh * v;
 int j1 = j0 + v;
 int j2 = j1 + v;
 int j3 = j2 + v;
 int je = j1;
 for (; j0 < je; j0 += 4, j1 += 4, j2 += 4, j3 += 4) {
 const __m128i T0 = _mm_loadu_si128((__m128i *)(a + j0));
 const __m128i T1 = _mm_loadu_si128((__m128i *)(a + j1));
 const __m128i T2 = _mm_loadu_si128((__m128i *)(a + j2));
 const __m128i T3 = _mm_loadu_si128((__m128i *)(a + j3));
 const __m128i MT1 = montgomery_mul_128(T1, XX, r, m1);
 const __m128i MT2 = montgomery_mul_128(T2, WW, r, m1);
 const __m128i MT3 = montgomery_mul_128(T3, WX, r, m1);
 const __m128i T0P2 = montgomery_add_128(T0, MT2, m2, m0);
 const __m128i T1P3 = montgomery_add_128(MT1, MT3, m2, m0);
 const __m128i T0M2 = montgomery_sub_128(T0, MT2, m2, m0);
 const __m128i T1M3 = montgomery_mul_128(
 montgomery_sub_128(MT1, MT3, m2, m0), Imag, r, m1);
 _mm_storeu_si128((__m128i *)(a + j0),
 montgomery_add_128(T0P2, T1P3, m2, m0));
 _mm_storeu_si128((__m128i *)(a + j1),
 montgomery_sub_128(T0P2, T1P3, m2, m0));
 _mm_storeu_si128((__m128i *)(a + j2),
 montgomery_add_128(T0M2, T1M3, m2, m0));
 _mm_storeu_si128((__m128i *)(a + j3),
 montgomery_sub_128(T0M2, T1M3, m2, m0));
 }
 }
 xx *= dw[__builtin_ctz((jh += 4))];
 }
 } else {
 const __m256i m0 = _mm256_set1_epi32(0);
 const __m256i m1 = _mm256_set1_epi32(mod);
 const __m256i m2 = _mm256_set1_epi32(mod + mod);
 const __m256i r = _mm256_set1_epi32(mint::r);
 const __m256i Imag = _mm256_set1_epi32(imag.a);
 mint ww = one, xx = one, wx = one;
 for (int jh = 0; jh < u;) {
 if (jh == 0) {
 int j0 = 0;
 int j1 = v;
 int j2 = j1 + v;
 int j3 = j2 + v;
 int je = v;
 for (; j0 < je; j0 += 8, j1 += 8, j2 += 8, j3 += 8) {
 const __m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
 const __m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
 const __m256i T2 = _mm256_loadu_si256((__m256i *)(a + j2));
 const __m256i T3 = _mm256_loadu_si256((__m256i *)(a + j3));
 const __m256i T0P2 = montgomery_add_256(T0, T2, m2, m0);
 const __m256i T1P3 = montgomery_add_256(T1, T3, m2, m0);
 const __m256i T0M2 = montgomery_sub_256(T0, T2, m2, m0);
 const __m256i T1M3 = montgomery_mul_256(
 montgomery_sub_256(T1, T3, m2, m0), Imag, r, m1);
 _mm256_storeu_si256((__m256i *)(a + j0),
 montgomery_add_256(T0P2, T1P3, m2, m0));
 _mm256_storeu_si256((__m256i *)(a + j1),
 montgomery_sub_256(T0P2, T1P3, m2, m0));
 _mm256_storeu_si256((__m256i *)(a + j2),
 montgomery_add_256(T0M2, T1M3, m2, m0));
 _mm256_storeu_si256((__m256i *)(a + j3),
 montgomery_sub_256(T0M2, T1M3, m2, m0));
 }
 } else {
 ww = xx * xx, wx = ww * xx;
 const __m256i WW = _mm256_set1_epi32(ww.a);
 const __m256i WX = _mm256_set1_epi32(wx.a);
 const __m256i XX = _mm256_set1_epi32(xx.a);
 int j0 = jh * v;
 int j1 = j0 + v;
 int j2 = j1 + v;
 int j3 = j2 + v;
 int je = j1;
 for (; j0 < je; j0 += 8, j1 += 8, j2 += 8, j3 += 8) {
 const __m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
 const __m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
 const __m256i T2 = _mm256_loadu_si256((__m256i *)(a + j2));
 const __m256i T3 = _mm256_loadu_si256((__m256i *)(a + j3));
 const __m256i MT1 = montgomery_mul_256(T1, XX, r, m1);
 const __m256i MT2 = montgomery_mul_256(T2, WW, r, m1);
 const __m256i MT3 = montgomery_mul_256(T3, WX, r, m1);
 const __m256i T0P2 = montgomery_add_256(T0, MT2, m2, m0);
 const __m256i T1P3 = montgomery_add_256(MT1, MT3, m2, m0);
 const __m256i T0M2 = montgomery_sub_256(T0, MT2, m2, m0);
 const __m256i T1M3 = montgomery_mul_256(
 montgomery_sub_256(MT1, MT3, m2, m0), Imag, r, m1);
 _mm256_storeu_si256((__m256i *)(a + j0),
 montgomery_add_256(T0P2, T1P3, m2, m0));
 _mm256_storeu_si256((__m256i *)(a + j1),
 montgomery_sub_256(T0P2, T1P3, m2, m0));
 _mm256_storeu_si256((__m256i *)(a + j2),
 montgomery_add_256(T0M2, T1M3, m2, m0));
 _mm256_storeu_si256((__m256i *)(a + j3),
 montgomery_sub_256(T0M2, T1M3, m2, m0));
 }
 }
 xx *= dw[__builtin_ctz((jh += 4))];
 }
 }
 u <<= 2;
 v >>= 2;
 }
 }

 __attribute__((target("avx2"))) void intt(mint *a, int n,
 int normalize = true) {
 int k = n ? __builtin_ctz(n) : 0;
 if (k == 0) return;
 if (k == 1) {
 mint a1 = a[1];
 a[1] = a[0] - a[1];
 a[0] = a[0] + a1;
 if (normalize) {
 a[0] *= mint(2).inverse();
 a[1] *= mint(2).inverse();
 }
 return;
 }
 int u = 1 << (k - 2);
 int v = 1;
 mint one = mint(1);
 mint imag = dy[1];
 while (u) {
 if (v == 1) {
 mint ww = one, xx = one, yy = one;
 u <<= 2;
 for (int jh = 0; jh < u;) {
 ww = xx * xx, yy = xx * imag;
 mint t0 = a[jh + 0], t1 = a[jh + 1];
 mint t2 = a[jh + 2], t3 = a[jh + 3];
 mint t0p1 = t0 + t1, t2p3 = t2 + t3;
 mint t0m1 = (t0 - t1) * xx, t2m3 = (t2 - t3) * yy;
 a[jh + 0] = t0p1 + t2p3, a[jh + 2] = (t0p1 - t2p3) * ww;
 a[jh + 1] = t0m1 + t2m3, a[jh + 3] = (t0m1 - t2m3) * ww;
 xx *= dy[__builtin_ctz(jh += 4)];
 }
 } else if (v == 4) {
 const __m128i m0 = _mm_set1_epi32(0);
 const __m128i m1 = _mm_set1_epi32(mod);
 const __m128i m2 = _mm_set1_epi32(mod + mod);
 const __m128i r = _mm_set1_epi32(mint::r);
 const __m128i Imag = _mm_set1_epi32(imag.a);
 mint ww = one, xx = one, yy = one;
 u <<= 2;
 for (int jh = 0; jh < u;) {
 if (jh == 0) {
 int j0 = 0;
 int j1 = v;
 int j2 = v + v;
 int j3 = j2 + v;
 for (; j0 < v; j0 += 4, j1 += 4, j2 += 4, j3 += 4) {
 const __m128i T0 = _mm_loadu_si128((__m128i *)(a + j0));
 const __m128i T1 = _mm_loadu_si128((__m128i *)(a + j1));
 const __m128i T2 = _mm_loadu_si128((__m128i *)(a + j2));
 const __m128i T3 = _mm_loadu_si128((__m128i *)(a + j3));
 const __m128i T0P1 = montgomery_add_128(T0, T1, m2, m0);
 const __m128i T2P3 = montgomery_add_128(T2, T3, m2, m0);
 const __m128i T0M1 = montgomery_sub_128(T0, T1, m2, m0);
 const __m128i T2M3 = montgomery_mul_128(
 montgomery_sub_128(T2, T3, m2, m0), Imag, r, m1);
 _mm_storeu_si128((__m128i *)(a + j0),
 montgomery_add_128(T0P1, T2P3, m2, m0));
 _mm_storeu_si128((__m128i *)(a + j2),
 montgomery_sub_128(T0P1, T2P3, m2, m0));
 _mm_storeu_si128((__m128i *)(a + j1),
 montgomery_add_128(T0M1, T2M3, m2, m0));
 _mm_storeu_si128((__m128i *)(a + j3),
 montgomery_sub_128(T0M1, T2M3, m2, m0));
 }
 } else {
 ww = xx * xx, yy = xx * imag;
 const __m128i WW = _mm_set1_epi32(ww.a);
 const __m128i XX = _mm_set1_epi32(xx.a);
 const __m128i YY = _mm_set1_epi32(yy.a);
 int j0 = jh * v;
 int j1 = j0 + v;
 int j2 = j1 + v;
 int j3 = j2 + v;
 int je = j1;
 for (; j0 < je; j0 += 4, j1 += 4, j2 += 4, j3 += 4) {
 const __m128i T0 = _mm_loadu_si128((__m128i *)(a + j0));
 const __m128i T1 = _mm_loadu_si128((__m128i *)(a + j1));
 const __m128i T2 = _mm_loadu_si128((__m128i *)(a + j2));
 const __m128i T3 = _mm_loadu_si128((__m128i *)(a + j3));
 const __m128i T0P1 = montgomery_add_128(T0, T1, m2, m0);
 const __m128i T2P3 = montgomery_add_128(T2, T3, m2, m0);
 const __m128i T0M1 = montgomery_mul_128(
 montgomery_sub_128(T0, T1, m2, m0), XX, r, m1);
 __m128i T2M3 = montgomery_mul_128(
 montgomery_sub_128(T2, T3, m2, m0), YY, r, m1);
 _mm_storeu_si128((__m128i *)(a + j0),
 montgomery_add_128(T0P1, T2P3, m2, m0));
 _mm_storeu_si128(
 (__m128i *)(a + j2),
 montgomery_mul_128(montgomery_sub_128(T0P1, T2P3, m2, m0), WW,
 r, m1));
 _mm_storeu_si128((__m128i *)(a + j1),
 montgomery_add_128(T0M1, T2M3, m2, m0));
 _mm_storeu_si128(
 (__m128i *)(a + j3),
 montgomery_mul_128(montgomery_sub_128(T0M1, T2M3, m2, m0), WW,
 r, m1));
 }
 }
 xx *= dy[__builtin_ctz(jh += 4)];
 }
 } else {
 const __m256i m0 = _mm256_set1_epi32(0);
 const __m256i m1 = _mm256_set1_epi32(mod);
 const __m256i m2 = _mm256_set1_epi32(mod + mod);
 const __m256i r = _mm256_set1_epi32(mint::r);
 const __m256i Imag = _mm256_set1_epi32(imag.a);
 mint ww = one, xx = one, yy = one;
 u <<= 2;
 for (int jh = 0; jh < u;) {
 if (jh == 0) {
 int j0 = 0;
 int j1 = v;
 int j2 = v + v;
 int j3 = j2 + v;
 for (; j0 < v; j0 += 8, j1 += 8, j2 += 8, j3 += 8) {
 const __m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
 const __m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
 const __m256i T2 = _mm256_loadu_si256((__m256i *)(a + j2));
 const __m256i T3 = _mm256_loadu_si256((__m256i *)(a + j3));
 const __m256i T0P1 = montgomery_add_256(T0, T1, m2, m0);
 const __m256i T2P3 = montgomery_add_256(T2, T3, m2, m0);
 const __m256i T0M1 = montgomery_sub_256(T0, T1, m2, m0);
 const __m256i T2M3 = montgomery_mul_256(
 montgomery_sub_256(T2, T3, m2, m0), Imag, r, m1);
 _mm256_storeu_si256((__m256i *)(a + j0),
 montgomery_add_256(T0P1, T2P3, m2, m0));
 _mm256_storeu_si256((__m256i *)(a + j2),
 montgomery_sub_256(T0P1, T2P3, m2, m0));
 _mm256_storeu_si256((__m256i *)(a + j1),
 montgomery_add_256(T0M1, T2M3, m2, m0));
 _mm256_storeu_si256((__m256i *)(a + j3),
 montgomery_sub_256(T0M1, T2M3, m2, m0));
 }
 } else {
 ww = xx * xx, yy = xx * imag;
 const __m256i WW = _mm256_set1_epi32(ww.a);
 const __m256i XX = _mm256_set1_epi32(xx.a);
 const __m256i YY = _mm256_set1_epi32(yy.a);
 int j0 = jh * v;
 int j1 = j0 + v;
 int j2 = j1 + v;
 int j3 = j2 + v;
 int je = j1;
 for (; j0 < je; j0 += 8, j1 += 8, j2 += 8, j3 += 8) {
 const __m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
 const __m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
 const __m256i T2 = _mm256_loadu_si256((__m256i *)(a + j2));
 const __m256i T3 = _mm256_loadu_si256((__m256i *)(a + j3));
 const __m256i T0P1 = montgomery_add_256(T0, T1, m2, m0);
 const __m256i T2P3 = montgomery_add_256(T2, T3, m2, m0);
 const __m256i T0M1 = montgomery_mul_256(
 montgomery_sub_256(T0, T1, m2, m0), XX, r, m1);
 const __m256i T2M3 = montgomery_mul_256(
 montgomery_sub_256(T2, T3, m2, m0), YY, r, m1);
 _mm256_storeu_si256((__m256i *)(a + j0),
 montgomery_add_256(T0P1, T2P3, m2, m0));
 _mm256_storeu_si256(
 (__m256i *)(a + j2),
 montgomery_mul_256(montgomery_sub_256(T0P1, T2P3, m2, m0), WW,
 r, m1));
 _mm256_storeu_si256((__m256i *)(a + j1),
 montgomery_add_256(T0M1, T2M3, m2, m0));
 _mm256_storeu_si256(
 (__m256i *)(a + j3),
 montgomery_mul_256(montgomery_sub_256(T0M1, T2M3, m2, m0), WW,
 r, m1));
 }
 }
 xx *= dy[__builtin_ctz(jh += 4)];
 }
 }
 u >>= 4;
 v <<= 2;
 }
 if (k & 1) {
 v = 1 << (k - 1);
 if (v < 8) {
 for (int j = 0; j < v; ++j) {
 mint ajv = a[j] - a[j + v];
 a[j] += a[j + v];
 a[j + v] = ajv;
 }
 } else {
 const __m256i m0 = _mm256_set1_epi32(0);
 const __m256i m2 = _mm256_set1_epi32(mod + mod);
 int j0 = 0;
 int j1 = v;
 for (; j0 < v; j0 += 8, j1 += 8) {
 const __m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
 const __m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
 __m256i naj = montgomery_add_256(T0, T1, m2, m0);
 __m256i najv = montgomery_sub_256(T0, T1, m2, m0);
 _mm256_storeu_si256((__m256i *)(a + j0), naj);
 _mm256_storeu_si256((__m256i *)(a + j1), najv);
 }
 }
 }
 if (normalize) {
 mint invn = mint(n).inverse();
 for (int i = 0; i < n; i++) a[i] *= invn;
 }
 }

 __attribute__((target("avx2"))) void inplace_multiply(
 int l1, int l2, int zero_padding = true) {
 int l = l1 + l2 - 1;
 int M = 4;
 while (M < l) M <<= 1;
 if (zero_padding) {
 for (int i = l1; i < M; i++) ntt_inner::_buf1[i] = 0;
 for (int i = l2; i < M; i++) ntt_inner::_buf2[i] = 0;
 }
 const __m256i m0 = _mm256_set1_epi32(0);
 const __m256i m1 = _mm256_set1_epi32(mod);
 const __m256i r = _mm256_set1_epi32(mint::r);
 const __m256i N2 = _mm256_set1_epi32(mint::n2);
 for (int i = 0; i < l1; i += 8) {
 __m256i a = _mm256_loadu_si256((__m256i *)(ntt_inner::_buf1 + i));
 __m256i b = montgomery_mul_256(a, N2, r, m1);
 _mm256_storeu_si256((__m256i *)(ntt_inner::_buf1 + i), b);
 }
 for (int i = 0; i < l2; i += 8) {
 __m256i a = _mm256_loadu_si256((__m256i *)(ntt_inner::_buf2 + i));
 __m256i b = montgomery_mul_256(a, N2, r, m1);
 _mm256_storeu_si256((__m256i *)(ntt_inner::_buf2 + i), b);
 }
 ntt(buf1, M);
 ntt(buf2, M);
 for (int i = 0; i < M; i += 8) {
 __m256i a = _mm256_loadu_si256((__m256i *)(ntt_inner::_buf1 + i));
 __m256i b = _mm256_loadu_si256((__m256i *)(ntt_inner::_buf2 + i));
 __m256i c = montgomery_mul_256(a, b, r, m1);
 _mm256_storeu_si256((__m256i *)(ntt_inner::_buf1 + i), c);
 }
 intt(buf1, M, false);
 const __m256i INVM = _mm256_set1_epi32((mint(M).inverse()).a);
 for (int i = 0; i < l; i += 8) {
 __m256i a = _mm256_loadu_si256((__m256i *)(ntt_inner::_buf1 + i));
 __m256i b = montgomery_mul_256(a, INVM, r, m1);
 __m256i c = my256_mulhi_epu32(my256_mullo_epu32(b, r), m1);
 __m256i d = _mm256_and_si256(_mm256_cmpgt_epi32(c, m0), m1);
 __m256i e = _mm256_sub_epi32(d, c);
 _mm256_storeu_si256((__m256i *)(ntt_inner::_buf1 + i), e);
 }
 }

 void ntt(vector<mint> &a) {
 int M = (int)a.size();
 for (int i = 0; i < M; i++) buf1[i].a = a[i].a;
 ntt(buf1, M);
 for (int i = 0; i < M; i++) a[i].a = buf1[i].a;
 }

 void intt(vector<mint> &a) {
 int M = (int)a.size();
 for (int i = 0; i < M; i++) buf1[i].a = a[i].a;
 intt(buf1, M, true);
 for (int i = 0; i < M; i++) a[i].a = buf1[i].a;
 }

 vector<mint> multiply(const vector<mint> &a, const vector<mint> &b) {
 if (a.size() == 0 && b.size() == 0) return vector<mint>{};
 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;
 }
 assert(l <= ntt_inner::SZ_FFT_BUF);
 int M = 4;
 while (M < l) M <<= 1;
 for (int i = 0; i < (int)a.size(); ++i) buf1[i].a = a[i].a;
 for (int i = (int)a.size(); i < M; ++i) buf1[i].a = 0;
 for (int i = 0; i < (int)b.size(); ++i) buf2[i].a = b[i].a;
 for (int i = (int)b.size(); i < M; ++i) buf2[i].a = 0;
 ntt(buf1, M);
 ntt(buf2, M);
 for (int i = 0; i < M; ++i)
 buf1[i].a = mint::reduce(uint64_t(buf1[i].a) * buf2[i].a);
 intt(buf1, M, false);
 vector<mint> s(l);
 mint invm = mint(M).inverse();
 for (int i = 0; i < l; ++i) s[i] = buf1[i] * invm;
 return s;
 }

 void ntt_doubling(vector<mint> &a) {
 int M = (int)a.size();
 for (int i = 0; i < M; i++) buf1[i].a = a[i].a;
 intt(buf1, M);
 mint r = 1, zeta = mint(pr).pow((mint::get_mod() - 1) / (M << 1));
 for (int i = 0; i < M; i++) buf1[i] *= r, r *= zeta;
 ntt(buf1, M);
 a.resize(2 * M);
 for (int i = 0; i < M; i++) a[M + i].a = buf1[i].a;
 }
};


namespace ArbitraryNTT {
using i64 = int64_t;
using u128 = __uint128_t;
constexpr int32_t m0 = 167772161;
constexpr int32_t m1 = 469762049;
constexpr int32_t m2 = 754974721;
using mint0 = LazyMontgomeryModInt<m0>;
using mint1 = LazyMontgomeryModInt<m1>;
using mint2 = LazyMontgomeryModInt<m2>;
constexpr int r01 = mint1(m0).inverse().get();
constexpr int r02 = mint2(m0).inverse().get();
constexpr int r12 = mint2(m1).inverse().get();
constexpr int r02r12 = i64(r02) * r12 % m2;
constexpr i64 w1 = m0;
constexpr i64 w2 = i64(m0) * m1;

template <typename T, typename submint>
vector<submint> mul(const vector<T> &a, const vector<T> &b) {
 static NTT<submint> ntt;
 vector<submint> s(a.size()), t(b.size());
 for (int i = 0; i < (int)a.size(); ++i) s[i] = i64(a[i] % submint::get_mod());
 for (int i = 0; i < (int)b.size(); ++i) t[i] = i64(b[i] % submint::get_mod());
 return ntt.multiply(s, t);
}

template <typename T>
vector<int> multiply(const vector<T> &s, const vector<T> &t, int mod) {
 auto d0 = mul<T, mint0>(s, t);
 auto d1 = mul<T, mint1>(s, t);
 auto d2 = mul<T, mint2>(s, t);
 int n = d0.size();
 vector<int> ret(n);
 const int W1 = w1 % mod;
 const int W2 = w2 % mod;
 for (int i = 0; i < n; i++) {
 int n1 = d1[i].get(), n2 = d2[i].get(), a = d0[i].get();
 int b = i64(n1 + m1 - a) * r01 % m1;
 int c = (i64(n2 + m2 - a) * r02r12 + i64(m2 - b) * r12) % m2;
 ret[i] = (i64(a) + i64(b) * W1 + i64(c) * W2) % mod;
 }
 return ret;
}

template <typename mint>
vector<mint> multiply(const vector<mint> &a, const vector<mint> &b) {
 if (a.size() == 0 && b.size() == 0) return {};
 if (min<int>(a.size(), b.size()) < 128) {
 vector<mint> ret(a.size() + b.size() - 1);
 for (int i = 0; i < (int)a.size(); ++i)
 for (int j = 0; j < (int)b.size(); ++j) ret[i + j] += a[i] * b[j];
 return ret;
 }
 vector<int> s(a.size()), t(b.size());
 for (int i = 0; i < (int)a.size(); ++i) s[i] = a[i].get();
 for (int i = 0; i < (int)b.size(); ++i) t[i] = b[i].get();
 vector<int> u = multiply<int>(s, t, mint::get_mod());
 vector<mint> ret(u.size());
 for (int i = 0; i < (int)u.size(); ++i) ret[i] = mint(u[i]);
 return ret;
}

template <typename T>
vector<u128> multiply_u128(const vector<T> &s, const vector<T> &t) {
 if (s.size() == 0 && t.size() == 0) return {};
 if (min<int>(s.size(), t.size()) < 128) {
 vector<u128> ret(s.size() + t.size() - 1);
 for (int i = 0; i < (int)s.size(); ++i)
 for (int j = 0; j < (int)t.size(); ++j) ret[i + j] += i64(s[i]) * t[j];
 return ret;
 }
 auto d0 = mul<T, mint0>(s, t);
 auto d1 = mul<T, mint1>(s, t);
 auto d2 = mul<T, mint2>(s, t);
 int n = d0.size();
 vector<u128> ret(n);
 for (int i = 0; i < n; i++) {
 i64 n1 = d1[i].get(), n2 = d2[i].get();
 i64 a = d0[i].get();
 i64 b = (n1 + m1 - a) * r01 % m1;
 i64 c = ((n2 + m2 - a) * r02r12 + (m2 - b) * r12) % m2;
 ret[i] = a + b * w1 + u128(c) * w2;
 }
 return ret;
}
} 


namespace MultiPrecisionIntegerImpl {
struct TENS {
 static constexpr int offset = 30;
 constexpr TENS() : _tend() {
 _tend[offset] = 1;
 for (int i = 1; i <= offset; i++) {
 _tend[offset + i] = _tend[offset + i - 1] * 10.0;
 _tend[offset - i] = 1.0 / _tend[offset + i];
 }
 }
 long double ten_ld(int n) const {
 assert(-offset <= n and n <= offset);
 return _tend[n + offset];
 }

 private:
 long double _tend[offset * 2 + 1];
};
} 


struct MultiPrecisionInteger {
 using M = MultiPrecisionInteger;
 inline constexpr static MultiPrecisionIntegerImpl::TENS tens = {};

 static constexpr int D = 1000000000;
 static constexpr int logD = 9;
 bool neg;
 vector<int> dat;

 MultiPrecisionInteger() : neg(false), dat() {}

 MultiPrecisionInteger(bool n, const vector<int>& d) : neg(n), dat(d) {}

 template <typename I,
 enable_if_t<internal::is_broadly_integral_v<I>>* = nullptr>
 MultiPrecisionInteger(I x) : neg(false) {
 if constexpr (internal::is_broadly_signed_v<I>) {
 if (x < 0) neg = true, x = -x;
 }
 while (x) dat.push_back(x % D), x /= D;
 }

 MultiPrecisionInteger(const string& S) : neg(false) {
 assert(!S.empty());
 if (S.size() == 1u && S[0] == '0') return;
 int l = 0;
 if (S[0] == '-') ++l, neg = true;
 for (int ie = S.size(); l < ie; ie -= logD) {
 int is = max(l, ie - logD);
 long long x = 0;
 for (int i = is; i < ie; i++) x = x * 10 + S[i] - '0';
 dat.push_back(x);
 }
 }

 friend M operator+(const M& lhs, const M& rhs) {
 if (lhs.neg == rhs.neg) return {lhs.neg, _add(lhs.dat, rhs.dat)};
 if (_leq(lhs.dat, rhs.dat)) {
 
 auto c = _sub(rhs.dat, lhs.dat);
 bool n = _is_zero(c) ? false : rhs.neg;
 return {n, c};
 }
 auto c = _sub(lhs.dat, rhs.dat);
 bool n = _is_zero(c) ? false : lhs.neg;
 return {n, c};
 }
 friend M operator-(const M& lhs, const M& rhs) { return lhs + (-rhs); }

 friend M operator*(const M& lhs, const M& rhs) {
 auto c = _mul(lhs.dat, rhs.dat);
 bool n = _is_zero(c) ? false : (lhs.neg ^ rhs.neg);
 return {n, c};
 }
 friend pair<M, M> divmod(const M& lhs, const M& rhs) {
 auto dm = _divmod_newton(lhs.dat, rhs.dat);
 bool dn = _is_zero(dm.first) ? false : lhs.neg != rhs.neg;
 bool mn = _is_zero(dm.second) ? false : lhs.neg;
 return {M{dn, dm.first}, M{mn, dm.second}};
 }
 friend M operator/(const M& lhs, const M& rhs) {
 return divmod(lhs, rhs).first;
 }
 friend M operator%(const M& lhs, const M& rhs) {
 return divmod(lhs, rhs).second;
 }

 M& operator+=(const M& rhs) { return (*this) = (*this) + rhs; }
 M& operator-=(const M& rhs) { return (*this) = (*this) - rhs; }
 M& operator*=(const M& rhs) { return (*this) = (*this) * rhs; }
 M& operator/=(const M& rhs) { return (*this) = (*this) / rhs; }
 M& operator%=(const M& rhs) { return (*this) = (*this) % rhs; }

 M operator-() const {
 if (is_zero()) return *this;
 return {!neg, dat};
 }
 M operator+() const { return *this; }
 friend M abs(const M& m) { return {false, m.dat}; }
 bool is_zero() const { return _is_zero(dat); }

 friend bool operator==(const M& lhs, const M& rhs) {
 return lhs.neg == rhs.neg && lhs.dat == rhs.dat;
 }
 friend bool operator!=(const M& lhs, const M& rhs) {
 return lhs.neg != rhs.neg || lhs.dat != rhs.dat;
 }
 friend bool operator<(const M& lhs, const M& rhs) {
 if (lhs == rhs) return false;
 return _neq_lt(lhs, rhs);
 }
 friend bool operator<=(const M& lhs, const M& rhs) {
 if (lhs == rhs) return true;
 return _neq_lt(lhs, rhs);
 }
 friend bool operator>(const M& lhs, const M& rhs) {
 if (lhs == rhs) return false;
 return _neq_lt(rhs, lhs);
 }
 friend bool operator>=(const M& lhs, const M& rhs) {
 if (lhs == rhs) return true;
 return _neq_lt(rhs, lhs);
 }

 
 
 pair<long double, int> dfp() const {
 if (is_zero()) return {0, 0};
 int l = max<int>(0, _size() - 3);
 int b = logD * l;
 string prefix{};
 for (int i = _size() - 1; i >= l; i--) {
 prefix += _itos(dat[i], i != _size() - 1);
 }
 b += prefix.size() - 1;
 long double a = 0;
 for (auto& c : prefix) a = a * 10.0 + (c - '0');
 a *= tens.ten_ld(-((int)prefix.size()) + 1);
 a = clamp<long double>(a, 1.0, nextafterl(10.0, 1.0));
 if (neg) a = -a;
 return {a, b};
 }
 string to_string() const {
 if (is_zero()) return "0";
 string res;
 if (neg) res.push_back('-');
 for (int i = _size() - 1; i >= 0; i--) {
 res += _itos(dat[i], i != _size() - 1);
 }
 return res;
 }
 long double to_ld() const {
 auto [a, b] = dfp();
 if (-tens.offset <= b and b <= tens.offset) {
 return a * tens.ten_ld(b);
 }
 return a * powl(10, b);
 }
 long long to_ll() const {
 long long res = _to_ll(dat);
 return neg ? -res : res;
 }
 __int128_t to_i128() const {
 __int128_t res = _to_i128(dat);
 return neg ? -res : res;
 }

 friend istream& operator>>(istream& is, M& m) {
 string s;
 is >> s;
 m = M{s};
 return is;
 }

 friend ostream& operator<<(ostream& os, const M& m) {
 return os << m.to_string();
 }

 
 static void _test_private_function(const M&, const M&);

 private:
 
 int _size() const { return dat.size(); }
 
 static bool _eq(const vector<int>& a, const vector<int>& b) { return a == b; }
 
 static bool _lt(const vector<int>& a, const vector<int>& b) {
 if (a.size() != b.size()) return a.size() < b.size();
 for (int i = a.size() - 1; i >= 0; i--) {
 if (a[i] != b[i]) return a[i] < b[i];
 }
 return false;
 }
 
 static bool _leq(const vector<int>& a, const vector<int>& b) {
 return _eq(a, b) || _lt(a, b);
 }
 
 static bool _neq_lt(const M& lhs, const M& rhs) {
 assert(lhs != rhs);
 if (lhs.neg != rhs.neg) return lhs.neg;
 bool f = _lt(lhs.dat, rhs.dat);
 if (f) return !lhs.neg;
 return lhs.neg;
 }
 
 static bool _is_zero(const vector<int>& a) { return a.empty(); }
 
 static bool _is_one(const vector<int>& a) {
 return (int)a.size() == 1 && a[0] == 1;
 }
 
 static void _shrink(vector<int>& a) {
 while (a.size() && a.back() == 0) a.pop_back();
 }
 
 void _shrink() {
 while (_size() && dat.back() == 0) dat.pop_back();
 }
 
 static vector<int> _add(const vector<int>& a, const vector<int>& b) {
 vector<int> c(max(a.size(), b.size()) + 1);
 for (int i = 0; i < (int)a.size(); i++) c[i] += a[i];
 for (int i = 0; i < (int)b.size(); i++) c[i] += b[i];
 for (int i = 0; i < (int)c.size() - 1; i++) {
 if (c[i] >= D) c[i] -= D, c[i + 1]++;
 }
 _shrink(c);
 return c;
 }
 
 static vector<int> _sub(const vector<int>& a, const vector<int>& b) {
 assert(_leq(b, a));
 vector<int> c{a};
 int borrow = 0;
 for (int i = 0; i < (int)a.size(); i++) {
 if (i < (int)b.size()) borrow += b[i];
 c[i] -= borrow;
 borrow = 0;
 if (c[i] < 0) c[i] += D, borrow = 1;
 }
 assert(borrow == 0);
 _shrink(c);
 return c;
 }
 
 static vector<int> _mul_fft(const vector<int>& a, const vector<int>& b) {
 if (a.empty() || b.empty()) return {};
 auto m = ArbitraryNTT::multiply_u128(a, b);
 vector<int> c;
 c.reserve(m.size() + 3);
 __uint128_t x = 0;
 for (int i = 0;; i++) {
 if (i >= (int)m.size() && x == 0) break;
 if (i < (int)m.size()) x += m[i];
 c.push_back(x % D);
 x /= D;
 }
 _shrink(c);
 return c;
 }
 
 static vector<int> _mul_naive(const vector<int>& a, const vector<int>& b) {
 if (a.empty() || b.empty()) return {};
 vector<long long> prod(a.size() + b.size() - 1 + 1);
 for (int i = 0; i < (int)a.size(); i++) {
 for (int j = 0; j < (int)b.size(); j++) {
 long long p = 1LL * a[i] * b[j];
 prod[i + j] += p;
 if (prod[i + j] >= (4LL * D * D)) {
 prod[i + j] -= 4LL * D * D;
 prod[i + j + 1] += 4LL * D;
 }
 }
 }
 vector<int> c(prod.size() + 1);
 long long x = 0;
 int i = 0;
 for (; i < (int)prod.size(); i++) x += prod[i], c[i] = x % D, x /= D;
 while (x) c[i] = x % D, x /= D, i++;
 _shrink(c);
 return c;
 }
 
 static vector<int> _mul(const vector<int>& a, const vector<int>& b) {
 if (_is_zero(a) || _is_zero(b)) return {};
 if (_is_one(a)) return b;
 if (_is_one(b)) return a;
 if (min<int>(a.size(), b.size()) <= 128) {
 return a.size() < b.size() ? _mul_naive(b, a) : _mul_naive(a, b);
 }
 return _mul_fft(a, b);
 }
 
 static pair<vector<int>, vector<int>> _divmod_li(const vector<int>& a,
 const vector<int>& b) {
 assert(0 <= (int)a.size() && (int)a.size() <= 2);
 assert((int)b.size() == 1);
 long long va = _to_ll(a);
 int vb = b[0];
 return {_integer_to_vec(va / vb), _integer_to_vec(va % vb)};
 }
 
 static pair<vector<int>, vector<int>> _divmod_ll(const vector<int>& a,
 const vector<int>& b) {
 assert(0 <= (int)a.size() && (int)a.size() <= 2);
 assert(1 <= (int)b.size() && (int)b.size() <= 2);
 long long va = _to_ll(a), vb = _to_ll(b);
 return {_integer_to_vec(va / vb), _integer_to_vec(va % vb)};
 }
 
 static pair<vector<int>, vector<int>> _divmod_1e9(const vector<int>& a,
 const vector<int>& b) {
 assert((int)b.size() == 1);
 if (b[0] == 1) return {a, {}};
 if ((int)a.size() <= 2) return _divmod_li(a, b);
 vector<int> quo(a.size());
 long long d = 0;
 int b0 = b[0];
 for (int i = a.size() - 1; i >= 0; i--) {
 d = d * D + a[i];
 assert(d < 1LL * D * b0);
 int q = d / b0, r = d % b0;
 quo[i] = q, d = r;
 }
 _shrink(quo);
 return {quo, d ? vector<int>{int(d)} : vector<int>{}};
 }
 
 static pair<vector<int>, vector<int>> _divmod_naive(const vector<int>& a,
 const vector<int>& b) {
 if (_is_zero(b)) {
 cerr << "Divide by Zero Exception" << endl;
 exit(1);
 }
 assert(1 <= (int)b.size());
 if ((int)b.size() == 1) return _divmod_1e9(a, b);
 if (max<int>(a.size(), b.size()) <= 2) return _divmod_ll(a, b);
 if (_lt(a, b)) return {{}, a};
 
 int norm = D / (b.back() + 1);
 vector<int> x = _mul(a, {norm});
 vector<int> y = _mul(b, {norm});
 int yb = y.back();
 vector<int> quo(x.size() - y.size() + 1);
 vector<int> rem(x.end() - y.size(), x.end());
 for (int i = quo.size() - 1; i >= 0; i--) {
 if (rem.size() < y.size()) {
 
 } else if (rem.size() == y.size()) {
 if (_leq(y, rem)) {
 quo[i] = 1, rem = _sub(rem, y);
 }
 } else {
 assert(y.size() + 1 == rem.size());
 long long rb = 1LL * rem[rem.size() - 1] * D + rem[rem.size() - 2];
 int q = rb / yb;
 vector<int> yq = _mul(y, {q});
 
 while (_lt(rem, yq)) q--, yq = _sub(yq, y);
 rem = _sub(rem, yq);
 while (_leq(y, rem)) q++, rem = _sub(rem, y);
 quo[i] = q;
 }
 if (i) rem.insert(begin(rem), x[i - 1]);
 }
 _shrink(quo), _shrink(rem);
 auto [q2, r2] = _divmod_1e9(rem, {norm});
 assert(_is_zero(r2));
 return {quo, q2};
 }

 
 static pair<vector<int>, vector<int>> _divmod_dc(const vector<int>& a,
 const vector<int>& b);

 
 static vector<int> _calc_inv(const vector<int>& a, int deg) {
 assert(!a.empty() && D / 2 <= a.back() and a.back() < D);
 int k = deg, c = a.size();
 while (k > 64) k = (k + 1) / 2;
 vector<int> z(c + k + 1);
 z.back() = 1;
 z = _divmod_naive(z, a).first;
 while (k < deg) {
 vector<int> s = _mul(z, z);
 s.insert(begin(s), 0);
 int d = min(c, 2 * k + 1);
 vector<int> t{end(a) - d, end(a)}, u = _mul(s, t);
 u.erase(begin(u), begin(u) + d);
 vector<int> w(k + 1), w2 = _add(z, z);
 copy(begin(w2), end(w2), back_inserter(w));
 z = _sub(w, u);
 z.erase(begin(z));
 k *= 2;
 }
 z.erase(begin(z), begin(z) + k - deg);
 return z;
 }

 static pair<vector<int>, vector<int>> _divmod_newton(const vector<int>& a,
 const vector<int>& b) {
 if (_is_zero(b)) {
 cerr << "Divide by Zero Exception" << endl;
 exit(1);
 }
 if ((int)b.size() <= 64) return _divmod_naive(a, b);
 if ((int)a.size() - (int)b.size() <= 64) return _divmod_naive(a, b);
 int norm = D / (b.back() + 1);
 vector<int> x = _mul(a, {norm});
 vector<int> y = _mul(b, {norm});
 int s = x.size(), t = y.size();
 int deg = s - t + 2;
 vector<int> z = _calc_inv(y, deg);
 vector<int> q = _mul(x, z);
 q.erase(begin(q), begin(q) + t + deg);
 vector<int> yq = _mul(y, {q});
 while (_lt(x, yq)) q = _sub(q, {1}), yq = _sub(yq, y);
 vector<int> r = _sub(x, yq);
 while (_leq(y, r)) q = _add(q, {1}), r = _sub(r, y);
 _shrink(q), _shrink(r);
 auto [q2, r2] = _divmod_1e9(r, {norm});
 assert(_is_zero(r2));
 return {q, q2};
 }

 
 
 static string _itos(int x, bool zero_padding) {
 assert(0 <= x && x < D);
 string res;
 for (int i = 0; i < logD; i++) {
 res.push_back('0' + x % 10), x /= 10;
 }
 if (!zero_padding) {
 while (res.size() && res.back() == '0') res.pop_back();
 assert(!res.empty());
 }
 reverse(begin(res), end(res));
 return res;
 }

 
 template <typename I,
 enable_if_t<internal::is_broadly_integral_v<I>>* = nullptr>
 static vector<int> _integer_to_vec(I x) {
 if constexpr (internal::is_broadly_signed_v<I>) {
 assert(x >= 0);
 }
 vector<int> res;
 while (x) res.push_back(x % D), x /= D;
 return res;
 }

 static long long _to_ll(const vector<int>& a) {
 long long res = 0;
 for (int i = (int)a.size() - 1; i >= 0; i--) res = res * D + a[i];
 return res;
 }

 static __int128_t _to_i128(const vector<int>& a) {
 __int128_t res = 0;
 for (int i = (int)a.size() - 1; i >= 0; i--) res = res * D + a[i];
 return res;
 }

 static void _dump(const vector<int>& a, string s = "") {
 if (!s.empty()) cerr << s << " : ";
 cerr << "{ ";
 for (int i = 0; i < (int)a.size(); i++) cerr << a[i] << ", ";
 cerr << "}" << endl;
 }
};

using bigint = MultiPrecisionInteger;












using namespace std;




template <typename T>
struct Binomial {
 vector<T> f, g, h;
 Binomial(int MAX = 0) {
 assert(T::get_mod() != 0 && "Binomial<mint>()");
 f.resize(1, T{1});
 g.resize(1, T{1});
 h.resize(1, T{1});
 if (MAX > 0) extend(MAX + 1);
 }

 void extend(int m = -1) {
 int n = f.size();
 if (m == -1) m = n * 2;
 m = min<int>(m, T::get_mod());
 if (n >= m) return;
 f.resize(m);
 g.resize(m);
 h.resize(m);
 for (int i = n; i < m; i++) f[i] = f[i - 1] * T(i);
 g[m - 1] = f[m - 1].inverse();
 h[m - 1] = g[m - 1] * f[m - 2];
 for (int i = m - 2; i >= n; i--) {
 g[i] = g[i + 1] * T(i + 1);
 h[i] = g[i] * f[i - 1];
 }
 }

 T fac(int i) {
 if (i < 0) return T(0);
 while (i >= (int)f.size()) extend();
 return f[i];
 }

 T finv(int i) {
 if (i < 0) return T(0);
 while (i >= (int)g.size()) extend();
 return g[i];
 }

 T inv(int i) {
 if (i < 0) return -inv(-i);
 while (i >= (int)h.size()) extend();
 return h[i];
 }

 T C(int n, int r) {
 if (n < 0 || n < r || r < 0) return T(0);
 return fac(n) * finv(n - r) * finv(r);
 }

 inline T operator()(int n, int r) { return C(n, r); }

 template <typename I>
 T multinomial(const vector<I>& r) {
 static_assert(is_integral<I>::value == true);
 int n = 0;
 for (auto& x : r) {
 if (x < 0) return T(0);
 n += x;
 }
 T res = fac(n);
 for (auto& x : r) res *= finv(x);
 return res;
 }

 template <typename I>
 T operator()(const vector<I>& r) {
 return multinomial(r);
 }

 T C_naive(int n, int r) {
 if (n < 0 || n < r || r < 0) return T(0);
 T ret = T(1);
 r = min(r, n - r);
 for (int i = 1; i <= r; ++i) ret *= inv(i) * (n--);
 return ret;
 }

 T P(int n, int r) {
 if (n < 0 || n < r || r < 0) return T(0);
 return fac(n) * finv(n - r);
 }

 
 T H(int n, int r) {
 if (n < 0 || r < 0) return T(0);
 return r == 0 ? 1 : C(n + r - 1, r);
 }
};




template <typename mint>
mint lagrange_interpolation(const vector<mint>& y, long long x,
 Binomial<mint>& C) {
 int N = (int)y.size() - 1;
 if (x <= N) return y[x];
 mint ret = 0;
 vector<mint> dp(N + 1, 1), pd(N + 1, 1);
 mint a = x, one = 1;
 for (int i = 0; i < N; i++) dp[i + 1] = dp[i] * a, a -= one;
 for (int i = N; i > 0; i--) pd[i - 1] = pd[i] * a, a += one;
 for (int i = 0; i <= N; i++) {
 mint tmp = y[i] * dp[i] * pd[i] * C.finv(i) * C.finv(N - i);
 ret += ((N - i) & 1) ? -tmp : tmp;
 }
 return ret;
}





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;
 }

 
 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;









template <typename mint>
struct ProductTree {
 using fps = FormalPowerSeries<mint>;
 const vector<mint> &xs;
 vector<fps> buf;
 int N, xsz;
 vector<int> l, r;
 ProductTree(const vector<mint> &xs_) : xs(xs_), xsz(xs.size()) {
 N = 1;
 while (N < (int)xs.size()) N *= 2;
 buf.resize(2 * N);
 l.resize(2 * N, xs.size());
 r.resize(2 * N, xs.size());
 fps::set_fft();
 if (fps::ntt_ptr == nullptr)
 build();
 else
 build_ntt();
 }

 void build() {
 for (int i = 0; i < xsz; i++) {
 l[i + N] = i;
 r[i + N] = i + 1;
 buf[i + N] = {-xs[i], 1};
 }
 for (int i = N - 1; i > 0; i--) {
 l[i] = l[(i << 1) | 0];
 r[i] = r[(i << 1) | 1];
 if (buf[(i << 1) | 0].empty())
 continue;
 else if (buf[(i << 1) | 1].empty())
 buf[i] = buf[(i << 1) | 0];
 else
 buf[i] = buf[(i << 1) | 0] * buf[(i << 1) | 1];
 }
 }

 void build_ntt() {
 fps f;
 f.reserve(N * 2);
 for (int i = 0; i < xsz; i++) {
 l[i + N] = i;
 r[i + N] = i + 1;
 buf[i + N] = {-xs[i] + 1, -xs[i] - 1};
 }
 for (int i = N - 1; i > 0; i--) {
 l[i] = l[(i << 1) | 0];
 r[i] = r[(i << 1) | 1];
 if (buf[(i << 1) | 0].empty())
 continue;
 else if (buf[(i << 1) | 1].empty())
 buf[i] = buf[(i << 1) | 0];
 else if (buf[(i << 1) | 0].size() == buf[(i << 1) | 1].size()) {
 buf[i] = buf[(i << 1) | 0];
 f.clear();
 copy(begin(buf[(i << 1) | 1]), end(buf[(i << 1) | 1]),
 back_inserter(f));
 buf[i].ntt_doubling();
 f.ntt_doubling();
 for (int j = 0; j < (int)buf[i].size(); j++) buf[i][j] *= f[j];
 } else {
 buf[i] = buf[(i << 1) | 0];
 f.clear();
 copy(begin(buf[(i << 1) | 1]), end(buf[(i << 1) | 1]),
 back_inserter(f));
 buf[i].ntt_doubling();
 f.intt();
 f.resize(buf[i].size(), mint(0));
 f.ntt();
 for (int j = 0; j < (int)buf[i].size(); j++) buf[i][j] *= f[j];
 }
 }
 for (int i = 0; i < 2 * N; i++) {
 buf[i].intt();
 buf[i].shrink();
 }
 }
};

template <typename mint>
vector<mint> InnerMultipointEvaluation(const FormalPowerSeries<mint> &f,
 const vector<mint> &xs,
 const ProductTree<mint> &ptree) {
 using fps = FormalPowerSeries<mint>;
 vector<mint> ret;
 ret.reserve(xs.size());
 auto rec = [&](auto self, fps a, int idx) {
 if (ptree.l[idx] == ptree.r[idx]) return;
 a %= ptree.buf[idx];
 if ((int)a.size() <= 64) {
 for (int i = ptree.l[idx]; i < ptree.r[idx]; i++)
 ret.push_back(a.eval(xs[i]));
 return;
 }
 self(self, a, (idx << 1) | 0);
 self(self, a, (idx << 1) | 1);
 };
 rec(rec, f, 1);
 return ret;
}

template <typename mint>
vector<mint> MultipointEvaluation(const FormalPowerSeries<mint> &f,
 const vector<mint> &xs) {
 if(f.empty() || xs.empty()) return vector<mint>(xs.size(), mint(0));
 return InnerMultipointEvaluation(f, xs, ProductTree<mint>(xs));
}






template <class mint>
FormalPowerSeries<mint> PolynomialInterpolation(const vector<mint> &xs,
 const vector<mint> &ys) {
 using fps = FormalPowerSeries<mint>;
 assert(xs.size() == ys.size());
 ProductTree<mint> ptree(xs);
 fps w = ptree.buf[1].diff();
 vector<mint> vs = InnerMultipointEvaluation<mint>(w, xs, ptree);
 auto rec = [&](auto self, int idx) -> fps {
 if (idx >= ptree.N) {
 if (idx - ptree.N < (int)xs.size())
 return {ys[idx - ptree.N] / vs[idx - ptree.N]};
 else
 return {mint(1)};
 }
 if (ptree.buf[idx << 1 | 0].empty())
 return {};
 else if (ptree.buf[idx << 1 | 1].empty())
 return self(self, idx << 1 | 0);
 return self(self, idx << 1 | 0) * ptree.buf[idx << 1 | 1] +
 self(self, idx << 1 | 1) * ptree.buf[idx << 1 | 0];
 };
 return rec(rec, 1);
}







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};
}













using namespace Nyaan;
using mint = LazyMontgomeryModInt<998244353>;

using vm = vector<mint>;
using vvm = vector<vm>;
Binomial<mint> C;
using fps = FormalPowerSeries<mint>;
using namespace Nyaan;

vm select(int r, int a, vm ys) {
 int N = sz(ys);
 fps xs(N);
 rep(i, N) xs[i] = i;
 auto f = PolynomialInterpolation(xs, ys);
 rep(i, N) xs[i] = a * i + r;
 return MultipointEvaluation<mint>(f, xs);
}
vm shift(vm ys) {
 int N = sz(ys);
 ys.push_back(lagrange_interpolation(ys, N, C));
 auto res = mkrui(ys);
 res.erase(begin(res));
 return res;
}

void q() {
 inl(N);
 vl A(N - 1);
 in(A);
 bigint M;
 in(M);
 vm ys{1};
 each(a, A) {
 auto [q, r] = divmod(M, a);
 ys = select(r.to_ll(), a, ys);
 ys = shift(ys);
 M = q;
 }
 trc(ys, M);
 out(lagrange_interpolation(ys, (M % 998244353).to_ll(), C));
}

void Nyaan::solve() {
 int t = 1;
 
 while (t--) q();
}