# include <bits/stdc++.h>
# ifndef ONLINE_JUDGE
# define LOCAL
# endif
# if __cplusplus >= 201100LL
# else
# error Please use C++11 Or Higher CPP version
# endif
using ll = long long;
using ull = unsigned long long;
using i128 = __int128;
using u128 = unsigned __int128;
using arr3 = std :: array < int, 3 >;
using arr4 = std :: array < int, 4 >;
using pii = std :: pair < int, int >;
# if __cplusplus >= 201400LL
# define rep(i, f, t, ...) for ( std :: decay < decltype ( f + t ) > :: type i = f, _END_##i = t, ##__VA_ARGS__; i <= _END_##i; ++i )
# define per(i, f, t, ...) for ( std :: decay < decltype ( f + t ) > :: type i = f, _END_##i = t, ##__VA_ARGS__; i >= _END_##i; --i )
# else
# define rep(i, f, t, ...) for ( decltype ( f + t ) > i = f, _END_##i = t, ##__VA_ARGS__; i <= _END_##i; ++i )
# define per(i, f, t, ...) for ( decltype ( f + t ) > i = f, _END_##i = t, ##__VA_ARGS__; i >= _END_##i; --i )
# endif
# define emb emplace_back
# define pb push_back
# define mkp make_pair
# define FILEIO(filename) freopen(filename ".in", "r", stdin), freopen(filename ".out", "w", stdout)

# ifndef INLINE_V
# if __cplusplus >= 201700LL
# define INLINE_V inline
# else
# define INLINE_V
# endif
# endif
# if __cplusplus >= 201400LL
# define EXPR constexpr
# else
# define EXPR
# endif

# if __cplusplus >= 201400LL
template < class T, class U >
inline constexpr auto ceil ( T && x, U && y ) {
  return ( x > 0 ? ( x + y - 1 ) / y : x / y );
}

template < class T, class U >
inline constexpr auto floor ( T && x, U && y ) {
  return ( x > 0 ? x / y : ( x - y + 1 ) / y );
}

template < class T, class U >
inline constexpr auto divmod ( T && x, U && y ) {
  auto && q = floor ( x, y );
  return std :: pair { q, x - q * y };
}

# else

template < class T, class U >
inline T ceil ( T && x, U && y ) {
  return ( x > 0 ? ( x + y - 1 ) / y : x / y );
}

template < class T, class U >
inline T floor ( T && x, U && y ) {
  return ( x > 0 ? x / y : ( x - y + 1 ) / y );
}

template < class T, class U >
inline pair < T, T > divmod ( T && x, U && y ) {
  T && q = floor ( x, y );
  return std :: pair { q, x - q * y };
}

# endif

# if __cplusplus >= 202000LL

# define ALL( vc ) std :: begin ( vc ), std :: end ( vc )

template < class T >
requires ( std :: is_signed_v < T > && std :: is_integral_v < T > )
INLINE_V constexpr static T inf = std :: numeric_limits < T > :: max ( ) >> 1;

template <std :: ranges :: forward_range R>
inline constexpr auto Sum ( R && r ) {
  return std :: accumulate ( std :: ranges :: begin ( r ), std :: ranges :: end ( r ), 0 );
}

template < std :: ranges :: forward_range R, typename T >
inline constexpr auto Sum ( R && r, T init) {
  return std :: accumulate ( std :: ranges :: begin ( r ), std :: ranges :: end ( r ), init );
}

template < std :: ranges :: forward_range R >
inline constexpr int SZ ( R && x ) { return std :: size ( x ); }

# else

# define ALL( vc ) vc.begin ( ), vc.end ( )

template < class R >
inline EXPR auto Sum ( R && r ) {
  return std :: accumulate ( ALL ( r ), 0);
}

template < class R, typename T >
inline EXPR auto Sum( R && r, T init ) {
  return std :: accumulate ( ALL ( r ), init );
}

template < class R >
inline EXPR auto Sum ( R f, R t ) {
  return std ::accumulate ( f, t );
}

template < class R, typename T >
inline EXPR auto Sum ( R f, R t, T init ) {
  return std :: accumulate ( f, t, init );
}

template < class T >
inline EXPR int SZ ( T && x ) { return static_cast < int > ( x.size ( ) ) ; }

template < class T >
INLINE_V constexpr static T inf = std :: numeric_limits < T > :: max ( ) >> 1;

# endif

template < class Os, class T >
inline Os & operator << ( Os & os, T & t ) {
  for ( auto ele : t )
    os << ele << " ";
  os << "\n";
  return os;
}

# ifndef __GEORGEYUCJR_READ__
# define __GEORGEYUCJR_READ__

# if __cplusplus >= 202000LL && !defined(MACOS) && defined(__unix__)

# define CHECK_EOF 0
# if CHECK_EOF
# define ECHK0 *ptr ? *ptr++ : -1
# define ECHK1                                                                                                                                        \
  if (ch == -1)                                                                                                                                      \
    return set(false), *this;
# define ECHK2                                                                                                                                        \
  if (!*ptr)                                                                                                                                         \
    return set(false), *this;
# define ECHK3 and ch != -1
# define ECHK4 and*ptr
# else
# define ECHK0 *ptr++
# define ECHK1
# define ECHK2
# define ECHK3
# define ECHK4
# endif

# if !defined(MACOS)
# define USE_MMAP
# include <sys/mman.h>
# include <sys/stat.h>
# endif

# if __cpluspls >= 202000LL && !defined(MACOS)
# define DISABLE_RANGE
# endif

# if CHECK_EOF || !defined(USE_MMAP)
# define EV -1
# else
# define EV 0
# endif

namespace FastIO {
template <typename T> struct make_unsigned : public std :: make_unsigned<T> {};
template <> struct make_unsigned<i128> {
  using type = u128;
};
template <typename T>
concept tupleLike = requires(T t) { std :: tuple_size_v<decltype(std :: tuple_cat(t))>; };

constexpr ull pw10(int t) { return t == 0 ? 1 : pw10(t - 1) * 10; }

std :: mt19937 mrand(std :: random_device{}());
std :: mt19937_64 mrand64(std :: random_device{}());

constexpr std :: size_t bufSize = 1 << 20;

class IO;
class In {
  friend class IO;

private:
  FILE *inFile;
  bool status = true;
# ifdef USE_MMAP
  struct stat st;
  char *ptr;
  int fd;
# elif !defined(LOCAL)
  char buf[bufSize], *p1, *p2;
# endif

public:
# ifdef LOCAL
  inline int getch() { return fgetc_unlocked(inFile); }
  inline void input(FILE *f) { inFile = f, set(); }
# elif defined(USE_MMAP)
  inline int getch() { return ECHK0; }
  inline void input(FILE *f) {
    inFile = f;
    if (inFile)
      fd = fileno(inFile), fstat(fd, &st), ptr = (char *)mmap(nullptr, st.st_size, PROT_READ, MAP_PRIVATE, fd, 0), set();
  }
  static constexpr auto n = []() {
    std :: array<uint32_t, 0x10000> n{};
    for (int i = 0; i != 0x10000; ++i)
      n[i] = -1;
    constexpr uint32_t e0 = 0x01, e1 = 0x100;
    int x = 0;
    for (uint32_t i = 0, c0 = 0x3030; i != 10; i++, c0 += e0)
      for (uint32_t j = 0, c1 = c0; j != 10; j++, c1 += e1)
        n[c1] = x++;
    return n;
  }();
# else
  inline int getch() { return (p1 == p2 ? (p2 = (p1 = buf) + fread(buf, 1, bufSize, inFile)) : 0), p1 == p2 ? -1 : *p1++; }
  inline void input(FILE *f) { inFile = f, p1 = p2 = buf, set(); }
# endif
  inline void input(const char *s) { input(fopen(s, "rb")); }
  inline void set(bool s = true) { status = s; }
  inline bool get() const { return status; }
};
class Out {
  friend class IO;

public:
  FILE *outFile;
# ifndef LOCAL
  char pbuf[bufSize], *pp;
# endif

public:
# ifdef LOCAL
  inline void flush() { outFile ? fflush(outFile) : 0; }
  inline void putch(char c) { fputc_unlocked(c, outFile); }
# else
  inline void flush() { (outFile ? fwrite(pbuf, 1, pp - pbuf, outFile), fflush(outFile) : 0), pp = pbuf; }
  inline void putch(char c) { ((pp - pbuf == bufSize) ? fwrite(pbuf, 1, bufSize, outFile), pp = pbuf : 0), *pp++ = c; }
# endif
  inline void output(const char *s) { flush(), outFile = fopen(s, "wb"); }
  inline void output(FILE *f) { flush(), outFile = f; }
};
class IO : public In, public Out {
private:
  std :: size_t precision = 12;

public:
  IO(FILE * = nullptr, FILE * = nullptr);
  ~IO() { flush(); }
  static constexpr bool blank(char);
  template <typename... Args>
  requires(sizeof...(Args) > 1)
  IO &read(Args &...);
  template <typename T>
  requires(std :: is_signed_v<T> && std :: is_integral_v<T>) || std :: is_same_v<std :: decay_t<T>, i128>
  IO &read(T &);
  template <typename T>
  requires(std :: is_unsigned_v<T> && std :: is_integral_v<T>) || std :: is_same_v<std :: decay_t<T>, u128>
  IO &read(T &);
  template <typename T>
  requires std :: is_floating_point_v<T>
  IO &read(T &);
  template <typename T> T read();
  IO &read(char &);
  IO &read(char *);
  IO &read(std :: string &);
  IO &readline(char *);
  IO &readline(std :: string &);
  template <tupleLike T> IO &read(T &&);
  template <std :: ranges::input_range R> IO &read(R &&r) { return readArray(std :: forward<R>(r)); }
  void setprecision(std :: size_t n = 6u) { precision = n; }
  template <typename... Args>
  requires(sizeof...(Args) > 1)
  void write(Args &&...);
  inline void write() const {}
  template <typename T>
  requires(std :: is_signed_v<T> && std :: is_integral_v<T>) || std :: is_same_v<std :: decay_t<T>, i128>
  void write(T);
  template <typename T>
  requires(std :: is_unsigned_v<T> && std :: is_integral_v<T>)
  void write(T);
  void write(u128);
  inline void write(char);
  template <typename T>
  requires std :: is_floating_point_v<T>
  void write(T);
  inline void write(bool);
  void write(char *);
  void write(const char *);
  void write(const std :: string &);
  template <typename T> void write(std :: initializer_list<T>);
  template <tupleLike T> void write(T &&);
  template <std :: ranges::input_range R> void write(R &&);
  template <typename... Args> void writeln(Args &&...);
  template <typename T> inline void writespc(T && x);
  operator bool() const { return get(); }
} io(stdin, stdout);
template <typename... Args> inline void writeln(Args &&...x) { io.writeln(std :: forward<Args>(x)...); }
template <typename T> inline void writeln(std :: initializer_list<T> x) { io.writeln(x); }
#pragma endregion

# define isdigit(x) ((x) >= '0' && (x) <= '9')
IO::IO(FILE *in, FILE *out) { input(in), output(out); }
constexpr bool IO::blank(char c) { return c == ' ' || c == '\n' || c == '\r'; }
template <typename... Args>
requires(sizeof...(Args) > 1)
IO &IO::read(Args &...args) {
  if constexpr (CHECK_EOF)
    (read(args) && ...);
  else
    (read(args), ...);
  return *this;
}
template <typename T>
requires(std :: is_signed_v<T> && std :: is_integral_v<T>) || std :: is_same_v<std :: decay_t<T>, i128>
IO &IO::read(T &x) {
  x = 0;
  static typename make_unsigned<T>::type t;
  bool sign = false;
# ifndef USE_MMAP
  int ch = getch();
  while (!isdigit(ch) ECHK3)
    sign = (ch == '-'), ch = getch();
  ECHK1
  t = 0;
  while (isdigit(ch))
    t = t * 10 + (ch ^ 48), ch = getch();
# else
  while (!isdigit(*ptr) ECHK4)
    sign = *ptr++ == '-';
  ECHK2
  t = *ptr++ ^ 48;
  while (~n[*reinterpret_cast<std :: uint16_t *&>(ptr)])
    t = t * 100 + n[*reinterpret_cast<std :: uint16_t *&>(ptr)++];
  if (isdigit(*ptr))
    t = t * 10 + (*ptr++ ^ 48);
# endif
  x = sign ? (~t + 1) : t;
  return *this;
}
template <typename T>
requires(std :: is_unsigned_v<T> && std :: is_integral_v<T>) || std :: is_same_v<std :: decay_t<T>, u128>
IO &IO::read(T &x) {
  x = 0;
# ifndef USE_MMAP
  int ch = getch();
  while (!isdigit(ch) ECHK3)
    ch = getch();
  ECHK1
  while (isdigit(ch))
    x = (x << 1) + (x << 3) + (ch ^ 48), ch = getch();
# else
  while (!isdigit(*ptr) ECHK4)
    ptr++;
  ECHK2
  x = *ptr++ ^ 48;
  while (~n[*reinterpret_cast<std :: uint16_t *&>(ptr)])
    x = x * 100 + n[*reinterpret_cast<std :: uint16_t *&>(ptr)++];
  if (isdigit(*ptr))
    x = x * 10 + (*ptr++ ^ 48);
# endif
  return *this;
}
template <typename T>
requires std :: is_floating_point_v<T>
IO &IO::read(T &x) {
  x = 0;
  T t = 1;
  bool sign = false;
  int ch = getch();
  while (!isdigit(ch) ECHK3)
    sign = (ch == '-'), ch = getch();
  ECHK1
  while (isdigit(ch))
    x = x * 10 + (ch ^ 48), ch = getch();
  if (ch == '.')
    for (ch = getch(); isdigit(ch); ch = getch())
      x += (t /= 10) * (ch ^ 48);
  x = sign ? -x : x;
  return *this;
}
template <typename T> T IO::read() {
  static std :: decay_t<T> x;
  return read(x), x;
}
IO &IO::read(char &ch) {
  do
    ch = getch();
  while (blank(ch));
  ECHK1
  return *this;
}
IO &IO::read(char *s) {
  int ch = getch();
  while (blank(ch))
    ch = getch();
  ECHK1
  while (!blank(ch) && ch != EV)
    *s++ = ch, ch = getch();
  *s = 0;
  return *this;
}
IO &IO::read(std :: string &s) {
  int ch = getch();
  while (blank(ch))
    ch = getch();
  ECHK1
  s.erase();
  while (!blank(ch) && ch != EV)
    s.append(1, ch), ch = getch();
  return *this;
}
IO &IO::readline(char *s) {
  char *t = s;
  int ch = getch();
  while (ch != '\n' && ch != EV)
    *s++ = ch, ch = getch();
  *s = 0;
  if (s == t && ch != '\n')
    set(false);
  return *this;
}
IO &IO::readline(std :: string &s) {
  s.erase();
  int ch = getch();
  while (ch != '\n' && ch != EV)
    s.append(1, ch), ch = getch();
  if (s.empty() && ch != '\n')
    set(false);
  return *this;
}
template <tupleLike T> IO &IO::read(T &&t) {
  std :: apply([&](auto & ...t) { (read(t), ...); }, t);
  return *this;
}
template <typename... Args>
requires(sizeof...(Args) > 1)
void IO::write(Args &&...args) {
  (write(std :: forward<Args>(args)), ...);
}
template <typename T>
requires(std :: is_signed_v<T> && std :: is_integral_v<T>) || std :: is_same_v<std :: decay_t<T>, i128>
void IO::write(T x) {
  static typename make_unsigned<T>::type y;
  y = x;
  if (x < 0)
    putch('-'), write(~y + 1);
  else
    write(y);
}
constexpr auto D = []() {
  constexpr uint32_t e0 = 0x1, e1 = 0x100, e2 = 0x10000, e3 = 0x1000000;
  std :: array<uint32_t, 10000> m{};
  int x = 0;
  for (uint32_t i = 0, c0 = 0x30303030; i != 10; i++, c0 += e0)
    for (uint32_t j = 0, c1 = c0; j != 10; j++, c1 += e1)
      for (uint32_t k = 0, c2 = c1; k != 10; k++, c2 += e2)
        for (uint32_t l = 0, c3 = c2; l != 10; l++, c3 += e3)
          m[x++] = c3;
  return m;
}();
template <typename T>
requires(std :: is_unsigned_v<T> && std :: is_integral_v<T>)
void IO::write(T x) {
# ifndef LOCAL
  if (std :: end(pbuf) - pp < 64)
    flush();

  auto L = [&](int x) { return x == 1 ? 0 : pw10(x - 1); };
  auto R = [&](int x) { return pw10(x) - 1; };

# define de(t)                                                                                                                                        \
  case L(t)... R(t):                                                                                                                                 \
    *reinterpret_cast<uint32_t *>(pp) = D[x / pw10((t)-4)];                                                                                          \
    pp += 4;                                                                                                                                         \
    x %= pw10((t)-4);

  ull y = x;
  switch (y) {
    de(18);
    de(14);
    de(10);
    de(6);
  case L(2)... R(2):
    *reinterpret_cast<uint32_t *>(pp) = D[x * 100];
    pp += 2;
    break;

    de(17);
    de(13);
    de(9);
    de(5);
  case L(1)... R(1):
    *pp = static_cast<char>('0' + x);
    pp += 1;
    break;

  default:
    *reinterpret_cast<uint32_t *>(pp) = D[x / pw10(16)];
    pp += 4;
    x %= pw10(16);
    de(16);
    de(12);
    de(8);
  case L(4)... R(4):
    *reinterpret_cast<uint32_t *>(pp) = D[x];
    pp += 4;
    break;

    de(19);
    de(15);
    de(11);
    de(7);
  case L(3)... R(3):
    *reinterpret_cast<uint32_t *>(pp) = D[x * 10];
    pp += 3;
    break;
  }
# else
  write(u128(x));
# endif
}
constexpr u128 _pw10(int x) { return x == 0 ? 1 : _pw10(x - 1) * 10; }
void IO::write(u128 x) {
  if (std :: end(pbuf) - pp < 64)
    flush();
  constexpr auto pw10 = _pw10;
  auto L = [&](int x) { return x == 1 ? 0 : pw10(x - 1); };
  auto R = [&](int x) { return pw10(x) - 1; };

# define de(t)                                                                                                                                        \
  case L(t)... R(t):                                                                                                                                 \
    *reinterpret_cast<uint32_t *>(pp) = D[x / pw10((t)-4)];                                                                                          \
    pp += 4;                                                                                                                                         \
    x %= pw10((t)-4);

  switch (x) {
    de(38);
    de(34);
    de(30);
    de(26);
    de(22);
    de(18);
    de(14);
    de(10);
    de(6);
  case L(2)... R(2):
    *reinterpret_cast<uint32_t *>(pp) = D[x * 100];
    pp += 2;
    break;

    de(37);
    de(33);
    de(29);
    de(25);
    de(21);
    de(17);
    de(13);
    de(9);
    de(5);
  case L(1)... R(1):
    *pp = static_cast<char>('0' + x);
    pp += 1;
    break;

    de(36);
    de(32);
    de(28);
    de(24);
    de(20);
    de(16);
    de(12);
    de(8);
  case L(4)... R(4):
    *reinterpret_cast<uint32_t *>(pp) = D[x];
    pp += 4;
    break;

  default:
    *reinterpret_cast<uint32_t *>(pp) = D[x / pw10(35)];
    pp += 4;
    x %= pw10(35);
    de(35);
    de(31);
    de(27);
    de(23);
    de(19);
    de(15);
    de(11);
    de(7);
  case L(3)... R(3):
    *reinterpret_cast<uint32_t *>(pp) = D[x * 10];
    pp += 3;
    break;
  }
}
void IO::write(bool x) { putch(x ^ 48); }
void IO::write(char c) { putch(c); }
template <typename T>
requires std :: is_floating_point_v<T>
void IO::write(T x) {
  static char buf[512];
  *std :: to_chars(buf, buf + 512, x, std :: chars_format::fixed, precision).ptr = 0;
  write(buf);
}
void IO::write(char *s) {
  while (*s)
    putch(*s++);
}
void IO::write(const char *s) {
  while (*s)
    putch(*s++);
}
void IO::write(const std :: string &s) { write(s.data()); }
template <typename T> void IO::write(std :: initializer_list<T> t) {
  auto first = std :: begin(t), last = std :: end(t);
  if (first != last)
    for (write(*first++); first != last; ++first) {
      putch(' ');
      write(*first);
    }
}
template <tupleLike T> void IO::write(T &&t) {
  [&]<std :: size_t... I>(std :: index_sequence<I...>) {
    (..., (I == 0 ? void(0) : putch(' '), write(std :: get<I>(t))));
  }(std :: make_index_sequence<std :: tuple_size_v<std :: remove_reference_t<T>>>());
}
template <std :: ranges::input_range R> void IO::write(R &&r) {
  if constexpr (std :: is_same_v<std :: decay_t<R>, std :: string>)
    return write(r.data());
  auto first = std :: ranges::begin(r), last = std :: ranges::end(r);
  if (first != last)
    for (write(*first++); first != last; ++first) {
      putch(' ');
      write(*first);
    }
}
template <typename... Args> void IO::writeln(Args &&...x) { write(std :: forward<Args>(x)...), putch('\n'); }
template <typename T> inline void IO::writespc(T && x) { write(x), putch('\n'); }
} // namespace FastIO
using namespace FastIO;

# else

# ifndef INLINE_V
# if __cplusplus >= 201700LL
# define INLINE_V inline
# else
# define INLINE_V
# endif
# endif

# ifndef LUOGU
# define auto_att __attribute__((target("avx2"), optimize(3, "Ofast", "unroll-loops", "inline")))
# else
# define auto_att
# endif

namespace FastIO {
# define endl '\n'
INLINE_V static constexpr size_t buf_size = 1 << 18;
INLINE_V static constexpr size_t integer_size = 20;
INLINE_V static constexpr size_t block_size = 10000;

static char inbuf[buf_size + 1] = {};
static char outbuf[buf_size + 1] = {};
static char block_str[block_size * 4 + 1] = {};

static constexpr uint64_t power10[] = {1,
                                       10,
                                       100,
                                       1000,
                                       10000,
                                       100000,
                                       1000000,
                                       10000000,
                                       100000000,
                                       1000000000,
                                       10000000000,
                                       100000000000,
                                       1000000000000,
                                       10000000000000,
                                       100000000000000,
                                       1000000000000000,
                                       10000000000000000,
                                       100000000000000000,
                                       1000000000000000000,
                                       10000000000000000000u
                                      };

struct Scanner {
# ifndef LOCAL
private:
  size_t pos, end;

  inline void load() {
    end = fread(inbuf, 1, buf_size, stdin);
    inbuf[end] = '\0';
  }
  inline void reload() {
    size_t len = end - pos;
    memmove(inbuf, inbuf + pos, len);
    end = len + fread(inbuf + len, 1, buf_size - len, stdin);
    inbuf[end] = '\0';
    pos = 0;
  }
  inline void skip_space() {
    while (inbuf[pos] <= ' ') {
      if (__builtin_expect(++pos == end, 0))
        reload();
    }
  }
  inline char get_next() { return inbuf[pos++]; }
  inline char get_next_nonspace() {
    skip_space();
    return inbuf[pos++];
  }

public:
  Scanner() { load(); }

  auto_att inline void read(char &c) { c = get_next_nonspace(); }
  auto_att inline void read(std :: string &s) {
    skip_space();
    s = "";
    do {
      size_t start = pos;
      while (inbuf[pos] > ' ')
        ++pos;
      s += std :: string(inbuf + start, inbuf + pos);
      if (inbuf[pos] != '\0')
        break;
      reload();
    } while (true);
  }

  template <class T> typename std :: enable_if<std :: is_integral<T>::value, void>::type read(T &x) {
    char c = get_next_nonspace();
    if (__builtin_expect(pos + integer_size >= end, 0))
      reload();
    bool neg = false;
    if (c == '-')
      neg = true, x = 0;
    else
      x = c & 15;
    while ((c = get_next()) >= '0')
      x = (x << 3) + (x << 1) + (c & 15);
    if (neg)
      x = -x;
  }
# else
  template <typename _Tp> inline void read(_Tp &x) {
    char ch = getchar();
    int f = 1;
    x = 0;
    for (; ch < '0' || ch > '9'; ch = getchar())
      f = ch == '-' ? -1 : 1;
    for (; ch >= '0' && ch <= '9'; ch = getchar())
      x = (x << 3) + (x << 1) + (ch ^ 48);
    x *= f;
  }
  inline void read(char &c) { c = getchar(); }
  inline void read(char *x) {
    int i = 0;
    char c = ' ';
    for (; c <= ' ';)
      c = getchar();
    for (; c > ' ';)
      x[i++] = c, c = getchar();
    x[i] = 0;
  }
  inline void read(std :: string &x) {
    x.clear();
    char c = ' ';
    for (; c <= ' ';)
      c = getchar();
    for (; c > ' ';)
      x += c, c = getchar();
  }
# endif
  template <class Head, class... Others> void read(Head &head, Others &...others) {
    read(head);
    read(others...);
  }
  template <class T1, class T2> inline void read(std :: pair<T1, T2> &x) { read(x.first), read(x.second); }
} is;

struct Printer {
# ifndef LOCAL
private:
  size_t pos = 0;

  auto_att inline void flush() {
    fwrite(outbuf, 1, pos, stdout);
    pos = 0;
  }

  inline void pre_calc() {
    for (size_t i = 0; i < block_size; ++i) {
      size_t j = 4, k = i;
      while (j--) {
        block_str[i * 4 + j] = k % 10 | 48;
        k /= 10;
      }
    }
  }

  static EXPR size_t get_integer_size(uint64_t n) {
    if (n >= power10[10]) {
      if (n >= power10[19])
        return 20;
      if (n >= power10[18])
        return 19;
      if (n >= power10[17])
        return 18;
      if (n >= power10[16])
        return 17;
      if (n >= power10[15])
        return 16;
      if (n >= power10[14])
        return 15;
      if (n >= power10[13])
        return 14;
      if (n >= power10[12])
        return 13;
      if (n >= power10[11])
        return 12;
      return 11;
    } else {
      if (n >= power10[9])
        return 10;
      if (n >= power10[8])
        return 9;
      if (n >= power10[7])
        return 8;
      if (n >= power10[6])
        return 7;
      if (n >= power10[5])
        return 6;
      if (n >= power10[4])
        return 5;
      if (n >= power10[3])
        return 4;
      if (n >= power10[2])
        return 3;
      if (n >= power10[1])
        return 2;
      return 1;
    }
  }

public:
  Printer() { pre_calc(); }
  ~Printer() { flush(); }
  inline void write(char c) {
    outbuf[pos++] = c;
    if (__builtin_expect(pos == buf_size, 0))
      flush();
  }
  inline void write(const char *s) {
    while (*s != 0) {
      outbuf[pos++] = *s++;
      // if (pos == buf_size) flush();
      if (__builtin_expect(pos == buf_size, 0))
        flush();
    }
  }
  inline void write(const std :: string &s) {
    for (auto c : s) {
      outbuf[pos++] = c;
      // if (pos == buf_size) flush();
      if (__builtin_expect(pos == buf_size, 0))
        flush();
    }
  }

  template <class T> typename std :: enable_if<std :: is_integral<T>::value, void>::type write(T x) {
    if (__builtin_expect(pos + integer_size >= buf_size, 0))
      flush();
    if (x < 0)
      write('-'), x = -x;
    size_t digit = get_integer_size(x);
    size_t len = digit;
    while (len >= 4) {
      len -= 4;
      memcpy(outbuf + pos + len, block_str + ((x % block_size) << 2), 4);
      x /= block_size;
    }
    memcpy(outbuf + pos, block_str + (x << 2) + (4 - len), len);
    pos += digit;
  }
# else
  inline void write(char c) { putchar(c); }
  inline void write(const char *x) {
    for (int i = 0; x[i]; ++i)
      write(x[i]);
  }
  inline void write(std::string s) {
    for (auto c : s)
      write(c);
  }
  template <typename _Tp> inline void write(_Tp x) {
    if (x < 0)
      putchar('-'), x = -x;
    if (x > 9)
      write(x / 10);
    putchar(x % 10 ^ 48);
  }
# endif
  template <class Head, class... Others> void write(const Head &head, const Others &...others) {
    write(head);
    write(' ');
    write(others...);
  }

  template <class... Args> void writeln(const Args &...args) {
    write(args...);
    write('\n');
  }
  template < class T >
  inline void writespc ( const T & x ) {
    write ( x );
    write ( " " );
  }
  template <class T1, class T2> inline void write(const std :: pair<T1, T2> &x) { write(x.first, x.second); }
} os;

struct IO : public Scanner, public Printer {
} io;
}; // namespace FastIO
using namespace FastIO;
# endif
# endif
using namespace std;

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

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

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

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

  NTT() { setwy(level); }

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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


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

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

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

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

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

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

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

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

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

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

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

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

#ifndef ATCODER_MODINT_HPP
#define ATCODER_MODINT_HPP 1

#include <type_traits>

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

#ifndef ATCODER_INTERNAL_MATH_HPP
#define ATCODER_INTERNAL_MATH_HPP 1

#include <utility>

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

namespace atcoder {

namespace internal {

// @param m `1 <= m`
// @return x mod m
constexpr long long safe_mod(long long x, long long m) {
  x %= m;
  if (x < 0)
    x += m;
  return x;
}

// Fast modular multiplication by barrett reduction
// Reference: https://en.wikipedia.org/wiki/Barrett_reduction
// NOTE: reconsider after Ice Lake
struct barrett {
  unsigned int _m;
  unsigned long long im;

  // @param m `1 <= m < 2^31`
  barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}

  // @return m
  unsigned int umod() const { return _m; }

  // @param a `0 <= a < m`
  // @param b `0 <= b < m`
  // @return `a * b % m`
  unsigned int mul(unsigned int a, unsigned int b) const {
    // [1] m = 1
    // a = b = im = 0, so okay

    // [2] m >= 2
    // im = ceil(2^64 / m)
    // -> im * m = 2^64 + r (0 <= r < m)
    // let z = a*b = c*m + d (0 <= c, d < m)
    // a*b * im = (c*m + d) * im = c*(im*m) + d*im = c*2^64 + c*r + d*im
    // c*r + d*im < m * m + m * im < m * m + 2^64 + m <= 2^64 + m * (m + 1) <
    // 2^64 * 2
    // ((ab * im) >> 64) == c or c + 1
    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;
  }
};

// @param n `0 <= n`
// @param m `1 <= m`
// @return `(x ** n) % m`
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;
}

// Reference:
// M. Forisek and J. Jancina,
// Fast Primality Testing for Integers That Fit into a Machine Word
// @param n `0 <= n`
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);

// @param b `1 <= b`
// @return pair(g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g
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};

  // Contracts:
  // [1] s - m0 * a = 0 (mod b)
  // [2] t - m1 * a = 0 (mod b)
  // [3] s * |m1| + t * |m0| <= b
  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

    // [3]:
    // (s - t * u) * |m1| + t * |m0 - m1 * u|
    // <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u)
    // = s * |m1| + t * |m0| <= b

    auto tmp = s;
    s = t;
    t = tmp;
    tmp = m0;
    m0 = m1;
    m1 = tmp;
  }
  // by [3]: |m0| <= b/g
  // by g != b: |m0| < b/g
  if (m0 < 0)
    m0 += b / s;
  return {s, m0};
}

// Compile time primitive root
// @param m must be prime
// @return primitive root (and minimum in now)
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);

}  // namespace internal

}  // namespace atcoder

#endif  // ATCODER_INTERNAL_MATH_HPP

#ifndef ATCODER_INTERNAL_TYPE_TRAITS_HPP
#define ATCODER_INTERNAL_TYPE_TRAITS_HPP 1

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

#endif  // ATCODER_INTERNAL_TYPE_TRAITS_HPP

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

#endif  // ATCODER_MODINT_HPP

template < class Mint > class Comb {
private: std :: vector < Mint > fac; std :: vector < Mint > ifac;
public: inline void rsz ( int _n ) {
    int prv_n = ( int ) fac.size ( ) - 1; if (prv_n >= _n) return ; fac.resize(_n + 1), ifac.resize(_n + 1);
    for (int i = prv_n + 1; i <= _n; ++i) fac[i] = fac[i - 1] * Mint::raw(i); ifac[_n] = fac[_n].inv();
    for (int i = _n; i > prv_n; --i) ifac[i - 1] = ifac[i] * Mint::raw(i);
  } Comb ( int n = 1 ) { fac.assign(2, Mint(1)), ifac.assign(2, Mint(1)), rsz(n); } inline Mint Fac(int n) {
    if ( n >= (int)fac.size ( ) ) rsz ( n + 5 ); return fac[n];
  } inline Mint InvFac(int n) {
    if (n >= (int)fac.size()) rsz(n + 5); return ifac[n];
  } inline Mint Inv(int n) {
    if (n >= (int)fac.size()) rsz(n + 5); return ifac[n] * fac[n - 1];
  } inline Mint C(int n, int m) {
    if (n < 0 || n < m || m < 0) return Mint(0); if (n >= (int)fac.size()) rsz(n + 5); return fac[n] * ifac[m] * ifac[n - m];
  } inline Mint A(int n, int m) {
    if (n < 0 || n < m || m < 0) return Mint(0); if (n >= (int)fac.size()) rsz(n + 5); return fac[n] * ifac[n - m];
  } inline Mint operator()(int n, int m) { return C(n, m); }
};

using mint = atcoder :: modint998244353;
Comb < mint > comb;
using fps = FormalPowerSeries < mint >;

# ifndef __GEORGEYUCJR_READ__
# ifdef LOCAL
# include <georgeyucjr/debug.h>
using namespace georgeyucjr;
# else
# define write(...) void ( 36 )
# define bug(...) void ( 36 )
# endif
# else
# warning Do not use debug.hpp.
# endif

int n, m;

int gcd(int x, int y) {return y ? gcd(y, x % y) : x;}

signed main() {
  io.read (n, m);
  comb.rsz(2 * n + 5);
  auto clc = [&](int x) { assert(x >= 0); return comb(2 * x, x) * comb.Inv(x + 1); };
  fps pre(2 * n + 1);
  fps c(n + 1);
  rep(i, 1, n) c[i] = clc(i - 1) * m * mint(m - 1).pow(i - 1);
  fps f = (fps{1} - c).inv(n + 1);
  rep(i, 0, n) pre[i * 2] = f[i];
  int E = n / 2 * 2 + 10;
  fps d(E + 1);
  rep(i, 0, E / 2) d[i * 2] = clc(i) * mint(m - 1).pow(i);
  d = d << 2;

  fps f1 = fps{1} - d * m;
  fps f2 = fps{1} - d * (m - 1);

  fps mus = (((f2 * f2).pre(E) - fps{0, 0, m - 1}) * f1).pre(E);
  fps num = (f2 << 1) * m;
  fps g = num * mus.inv();
  for (int i = 1; i <= n; i += 2) pre[i] = g[i];
  mint ans = 0;
  rep(i, 1, 2 * n) ans += pre[gcd(i, 2 * n)];
  io.writeln((ans / (2 * n)).val());
# ifdef LOCAL
  cerr << "Time Information : " << endl;
  cerr << "Time : " << clock ( ) * 1. / CLOCKS_PER_SEC * 1000. << "ms" << endl;
# endif
  return 0;
}