// #define MULTI_TESTCASE
// #define AOJ_TESTCASE

// #define FAST_CIO
// #define INTERACTIVE

#define INF 4'000'000'000'000'000'037LL
#define EPS 1e-11

// https://github.com/miscalculation53/library/tree/wip/template/template_all.hpp

// https://github.com/miscalculation53/library/tree/wip/template/template_all_but_modint.hpp

// https://github.com/miscalculation53/library/tree/wip/template/template_types.hpp

#include <bits/stdc++.h>
using namespace std;

// https://github.com/miscalculation53/library/tree/wip/template/template_float128.hpp

using namespace std;

#if __cplusplus > 202002L && defined(__STDCPP_FLOAT128_T__)
#include <stdfloat>
using f128 = std::float128_t;
#elif defined(__SIZEOF_FLOAT128__)
using f128 = __float128;
#else
#error "f128 requires a compiler with IEEE binary128 support, such as GCC on x86_64."
#endif
#define LIBRARY_HAS_FLOAT128 1

#if __cplusplus > 202002L && defined(__STDCPP_FLOAT128_T__) && \
    (defined(_GLIBCXX_HAVE_FLOAT128_MATH) || defined(_GLIBCXX_LDOUBLE_IS_IEEE_BINARY128))
#define LIBRARY_USE_QUADMATH 0
#else
#include <quadmath.h>
#define LIBRARY_USE_QUADMATH 1

//! https://gcc.gnu.org/onlinedocs/libquadmath/Math-Library-Routines.html
#define _p_F128_UNARY(name) \
  inline f128 name(f128 x) { return f128(name##q(__float128(x))); }
_p_F128_UNARY(acos)
_p_F128_UNARY(acosh)
_p_F128_UNARY(asin)
_p_F128_UNARY(asinh)
_p_F128_UNARY(atan)
_p_F128_UNARY(atanh)
_p_F128_UNARY(cbrt)
_p_F128_UNARY(ceil)
_p_F128_UNARY(cos)
_p_F128_UNARY(cosh)
_p_F128_UNARY(erf)
_p_F128_UNARY(erfc)
_p_F128_UNARY(exp)
_p_F128_UNARY(exp2)
_p_F128_UNARY(expm1)
_p_F128_UNARY(fabs)
_p_F128_UNARY(floor)
_p_F128_UNARY(lgamma)
_p_F128_UNARY(log)
_p_F128_UNARY(log10)
_p_F128_UNARY(log1p)
_p_F128_UNARY(log2)
_p_F128_UNARY(logb)
_p_F128_UNARY(nearbyint)
_p_F128_UNARY(rint)
_p_F128_UNARY(round)
_p_F128_UNARY(sin)
_p_F128_UNARY(sinh)
_p_F128_UNARY(sqrt)
_p_F128_UNARY(tan)
_p_F128_UNARY(tanh)
_p_F128_UNARY(tgamma)
_p_F128_UNARY(trunc)
#undef _p_F128_UNARY

#define _p_F128_BINARY(name) \
  inline f128 name(f128 x, f128 y) { return f128(name##q(__float128(x), __float128(y))); } \
  template <class T, enable_if_t<is_arithmetic_v<T>, int> = 0> \
  inline f128 name(f128 x, T y) { return name(x, f128(y)); } \
  template <class T, enable_if_t<is_arithmetic_v<T>, int> = 0> \
  inline f128 name(T x, f128 y) { return name(f128(x), y); }
_p_F128_BINARY(atan2)
_p_F128_BINARY(copysign)
_p_F128_BINARY(fdim)
_p_F128_BINARY(fmax)
_p_F128_BINARY(fmin)
_p_F128_BINARY(fmod)
_p_F128_BINARY(hypot)
_p_F128_BINARY(nextafter)
_p_F128_BINARY(pow)
_p_F128_BINARY(remainder)
#undef _p_F128_BINARY

inline f128 fma(f128 x, f128 y, f128 z)
{ return f128(fmaq(__float128(x), __float128(y), __float128(z))); }
inline f128 hypot(f128 x, f128 y, f128 z)
{ return hypot(hypot(x, y), z); }
namespace template_float128_detail
{
template <class T>
constexpr bool arithmetic = is_arithmetic_v<T> || is_same_v<T, f128>;
}
#define _p_F128_TERNARY_MIXED(name) \
  template <class T, class U, enable_if_t<template_float128_detail::arithmetic<T> && template_float128_detail::arithmetic<U>, int> = 0> \
  inline f128 name(f128 x, T y, U z) { return name(x, f128(y), f128(z)); } \
  template <class T, class U, enable_if_t<template_float128_detail::arithmetic<T> && !is_same_v<T, f128> && template_float128_detail::arithmetic<U>, int> = 0> \
  inline f128 name(T x, f128 y, U z) { return name(f128(x), y, f128(z)); } \
  template <class T, class U, enable_if_t<template_float128_detail::arithmetic<T> && !is_same_v<T, f128> && template_float128_detail::arithmetic<U> && !is_same_v<U, f128>, int> = 0> \
  inline f128 name(T x, U y, f128 z) { return name(f128(x), f128(y), z); }
inline f128 ldexp(f128 x, int exponent)
{ return f128(ldexpq(__float128(x), exponent)); }
inline f128 scalbn(f128 x, int exponent)
{ return f128(scalbnq(__float128(x), exponent)); }
inline f128 scalbln(f128 x, long exponent)
{ return f128(scalblnq(__float128(x), exponent)); }
inline f128 modf(f128 x, f128 *integer)
{
  __float128 value;
  const auto fraction = modfq(__float128(x), &value);
  *integer = f128(value);
  return f128(fraction);
}
inline f128 remquo(f128 x, f128 y, int *quotient)
{ return f128(remquoq(__float128(x), __float128(y), quotient)); }
inline long lrint(f128 x) { return lrintq(__float128(x)); }
inline long lround(f128 x) { return lroundq(__float128(x)); }
inline long long llrint(f128 x) { return llrintq(__float128(x)); }
inline long long llround(f128 x) { return llroundq(__float128(x)); }
inline int ilogb(f128 x) { return ilogbq(__float128(x)); }
#endif

inline f128 nexttoward(f128 x, long double y)
{ return nextafter(x, f128(y)); }

namespace template_float128_detail
{
inline bool parse(string_view s, f128 &x)
{
  if (!s.empty() && s.front() == '+')
  {
    s.remove_prefix(1);
    if (s.empty() || s.front() == '-') return false;
  }
#if LIBRARY_USE_QUADMATH
  string token(s);
  char *end;
  errno = 0;
  const __float128 value = strtoflt128(token.c_str(), &end);
  if (end == token.data() || end != token.data() + token.size()) return false;
  if (errno == ERANGE && (value == 0 || isinfq(value))) return false;
  x = f128(value);
  return true;
#else
  auto [p, ec] = from_chars(s.data(), s.data() + s.size(), x);
  return ec == errc{} && p == s.data() + s.size();
#endif
}

inline string format(f128 x, chars_format style, int precision)
{
  string s(128, '\0');
  while (true)
  {
#if LIBRARY_USE_QUADMATH
    const char *spec = style == chars_format::fixed ? "%.*Qf" :
                       style == chars_format::scientific ? "%.*Qe" : "%.*Qg";
    const int n = quadmath_snprintf(s.data(), s.size(), spec, precision, __float128(x));
    if (n < 0) abort();
    if (size_t(n) < s.size())
    {
      s.resize(size_t(n));
      return s;
    }
    s.resize(size_t(n) + 1);
#else
    auto [p, ec] = to_chars(s.data(), s.data() + s.size(), x, style, precision);
    if (ec == errc{})
    {
      s.resize(size_t(p - s.data()));
      return s;
    }
    s.resize(s.size() * 2);
#endif
  }
}
} 

using ld = decltype(EPS);

using ll = long long;
using uint = unsigned int;
using ull = unsigned long long;
using pll = pair<ll, ll>;
using tlll = tuple<ll, ll, ll>;
using tllll = tuple<ll, ll, ll, ll>;

#define vc vector
template <class T>
using vvc = vc<vc<T>>;
template <class T>
using vvvc = vc<vc<vc<T>>>;

using vb = vc<bool>;
using vl = vc<ll>;
using vpll = vc<pll>;
using vtlll = vc<tlll>;
using vtllll = vc<tllll>;
using vstr = vc<string>;
using vvb = vvc<bool>;
using vvl = vvc<ll>;

template <class T>
using pql = priority_queue<T, vc<T>, greater<T>>;
template <class T>
using pqg = priority_queue<T>;

using i128 = __int128_t;
using u128 = __uint128_t;
i128 stoi128(const string &s)
{
  const bool neg = s.front() == '-';
  u128 res = 0;
  for (size_t i = size_t(neg); i < s.size(); i++)
    res = 10 * res + unsigned(s[i] - '0');
  if (neg)
    return -i128(res - 1) - 1;
  return i128(res);
}
string i128tos(i128 x)
{
  if (x == 0) return "0";
  string sign = "", res = "";
  u128 ux;
  if (x < 0)
    ux = u128(-(x + 1)) + 1, sign = "-";
  else
    ux = u128(x);
  while (ux > 0)
  {
    res += char('0' + ux % 10);
    ux /= 10;
  }
  reverse(res.begin(), res.end());
  return sign + res;
}
istream &operator>>(istream &is, i128 &a)
{
  string s;
  if (!(is >> s)) return is;
  a = stoi128(s);
  return is;
}
ostream &operator<<(ostream &os, const i128 &a)
{
  os << i128tos(a);
  return os;
}

#define cauto const auto
// https://github.com/miscalculation53/library/tree/wip/template/template_rep.hpp

#define overload4(_1,_2,_3,_4,name,...) name
#define LIBRARY_REPEAT_LOOP(T,i,l,r,d) for (auto [i, library_repeat_end, library_repeat_step] = array<T, 3>{T(l), T(r), T(d)}; library_repeat_step > 0 ? i < library_repeat_end : i > library_repeat_end; i += library_repeat_step)

#define rep1(i,n) LIBRARY_REPEAT_LOOP(ll, i, 0, n, 1)
#define rep2(i,l,r) LIBRARY_REPEAT_LOOP(ll, i, l, r, 1)
#define rep3(i,l,r,d) LIBRARY_REPEAT_LOOP(ll, i, l, r, d)
#define rep(...) overload4(__VA_ARGS__, rep3, rep2, rep1)(__VA_ARGS__)
#define repi1(i,n) LIBRARY_REPEAT_LOOP(int, i, 0, n, 1)
#define repi2(i,l,r) LIBRARY_REPEAT_LOOP(int, i, l, r, 1)
#define repi3(i,l,r,d) LIBRARY_REPEAT_LOOP(int, i, l, r, d)
#define repi(...) overload4(__VA_ARGS__, repi3, repi2, repi1)(__VA_ARGS__)

#define fe(...) for (auto __VA_ARGS__)
#define fec(...) for (cauto &__VA_ARGS__)
#define fem(...) for (auto &__VA_ARGS__)
// https://github.com/miscalculation53/library/tree/wip/template/template_math.hpp

// https://github.com/miscalculation53/library/tree/wip/template/template_arithmetic.hpp

template <class T, class U, class P>
constexpr pair<T, U> &operator+=(pair<T, U> &a, const P &b);
template <class T, class U, class P>
constexpr pair<T, U> operator+(pair<T, U> a, const P &b);
template <class T, class U, class P>
constexpr pair<T, U> &operator-=(pair<T, U> &a, const P &b);
template <class T, class U, class P>
constexpr pair<T, U> operator-(pair<T, U> a, const P &b);
template <class T, class U>
constexpr pair<T, U> operator-(pair<T, U> a);

template <class T, size_t n, class A>
constexpr array<T, n> &operator+=(array<T, n> &a, const A &b);
template <class T, size_t n, class A>
constexpr array<T, n> operator+(array<T, n> a, const A &b);
template <class T, size_t n, class A>
constexpr array<T, n> &operator-=(array<T, n> &a, const A &b);
template <class T, size_t n, class A>
constexpr array<T, n> operator-(array<T, n> a, const A &b);
template <class T, size_t n>
constexpr array<T, n> operator-(array<T, n> a);

template <class... Ts, class Tp>
constexpr tuple<Ts...> &operator+=(tuple<Ts...> &a, const Tp &b);
template <class... Ts, class Tp>
constexpr tuple<Ts...> operator+(tuple<Ts...> a, const Tp &b);
template <class... Ts, class Tp>
constexpr tuple<Ts...> &operator-=(tuple<Ts...> &a, const Tp &b);
template <class... Ts, class Tp>
constexpr tuple<Ts...> operator-(tuple<Ts...> a, const Tp &b);
template <class... Ts>
constexpr tuple<Ts...> operator-(tuple<Ts...> a);

namespace internal
{

}; 

// https://github.com/miscalculation53/library/tree/wip/utils/is_integral_ext.hpp

template <class T>
constexpr bool is_integral_ext = is_integral_v<T> || is_same_v<T, i128> || is_same_v<T, u128>;

template <class T>
constexpr bool is_signed_ext = is_signed_v<T> || is_same_v<T, i128>;

template <class T>
constexpr bool is_unsigned_ext = is_unsigned_v<T> || is_same_v<T, u128>;
// https://github.com/miscalculation53/library/tree/wip/utils/integer_arithmetic.hpp

// https://github.com/miscalculation53/library/tree/wip/utils/make_unsigned_ext.hpp

template <class T>
struct make_unsigned_ext
{
  using type = make_unsigned_t<T>;
};
template <>
struct make_unsigned_ext<i128>
{
  using type = u128;
};
template <>
struct make_unsigned_ext<u128>
{
  using type = u128;
};
template <class T>
using make_unsigned_ext_t = typename make_unsigned_ext<T>::type;

template <class T>
struct make_signed_ext
{
  using type = make_signed_t<T>;
};
template <>
struct make_signed_ext<u128>
{
  using type = i128;
};
template <class T>
using make_signed_ext_t = typename make_signed_ext<T>::type;

template <class T>
constexpr bool is_negative(T x)
{
  static_assert(is_integral_ext<T>);
  if constexpr (is_signed_ext<T>) return x < 0;
  return false;
}

template <class T, bool = is_integral_ext<T>>
struct unsigned_arithmetic { using type = T; };

template <class T>
struct unsigned_arithmetic<T, true>
{
  static_assert(!is_same_v<T, bool>);
  using type = common_type_t<unsigned int, make_unsigned_ext_t<T>>;
};

template <class T>
using unsigned_arithmetic_t = typename unsigned_arithmetic<T>::type;

// https://github.com/miscalculation53/library/tree/wip/utils/default_infty.hpp

template <class T>
constexpr decltype(auto) default_infty();

namespace default_infty_detail
{
  template <class T>
  struct is_product : false_type {};
  template <class T, class U>
  struct is_product<pair<T, U>> : true_type {};
  template <class... Ts>
  struct is_product<tuple<Ts...>> : true_type {};
  template <class T, size_t n>
  struct is_product<array<T, n>> : true_type {};

  template <class T, class = void>
  struct has_infty : false_type {};

  template <class T>
  struct has_infty<T, void_t<decltype(T::infty())>> : true_type {};

  template <class T>
  inline constexpr bool unsupported = false;
}

namespace template_math_detail
{

} 

template <class T = ll, class U, class V, typename = enable_if_t<is_integral_ext<U> && is_integral_ext<V>>>
inline constexpr T divceil(U a, V b) { return T(a) / T(b) + (T(a) % T(b) && (T(a) ^ T(b)) >= 0); }
template <class T = ll, class U, class V, typename = enable_if_t<is_integral_ext<U> && is_integral_ext<V>>>
inline constexpr T safemod(U a, V b)
{
  const T x = T(a), y = T(b);
  if constexpr (is_signed_ext<T>)
    if (y == T(-1)) return T(0);
  const T r = x % y;
  return r != T(0) && is_negative(x) != is_negative(y) ? T(r + y) : r;
}

template <class T = ll, class U, class V>
constexpr T ipow(U a, V b)
{
  assert(b >= 0);
  if (b == 0)
    return 1;
  if (a == 0 || a == 1)
    return a;
  if (a < 0 && a == -1)
    return b & 1 ? -1 : 1;

  T res = 1, tmp = a;
  while (true)
  {
    if (b & 1)
      res *= tmp;
    b >>= 1;
    if (b == 0)
      break;
    tmp *= tmp;
  }
  return res;
}

//! https://misawa.github.io/others/avoid_errors/techniques_to_avoid_errors.html

// https://github.com/miscalculation53/library/tree/wip/template/template_vector.hpp

#define ALL(a) (a).begin(), (a).end()
template <class T = ll, class V>
inline T SZ(const V &x) { return x.size(); }
#define eb emplace_back

#define LMD(x,fx) ([&](const auto &x) { return fx; })
#define GEN_VEC(n,i,fi) (gen_vec(n, LMD(i, fi)))

namespace template_vector_detail
{
template <class T, size_t depth>
struct nested_vector
{
  using type = vc<typename nested_vector<T, depth - 1>::type>;
};
template <class T>
struct nested_vector<T, 0>
{
  using type = T;
};

}

// https://github.com/miscalculation53/library/tree/wip/template/template_algo.hpp

// https://github.com/miscalculation53/library/tree/wip/utils/resolved_infty.hpp

// https://github.com/miscalculation53/library/tree/wip/utils/resolved_value.hpp

template <class T>
vvc<T> top(const vvc<T> &a)
{
  if (a.empty())
    return {};
  const int n = a.size(), m = a[0].size();
  vvc<T> b(m, vc<T>(n));
  repi(i, n)
  {
    assert(SZ<int>(a[i]) == m);
    repi(j, m) b[j][i] = a[i][j];
  }
  return b;
}
vstr top(const vstr &a)
{
  vvc<char> a_(a.size());
  repi(i, SZ<int>(a)) a_[i] = {ALL(a[i])};
  vvc<char> b_ = top(a_);
  vstr b(b_.size());
  repi(i, SZ<int>(b)) b[i] = {ALL(b_[i])};
  return b;
}

template <class T, class = void>
struct has_e0 : false_type {};
template <class T>
struct has_e0<T, void_t<decltype(T::e0())>> : true_type {};
template <class T>
inline constexpr bool has_e0_v = has_e0<T>::value;

template <class T>
struct MonoidAdd
{
  using S = T;
  static constexpr S e();
};
template <class T, auto infty = nullptr>
struct MonoidMin
{
  using S = T;
  static constexpr decltype(auto) e();
};
template <class T, auto infty = nullptr>
struct MonoidMax
{
  using S = T;
  static constexpr S e();
};

namespace internal
{
  template <class M, class I, class = void>
  struct HasMonoidPow : false_type
  {
  };
  template <class M, class I>
  struct HasMonoidPow<M, I, void_t<decltype(M::pow(declval<const typename M::S &>(), declval<I>()))>> : true_type
  {
  };
}

constexpr array<pll, 4> DRULgrid = {{{1, 0}, {0, 1}, {-1, 0}, {0, -1}}};
constexpr array<pll, 4> DRULplane = {{{0, -1}, {1, 0}, {0, 1}, {-1, 0}}};
// https://github.com/miscalculation53/library/tree/wip/template/template_binsearch.hpp

template <class T>
struct is_random_access_iterator
{
  static constexpr bool value = is_same_v<
    typename iterator_traits<T>::iterator_category,
    random_access_iterator_tag
  >;
};
template <class T>
constexpr bool is_random_access_iterator_v = is_random_access_iterator<T>::value;

#define DEFAULT_COMP ranges::less

namespace internal
{
};

// https://github.com/miscalculation53/library/tree/wip/template/template_bit.hpp

template <class T>
inline constexpr ull MASK(T k) { return (1ULL << k) - 1ULL; }

inline constexpr ll bit_width(ll x) { return std::bit_width((ull)x); }

inline constexpr ll bit_floor(ll x) { return std::bit_floor((ull)x); }

inline constexpr ll bit_ceil(ll x) { return std::bit_ceil((ull)x); }
inline constexpr ll countr_zero(ll x) { assert(x != 0); return std::countr_zero((ull)x); }
inline constexpr ll popcount(ll x) { return std::popcount((ull)x); }
inline constexpr bool has_single_bit(ll x) { return std::has_single_bit((ull)x); }

inline constexpr ull lsb_pos(ull x) { assert(x != 0); return countr_zero(x); }
inline constexpr ull msb_pos(ull x) { assert(x != 0); return bit_width(x) - 1; }
inline constexpr ull lsb_mask(ull x) { assert(x != 0); return x & -x; }
inline constexpr ull msb_mask(ull x) { assert(x != 0); return bit_floor(x); }

inline constexpr bool btest(ull x, uint k) { return (x >> k) & 1; }
inline constexpr bool bsubset(ull x, ull y) { return (x & y) == x; }
inline constexpr bool bsupset(ull x, ull y) { return (x & y) == y; }
inline constexpr ull bsetminus(ull x, ull y) { return x & ~y; }
// https://github.com/miscalculation53/library/tree/wip/template/template_inout.hpp

// https://github.com/miscalculation53/library/tree/wip/template/template_dump.hpp

#if defined(LOCAL) && LIBRARY_HAS_FLOAT128
namespace cpp_dump::_detail
{
  struct export_command;
  inline string export_var(const f128 &, const string &, size_t,
                           size_t, bool, const export_command &);
}
#endif

// https://github.com/miscalculation53/library/tree/wip/template/template_dump_map.hpp

#define CPP_DUMP_DEFINE_DATA(...) 
#define dump(...) ((void)0)
#define local(...) 
#define oj(...) __VA_ARGS__
#define local_oj(a,b) (b)

#if LIBRARY_HAS_FLOAT128
inline istream &operator>>(istream &is, f128 &x)
{
  string s;
  if (is >> s)
  {
    f128 value;
    if (template_float128_detail::parse(s, value)) x = value;
    else is.setstate(ios::failbit);
  }
  return is;
}

inline ostream &operator<<(ostream &os, f128 x)
{
  const auto field = os.flags() & ios::floatfield;
  const auto format = field == ios::fixed ? chars_format::fixed :
                      field == ios::scientific ? chars_format::scientific : chars_format::general;
  string s = template_float128_detail::format(x, format, int(os.precision()));
  if ((os.flags() & ios::showpos) && !signbit(x)) s.insert(s.begin(), '+');
  if (os.flags() & ios::uppercase)
    for (char &c : s) c = char(toupper(static_cast<unsigned char>(c)));
  return os << s;
}
#endif

namespace fastio
{
  constexpr uint32_t SIZ = 1 << 17;
  inline char ibuf[SIZ + 1], obuf[SIZ];
  inline uint32_t pil = 0, pir = 0, por = 0;

  __attribute__((noinline)) inline void load()
  {
    const uint32_t remaining = pir - pil;
    if (remaining && pil) memmove(ibuf, ibuf + pil, remaining);
    pil = 0;
    pir = remaining + uint32_t(fread(ibuf + remaining, 1, SIZ - remaining, stdin));
    ibuf[pir] = '\x7f';
  }

  inline bool ascii_space(unsigned char c)
  {
    return c == ' ' || unsigned(c - '\t') < 5;
  }

  inline void find_token()
  {
    for (;;)
    {
      const char *p = ibuf + pil;
      const char *end = ibuf + pir;
      while (ascii_space(*p)) ++p;
      pil = uint32_t(p - ibuf);
      if (p != end) return;
      load();
      if (!pir) abort();
    }
  }

  __attribute__((noinline)) inline void flush()
  {
    size_t written = 0;
    while (written < por)
    {
      const size_t n = fwrite(obuf + written, 1, por - written, stdout);
      if (!n) abort();
      written += n;
    }
    por = 0;
  }

  inline void append_bytes(const char *data, size_t size)
  {
    while (size)
    {
      const size_t n = min(size, size_t(SIZ - por));
      memcpy(obuf + por, data, n);
      por += uint32_t(n);
      data += n;
      size -= n;
      if (por == SIZ) flush();
    }
  }

  inline void append_char(char c)
  {
    if (por == SIZ) flush();
    obuf[por++] = c;
  }

  template <class T>
  using integer_magnitude = conditional_t<(sizeof(T) > sizeof(ull)), u128,
                            conditional_t<(sizeof(T) > sizeof(uint)), ull, uint>>;

  inline uint decode_eight(uint64_t columns)
  {
    columns &= 0x0f0f0f0f0f0f0f0fULL;
    const uint64_t pairs = (columns & 0x00ff00ff00ff00ffULL) * 10
                         + ((columns >> 8) & 0x00ff00ff00ff00ffULL);
    const uint64_t quads = (pairs & 0x0000ffff0000ffffULL) * 100
                         + ((pairs >> 16) & 0x0000ffff0000ffffULL);
    return uint(quads) * 10000 + uint(quads >> 32);
  }

  inline constexpr uint powers_of_ten[8] = {1, 10, 100, 1000, 10000, 100000, 1000000, 10000000};

  template <class T>
  __attribute__((always_inline)) inline void rd1_integer(T &x)
  {
    using U = integer_magnitude<T>;
    const char *p = ibuf + pil;
    for (;;)
    {
      while (ascii_space(*p)) ++p;
      if (p != ibuf + pir) break;
      pil = pir;
      load();
      if (!pir) abort();
      p = ibuf;
    }
    bool negative = false;
    if (*p == '+') ++p;
    else if constexpr (numeric_limits<T>::is_signed)
    {
      if (*p == '-') negative = true, ++p;
    }
    U value = 0;
    for (;;)
    {
      while (size_t(ibuf + pir - p) >= 8)
      {
        uint64_t columns;
        memcpy(&columns, p, 8);

        const uint64_t stop = ((columns + 0x4646464646464646ULL) |
                               (columns - 0x3030303030303030ULL)) & 0x8080808080808080ULL;
        if (stop)
        {
          if constexpr (sizeof(U) > sizeof(ull))
          {
            const unsigned digits = unsigned(__builtin_ctzll(stop)) / 8;
            if (digits)
            {
              uint tail;
              if (digits <= 3)
              {
                tail = unsigned(p[0] - '0');
                if (digits >= 2) tail = tail * 10 + unsigned(p[1] - '0');
                if (digits == 3) tail = tail * 10 + unsigned(p[2] - '0');
              }
              else tail = decode_eight(columns << (8 * (8 - digits)));
              value = value * powers_of_ten[digits] + tail;
              p += digits;
            }
            goto integer_complete;
          }
          else
          {
            if (stop & 0x80808080ULL)
            {
              if (!(stop & 0x808080ULL))
              {
                value = value * 1000 + uint(columns & 15) * 100
                      + uint(columns >> 8 & 15) * 10 + uint(columns >> 16 & 15);
                p += 3;
              }
              else if (!(stop & 0x8080ULL))
              {
                value = value * 100 + uint(columns & 15) * 10 + uint(columns >> 8 & 15);
                p += 2;
              }
              else if (!(stop & 0x80ULL))
              {
                value = value * 10 + uint(columns & 15);
                ++p;
              }
              goto integer_complete;
            }
            break;

          }
        }
        columns &= 0x0f0f0f0f0f0f0f0fULL;
        const uint64_t pairs = (columns & 0x00ff00ff00ff00ffULL) * 10
                             + ((columns >> 8) & 0x00ff00ff00ff00ffULL);
        const uint64_t quads = (pairs & 0x0000ffff0000ffffULL) * 100
                             + ((pairs >> 16) & 0x0000ffff0000ffffULL);
        value = value * 100000000 + uint(quads) * 10000 + uint(quads >> 32);
        p += 8;
      }
      while (unsigned(*p - '0') < 10)
      {
        value = value * 10 + unsigned(*p - '0');
        ++p;
      }
      pil = uint32_t(p - ibuf);
      if (pil != pir) break;
      load();
      p = ibuf;
      if (!pir) break;
    }
integer_complete:
    pil = uint32_t(p - ibuf);
    if constexpr (numeric_limits<T>::is_signed)
    {
      if (negative && value) x = T(-T(value - 1) - 1);
      else x = T(value);
    }
    else x = T(value);
  }

  template <class T, enable_if_t<numeric_limits<T>::is_integer, int> = 0>
  __attribute__((always_inline)) inline void rd1(T &x) { rd1_integer(x); }
  inline void rd1(char &x)
  {
    find_token();
    x = ibuf[pil++];
  }
  inline void rd1(string &x)
  {
    find_token();
    x.clear();
    for (;;)
    {
      const char *p = ibuf + pil;
      const char *end = ibuf + pir;
      const char *first = p;
      while (p != end && !ascii_space(*p)) ++p;
      x.append(first, size_t(p - first));
      pil = uint32_t(p - ibuf);
      if (p != end) return;
      load();
      if (!pir) return;
    }
  }

  template <class T>
  void rd1_real(T &x)
  {
    string token;
    rd1(token);
    const char *first = token.data();
    const char *last = first + token.size();
    if (first != last && *first == '+') ++first;
    auto result = from_chars(first, last, x, chars_format::general);
    if (result.ec != errc{} || result.ptr != last)
    {
      char *end;
      if constexpr (is_same_v<T, float>) x = strtof(token.c_str(), &end);
      else if constexpr (is_same_v<T, double>) x = strtod(token.c_str(), &end);
      else x = strtold(token.c_str(), &end);
      if (end != token.data() + token.size()) abort();
    }
  }
  inline void rd1(float &x) { rd1_real(x); }
  inline void rd1(double &x) { rd1_real(x); }
  inline void rd1(long double &x) { rd1_real(x); }
#if LIBRARY_HAS_FLOAT128
  inline void rd1(f128 &x)
  {
    string token;
    rd1(token);
    if (!template_float128_detail::parse(token, x)) abort();
  }
#endif

  template <class A, class B> void rd1(pair<A, B> &x);
  template <class... T> void rd1(tuple<T...> &x);
  template <class T, size_t N> void rd1(array<T, N> &x);
  template <class T> void rd1(vc<T> &x);

  inline constexpr auto decimal_triplets = []
  {
    array<array<char, 3>, 1000> result{};
    for (unsigned i = 0; i < 1000; ++i)
    {
      result[i][0] = char('0' + i / 100);
      result[i][1] = char('0' + i / 10 % 10);
      result[i][2] = char('0' + i % 10);
    }
    return result;
  }();

  inline void store_nine(char *out, uint x)
  {
    const uint q = x / 1000, r = q / 1000;
    memcpy(out, decimal_triplets[r].data(), 3);
    memcpy(out + 3, decimal_triplets[q - r * 1000].data(), 3);
    memcpy(out + 6, decimal_triplets[x - q * 1000].data(), 3);
  }

  inline char *prepend_small(char *end, uint x)
  {
    while (x >= 1000)
    {
      const uint q = x / 1000;
      end -= 3;
      memcpy(end, decimal_triplets[x - q * 1000].data(), 3);
      x = q;
    }
    const unsigned digits = 1 + unsigned(x >= 10) + unsigned(x >= 100);
    end -= digits;
    memcpy(end, decimal_triplets[x].data() + 3 - digits, digits);
    return end;
  }

  inline char *prepend_wide(char *end, ull x)
  {
    while (x >= 1000000000ULL)
    {
      const ull q = x / 1000000000ULL;
      end -= 9;
      store_nine(end, uint(x - q * 1000000000ULL));
      x = q;
    }
    return prepend_small(end, uint(x));
  }

  inline char *store_leading_triplet(char *out, uint x)
  {
    if (x >= 100)
    {
      memcpy(out, decimal_triplets[x].data(), 3);
      return out + 3;
    }
    if (x >= 10)
    {
      memcpy(out, decimal_triplets[x].data() + 1, 2);
      return out + 2;
    }
    *out = char('0' + x);
    return out + 1;
  }

  inline char *store_short(char *out, uint x)
  {
    if (x >= 1000000)
    {
      const uint q = x / 1000000;
      const uint r = x - q * 1000000;
      out = store_leading_triplet(out, q);
      memcpy(out, decimal_triplets[r / 1000].data(), 3);
      memcpy(out + 3, decimal_triplets[r % 1000].data(), 3);
      return out + 6;
    }
    if (x >= 1000)
    {
      const uint q = x / 1000;
      out = store_leading_triplet(out, q);
      memcpy(out, decimal_triplets[x - q * 1000].data(), 3);
      return out + 3;
    }
    return store_leading_triplet(out, x);
  }

  template <class U>
  inline char *store_ordinary(char *out, U x)
  {
    if constexpr (sizeof(U) > sizeof(uint))
    {
      if (x >= 1000000000000000000ULL)
      {
        const uint hi = uint(x / 1000000000000000000ULL);
        const ull tail = x - ull(hi) * 1000000000000000000ULL;
        const uint middle = uint(tail / 1000000000ULL);
        out = store_leading_triplet(out, hi);
        store_nine(out, middle);
        store_nine(out + 9, uint(tail - ull(middle) * 1000000000ULL));
        return out + 18;
      }
    }
    if (x >= 1000000000)
    {
      const uint hi = uint(x / 1000000000);
      out = store_short(out, hi);
      store_nine(out, uint(x - U(hi) * 1000000000));
      return out + 9;
    }
    return store_short(out, uint(x));
  }

  template <class T, bool newline = false>
  __attribute__((always_inline)) inline void wt1_integer(T x)
  {
    using U = integer_magnitude<T>;
    U magnitude = U(x);
    bool negative = false;
    if constexpr (numeric_limits<T>::is_signed)
    {
      negative = x < 0;
      const U sign_mask = U(0) - U(negative);
      magnitude = (magnitude ^ sign_mask) + U(negative);
    }
    if constexpr (sizeof(U) <= sizeof(ull))
    {
      if (por > SIZ - 40) flush();
      char *out = obuf + por;
      
      *out = '-';
      out += negative;
      out = store_ordinary(out, magnitude);
      if constexpr (newline) *out++ = '\n';
      por = uint32_t(out - obuf);
      return;
    }
    char text[48];
    char *const end = text + sizeof(text);
    char *first = end;
    if constexpr (sizeof(U) > sizeof(ull))
    {
      constexpr ull base = 1000000000000000000ULL;
      while (magnitude >= base)
      {
        const U q = magnitude / base;
        const ull tail = ull(magnitude - q * base);
        const uint hi = uint(tail / 1000000000ULL);
        first -= 18;
        store_nine(first, hi);
        store_nine(first + 9, uint(tail - ull(hi) * 1000000000ULL));
        magnitude = q;
      }
      first = prepend_wide(first, ull(magnitude));
    }
    else if constexpr (sizeof(U) > sizeof(uint)) first = prepend_wide(first, magnitude);
    else first = prepend_small(first, magnitude);
    if (negative) *--first = '-';
    append_bytes(first, size_t(end - first));
    if constexpr (newline) append_char('\n');
  }

  inline void wt1(char x) { append_char(x); }
  inline void wt1(const char *x) { append_bytes(x, strlen(x)); }
  inline void wt1(string_view x) { append_bytes(x.data(), x.size()); }
  inline void wt1(const string &x) { append_bytes(x.data(), x.size()); }

  template <class T>
  void wt1_real(T x)
  {
    char text[numeric_limits<T>::max_exponent10 + 64];
    auto result = to_chars(text, text + sizeof(text), x, chars_format::fixed, 15);
    if (result.ec != errc{}) abort();
    append_bytes(text, size_t(result.ptr - text));
  }
  inline void wt1(float x) { wt1_real(x); }
  inline void wt1(double x) { wt1_real(x); }
  inline void wt1(long double x) { wt1_real(x); }
#if LIBRARY_HAS_FLOAT128
  inline void wt1(f128 x)
  {
    wt1(template_float128_detail::format(x, chars_format::fixed, 35));
  }
#endif

  template <class A, class B> void wt1(const pair<A, B> &x);
  template <class... T> void wt1(const tuple<T...> &x);
  template <class T, size_t N> void wt1(const array<T, N> &x);
  template <class T> void wt1(const vc<T> &x);

  template <class... T>
  void read(T &...x) { (rd1(x), ...); }
  template <class... T>
  void write(T &&...x) { (wt1(forward<T>(x)), ...); }
  template <class... T>
  void print(T &&...x)
  {
    if constexpr (sizeof...(T) == 1 &&
                  ((is_integral_v<decay_t<T>> && !is_same_v<decay_t<T>, char>) && ...))
      (wt1_integer<decay_t<T>, true>(forward<T>(x)), ...);
    else
    {
      size_t index = 0;
      ((index++ ? void(append_char(' ')) : void(), wt1(forward<T>(x))), ...);
      append_char('\n');
    }
  }
}

struct Dummy {
  Dummy() { atexit(fastio::flush); }
} dummy;

//! https://trap.jp/post/1224/

namespace internal
{

template <class... Ts>
void READnodump(Ts &...a) { fastio::read(a...); }

}; 

#define READ(...) internal::READnodump(__VA_ARGS__); dump(__VA_ARGS__)

#define LIBRARY_INOUT_TYPE(...) LIBRARY_INOUT_TYPE_IMPL(__VA_ARGS__)(__VA_ARGS__)
#define LIBRARY_INOUT_TYPE_IMPL(T,...) LIBRARY_INOUT_TYPE_SELECT(LIBRARY_INOUT_TYPE_PROBE T, LIBRARY_INOUT_TYPE_KEEP)
#define LIBRARY_INOUT_TYPE_PROBE(...) _, LIBRARY_INOUT_TYPE_REMOVE
#define LIBRARY_INOUT_TYPE_SELECT(...) LIBRARY_INOUT_TYPE_SECOND(__VA_ARGS__)
#define LIBRARY_INOUT_TYPE_SECOND(_0,_1,...) _1
#define LIBRARY_INOUT_TYPE_KEEP(...) __VA_ARGS__
#define LIBRARY_INOUT_TYPE_REMOVE(T) LIBRARY_INOUT_TYPE_KEEP T

#define IN(T,...) LIBRARY_INOUT_TYPE(T) __VA_ARGS__; READ(__VA_ARGS__)

#define CHAR(...) IN(char, __VA_ARGS__)
#define INT(...) IN(int, __VA_ARGS__)
#define LL(...) IN(ll, __VA_ARGS__)
#define STR(...) IN(string, __VA_ARGS__)
#define ARR(T,n,...) array<LIBRARY_INOUT_TYPE(T), n> __VA_ARGS__; READ(__VA_ARGS__)

#define READVEC(...) internal::READVECnodump(__VA_ARGS__); dump(__VA_ARGS__)
#define READVEC2(...) internal::READVEC2nodump(__VA_ARGS__); dump(__VA_ARGS__)

#define VEC(T,n,...) vc<LIBRARY_INOUT_TYPE(T)> __VA_ARGS__; READVEC(n, __VA_ARGS__)
#define VEC2(T,n,m,...) vvc<LIBRARY_INOUT_TYPE(T)> __VA_ARGS__; READVEC2(n, m, __VA_ARGS__)

#define READJAG(...) internal::READJAGnodump(__VA_ARGS__); dump(__VA_ARGS__)

#define JAG(T,n,...) vvc<LIBRARY_INOUT_TYPE(T)> __VA_ARGS__; READJAG(n, __VA_ARGS__)

#define ENDL '\n'

#define WRITE fastio::write
#define PRINT fastio::print

#define PRINTEXIT(...) do { PRINT(__VA_ARGS__); exit(0); } while (false)
#define PRINTRETURN(...) do { PRINT(__VA_ARGS__); return; } while (false)

#define PRINTVEXIT(...) do { PRINTV(__VA_ARGS__); exit(0); } while (false)
#define PRINTVRETURN(...) do { PRINTV(__VA_ARGS__); return; } while (false)

namespace internal
{

};

namespace internal
{

};

#define UNZIP(vt,...) auto [__VA_ARGS__] = unzip(vt)
#define ZIP(vt,...) auto vt = internal::zip_vectors(__VA_ARGS__)

// https://github.com/miscalculation53/library/tree/wip/template/template_random.hpp

mt19937_64 mt;

template <class T = ll, class U1, class U2>
T randrange(U1 l, U2 r)
{
  assert(T(l) < T(r));
  return uniform_int_distribution<T>(T(l), T(r) - 1)(mt);
}

bool randbool(double p)
{
  assert(0 <= p && p <= 1);
  return bernoulli_distribution(p)(mt);
}

namespace internal
{
}; 

// https://github.com/miscalculation53/library/tree/wip/math/modint/template_modint.hpp

// https://github.com/miscalculation53/library/tree/wip/math/modint/modint.hpp

/*!
 * Portions corresponding to AtCoder Library (CC0-1.0): modular arithmetic operators.
 * Upstream: atcoder/modint.hpp
 * https://github.com/atcoder/ac-library/tree/864245a00b00dd008d1abfdc239618fdb7d139da
 * License: https://github.com/atcoder/ac-library/blob/864245a00b00dd008d1abfdc239618fdb7d139da/LICENSE
 */

// https://github.com/miscalculation53/library/tree/wip/math/modint/modint_internal_static.hpp

// https://github.com/miscalculation53/library/tree/wip/utils/larger_int.hpp

namespace larger_int_detail
{
}

template <class T>
struct larger_int
{
private:
  static constexpr bool check();
  static_assert(check());

public:
  using type = T;
};

#define LARGER_INT(T,U) template <> struct larger_int<T> { using type = U; };

LARGER_INT(signed char, short)
LARGER_INT(short, int)
LARGER_INT(int, long long)
LARGER_INT(long, __int128_t)
LARGER_INT(long long, __int128_t)

LARGER_INT(unsigned char, unsigned short)
LARGER_INT(unsigned short, unsigned int)
LARGER_INT(unsigned int, unsigned long long)
LARGER_INT(unsigned long, __uint128_t)
LARGER_INT(unsigned long long, __uint128_t)

#undef LARGER_INT

template <class T>
struct Rational;

template <class T>
struct larger_int<Rational<T>>
{
  using type = Rational<typename larger_int<T>::type>;
};

template <class T>
using larger_int_t = typename larger_int<T>::type;
// https://github.com/miscalculation53/library/tree/wip/math/modint/modint_internal_isprime.hpp

/*!
 * Portions corresponding to AtCoder Library (CC0-1.0): primality testing.
 * Upstream: atcoder/internal_math.hpp
 * https://github.com/atcoder/ac-library/tree/864245a00b00dd008d1abfdc239618fdb7d139da
 * License: https://github.com/atcoder/ac-library/blob/864245a00b00dd008d1abfdc239618fdb7d139da/LICENSE
 */

namespace internal
{

template <class T>
constexpr ll powmod_constexpr(ll x, ll n, T m)
{
  if (m == 1)
    return 0;
  using U = make_unsigned_t<T>;
  using L = larger_int_t<U>;

  U r = 1, y = safemod(x, m);
  while (n)
  {
    if (n & 1)
      r = L(r) * y % m;
    y = L(y) * y % m;
    n >>= 1;
  }
  return r;
}

template <class T>
constexpr bool isprime_constexpr(T n)
{
  if constexpr (sizeof(T) > 4)
  {
    if (n <= INT_MAX)
      return isprime_constexpr<int>(n);
  }

  if (n <= 1)
    return false;
  if (n == 2 || n == 7 || n == 61)
    return true;
  if (n % 2 == 0)
    return false;

  ll d = n - 1;
  while (d % 2 == 0)
    d /= 2;

  using U = make_unsigned_t<T>;
  using L = larger_int_t<U>;

  auto miller_rabin = [&](const auto &bases) constexpr
  {
    for (ll a : bases)
    {
      ll t = d, y = powmod_constexpr(a, t, n);
      while (t != n - 1 && y != 1 && y != n - 1)
      {
        y = L(y) * y % n;
        t <<= 1;
      }
      if (y != n - 1 && t % 2 == 0)
        return false;
    }
    return true;
  };

  if constexpr(sizeof(T) <= 4)
  {
    constexpr ll bases[3] = {2, 7, 61};
    return miller_rabin(bases);
  }
  else
  {
    constexpr ll bases[7] = {2, 325, 9375, 28178, 450775, 9780504, 1795265022};
    return miller_rabin(bases);
  }
}

template <auto n>
constexpr bool isprime = isprime_constexpr(n);

};

namespace internal
{

template <auto M>
struct policy_static
{
  using mod_type = decltype(M);
  using value_type = make_unsigned_t<mod_type>;
  using calc_type = conditional_t<(M <= UINT_MAX), ull, larger_int_t<value_type>>;

  static constexpr bool is_prime = isprime_constexpr(M);

  static constexpr mod_type mod() { return M; }
  static constexpr value_type umod() { return M; }

  static constexpr value_type init(value_type v) { return v; }
  static constexpr mod_type val(value_type v) { return v; }
  static constexpr value_type mul(value_type a, value_type b)
  {
    return (value_type)((calc_type(a) * b) % M);
  }
};

};
// https://github.com/miscalculation53/library/tree/wip/math/modint/modint_internal_barrett32.hpp

/*!
 * Portions corresponding to AtCoder Library (CC0-1.0): Barrett reduction.
 * Upstream: atcoder/internal_math.hpp
 * https://github.com/atcoder/ac-library/tree/864245a00b00dd008d1abfdc239618fdb7d139da
 * License: https://github.com/atcoder/ac-library/blob/864245a00b00dd008d1abfdc239618fdb7d139da/LICENSE
 */

namespace internal
{

struct barrett32
{
  uint m = 0;
  ull im = 0;
  barrett32() = default;
  explicit barrett32(uint m) : m(m), im((ull)(-1) / m + 1) {}
  uint umod() const { return m; }
  uint mul(uint a, uint b) const
  {
    ull z = a;
    z *= b;
    ull x = ull((u128(z) * im) >> 64);
    ull y = x * m;
    return uint(z - y + (z < y ? m : 0));
  }
};

template <int id>
struct policy_barrett32
{
  using value_type = uint;
  using calc_type = ull;
  using mod_type = int;
  
  static constexpr bool is_prime = false;
  static inline barrett32 reducer{};
  static void set_mod(mod_type m)
  {
    assert(m >= 1 && "modint: set_mod(m) requires m >= 1");
    reducer = barrett32(m);
  }
  static mod_type mod() { return reducer.umod(); }
  static value_type umod() { return reducer.umod(); }
  static value_type init(value_type v) { return v; }
  static mod_type val(value_type v) { return v; }
  static value_type mul(value_type a, value_type b) { return reducer.mul(a, b); }
};

};
// https://github.com/miscalculation53/library/tree/wip/math/modint/modint_internal_montgomery64.hpp

namespace internal
{

inline constexpr ull inv64(ull a)
{
  ull x = a;
  while (a * x != 1) x *= 2 - a * x;
  return x;
}

struct montgomery64odd
{
  ull m = 0, im = 0, sq = 0;
  montgomery64odd() = default;
  explicit montgomery64odd(ull m) : m(m), im(inv64(m)), sq(-u128(m) % m) {}
  ull umod() const { return m; }
  ull reduce(u128 x) const
  {
    auto t = (x + u128(m) * (-im * ull(x))) >> 64;
    if (t >= m) t -= m;
    return (ull)t;
  }
  ull inv_reduce(i128 v) const { return reduce(u128(v % m + m) * sq); }
};

//! https://www.mathenachia.blog/even-mod-montgomery-impl/
struct montgomery64
{
  ull m = 0, mx = 0, imx = 0, d = 0, q = 0;
  uint b = 0;
  montgomery64() = default;
  explicit montgomery64(ull m) : m(m)
  {
    b = countr_zero(m), mx = m >> b; 
    imx = inv64(mx);
    d = powmod_constexpr((mx + 1) / 2, b, mx); 
    u128 sq = -u128(mx) % mx; 
    q = (1 + (((sq - 1) * d) << b)) % m;
  }
  ull umod() const { return m; }
  ull reduce(u128 x) const
  {
    if (b == 0)
    {
      auto t = (x + u128(mx) * (-imx * ull(x))) >> 64;
      if (t >= m) t -= m;
      return (ull)t;
    }
    ull p = x & MASK(b); 
    x = (x >> b) + p * d;
    ull y = p << (64 - b);
    auto t = (x + u128(mx) * (imx * (y - ull(x)))) >> (64 - b);
    if (t >= m) { t -= m; if (t >= m) t -= m; }
    return (ull)t;
  }
  ull inv_reduce(i128 v) const { return reduce(u128(v % m + m) * q); }
};

template <int id>
struct policy_montgomery64_odd
{
  using value_type = ull;
  using calc_type = u128;
  using mod_type = ll;

  static constexpr bool is_prime = false;
  static inline montgomery64odd reducer{};
  static void set_mod(mod_type m)
  {
    assert(m >= 1 && m % 2 == 1 && "modint: set_mod(m) requires a positive odd modulus");
    reducer = montgomery64odd(m);
  }
  static mod_type mod() { return reducer.umod(); }
  static value_type umod() { return reducer.umod(); }
  static value_type init(value_type v) { return reducer.inv_reduce(v); }
  static mod_type val(value_type v) { return reducer.reduce(v); }
  static value_type mul(value_type a, value_type b) { return reducer.reduce((calc_type)a * b); }
};

template <int id>
struct policy_montgomery64
{
  using value_type = ull;
  using calc_type = u128;
  using mod_type = ll;

  static constexpr bool is_prime = false;
  static inline montgomery64 reducer{};
  static mod_type mod() { return reducer.umod(); }
  static value_type init(value_type v) { return reducer.inv_reduce(v); }
};

};
// https://github.com/miscalculation53/library/tree/wip/math/extgcd.hpp

namespace internal
{

  template <class Policy, class PowInv = void>
  struct modint_impl
  {
    using V = typename Policy::value_type;
    using M = typename Policy::mod_type;
    using mint = modint_impl;

  private:
    V _v;

  public:
    static constexpr M mod()
    {
      M m = Policy::mod();
      assert(m > 0 && "modint: modulus is not set; call mint::set_mod(m) before use");
      return m;
    }

    template <class T = Policy>
    static auto set_mod(M m) -> decltype(T::set_mod(m)) { return T::set_mod(m); }

    modint_impl() : _v(0) { (void)mod(); }

    template <class T, typename = enable_if_t<is_integral_ext<T>>>
    modint_impl(T v)
    {
      const V m = mod();
      V rem;
      if constexpr (is_signed_ext<T>)
      {
        using S = make_signed_t<V>;
        S x = v % S(m);
        if (x < 0)
          x += m;
        rem = x;
      }
      else
        rem = V(v % m);
      _v = Policy::init(rem);
    };

    M val() const { return Policy::val(_v); }

    mint &operator+=(const mint &rhs)
    {
      _v += rhs._v;
      if (_v >= Policy::umod())
        _v -= Policy::umod();
      return *this;
    }
    mint &operator-=(const mint &rhs)
    {
      _v -= rhs._v;
      if (_v >= Policy::umod())
        _v += Policy::umod();
      return *this;
    }
    mint &operator*=(const mint &rhs)
    {
      _v = Policy::mul(_v, rhs._v);
      return *this;
    }

    template <class T>
    mint pow(T n) const
    {
      assert(n >= 0);
      if constexpr (!is_void_v<PowInv>)
        return raw(PowInv::get().pow(val(), n));
      mint x = *this, r = 1;
      while (n)
      {
        if (n & 1)
          r *= x;
        x *= x;
        n >>= 1;
      }
      return r;
    }
    mint inv() const
    {
      if constexpr (!is_void_v<PowInv>)
      {
        return raw(PowInv::get().inv(val()));
      }
      else if constexpr (Policy::is_prime)
      {
        return pow(mod() - 2);
      }
      else
      {
        auto [g, x, y] = extgcd<M>(val(), mod());
        assert(g == 1);
        return mint(x);
      }
    }

    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; }
    friend M safe_hash_key(const mint &x) { return x.val(); }
  
    friend void rd1(mint &x)
    {
      long long a;
      fastio::rd1(a);
      x = a;
    }
    friend void wt1(const mint &x)
    {
      fastio::wt1(x.val());
    }

  };

};

template <int mod>
using static_modint32 = internal::modint_impl<internal::policy_static<mod>>;
template <int id>
using dynamic_modint32 = internal::modint_impl<internal::policy_barrett32<id>>;
template <ll mod>
using static_modint64 = internal::modint_impl<internal::policy_static<mod>>;
template <int id>
using dynamic_modint64_odd = internal::modint_impl<internal::policy_montgomery64_odd<id>>;
template <int id>
using dynamic_modint64 = internal::modint_impl<internal::policy_montgomery64<id>>;

using modint998244353 = static_modint32<998244353>;
using modint1000000007 = static_modint32<1000000007>;
using modint = dynamic_modint32<-1>;
using modint61 = static_modint64<(1LL << 61) - 1>;
using modint64 = dynamic_modint64<-1>;

template <class T>
struct is_modint : std::false_type
{
};
template <class Policy, class PowInv>
struct is_modint<internal::modint_impl<Policy, PowInv>> : std::true_type
{
};
template <class T>
inline constexpr bool is_modint_v = is_modint<T>::value;

template <class T>
struct is_static_modint : false_type {};
template <int m, class PowInv>
struct is_static_modint<internal::modint_impl<internal::policy_static<m>, PowInv>> : true_type {};
template <ll m, class PowInv>
struct is_static_modint<internal::modint_impl<internal::policy_static<m>, PowInv>> : true_type {};
template <class T>
inline constexpr bool is_static_modint_v = is_static_modint<T>::value;

template <class T>
struct is_dynamic_modint : false_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint32<id>> : true_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint64_odd<id>> : true_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint64<id>> : true_type {};
template <class T>
inline constexpr bool is_dynamic_modint_v = is_dynamic_modint<T>::value;

template <typename, typename = void>
struct has_mod : std::false_type
{
};
template <typename T>
struct has_mod<T, std::void_t<decltype(T::mod())>> : std::true_type
{
};

template <class mint>
struct modint_less
{
};

template <class mint>
struct modint_hash
{
};
// https://github.com/miscalculation53/library/tree/wip/math/modint/power_table.hpp

template <class mint>
struct PowerTable
{
private:
  decltype(mint::mod()) mod;
  mint base;
  vc<mint> pw;

public:

  void reserve(int n);

};
// https://github.com/miscalculation53/library/tree/wip/math/modint/binomial.hpp

template <class T>
struct Binomial
{
private:
  inline static decltype(T::mod()) mod;
  
public:
  inline static vc<T> fac_, finv_, inv_;
  static void reserve(int n);
  static T inv(int n)
  {
    n %= T::mod();
    if (n < 0)
      n += T::mod();
    assert(n != 0);
    reserve(n);
    return inv_[n];
  }

};
// https://github.com/miscalculation53/library/tree/wip/math/modint/stom.hpp

// https://github.com/miscalculation53/library/tree/wip/math/modint/to_rational.hpp

// https://github.com/miscalculation53/library/tree/wip/math/svp2d.hpp

namespace cpp_dump
{
  template <class T>
  struct rat_value
  {
    T p, q;

    friend ostream &operator<<(ostream &os, const rat_value &x)
    {
      os << x.p;
      if (x.q != 1)
        os << '/' << x.q;
      return os;
    }
  };

  struct mint_to_rat_fn
  {
  };

  template <class T>
  struct rat_object
  {
    T value;
  };

  template <class T>
  inline constexpr bool rat_is_map = false;
  template <class... Args>
  inline constexpr bool rat_is_map<std::map<Args...>> = true;
  template <class... Args>
  inline constexpr bool rat_is_map<std::multimap<Args...>> = true;
  template <class... Args>
  inline constexpr bool rat_is_map<std::unordered_map<Args...>> = true;
  template <class... Args>
  inline constexpr bool rat_is_map<std::unordered_multimap<Args...>> = true;

  struct rat_closure : std::ranges::range_adaptor_closure<rat_closure>
  {

  };

  constexpr rat_closure rat()
  {
    return rat_closure{};
  }
}

using mint998 = modint998244353;
// using mint = modint1000000007;
using mint103 = static_modint32<100003>;
// using mint = modint;
using bi = Binomial<mint103>;

void init()
{
  oj(mt.seed(random_device()()));
}

unsigned xor128_x = 123456789, xor128_y = 362436069, xor128_z = 521288629, xor128_w = 88675123;
unsigned xor128() {
	unsigned t = xor128_x ^ (xor128_x << 11);
	xor128_x = xor128_y; xor128_y = xor128_z; xor128_z = xor128_w;
	return xor128_w = xor128_w ^ (xor128_w >> 19) ^ (t ^ (t >> 8));
}

// https://github.com/miscalculation53/library/tree/wip/math/prime/large/order_primitive_root.hpp

// https://github.com/miscalculation53/library/tree/wip/math/prime/prime_power.hpp

template <class P>
struct PrimePower
{
  P p;
  int e;
  P pe;

  PrimePower(P p, int e = 1) : p(p), e(e), pe(ipow(p, e)) {}
  PrimePower(P p, int e, P pe) : p(p), e(e), pe(pe) {}

};

tuple<int, ll, ll> ord_pow_div(ll n, ll m)
{
  assert(m >= 2);
  if (m == 2)
  {
    int e = countr_zero(n);
    return {e, 1LL << e, n >> e};
  }
  if (n % m != 0)
    return {0, 1, n};
  n /= m;
  if (n % m != 0)
    return {1, m, n};
  n /= m;
  ll m2 = m * m;
  auto [f, m2f, nn] = ord_pow_div(n, m2);
  int e = 2 + 2 * f;
  ll me = m2f * m2;
  if (nn % m == 0)
    e++, me *= m, nn /= m;
  return {e, me, nn};
}

template <class P>
vc<PrimePower<P>> factorized_mul
(const vc<PrimePower<P>> &fac1, const vc<PrimePower<P>> &fac2)
{
  const int n = fac1.size(), m = fac2.size();
  vc<PrimePower<P>> fac;
  fac.reserve(n + m);
  int i = 0, j = 0;
  while (i < n && j < m)
  {
    if (fac1[i].p < fac2[j].p)
      fac.emplace_back(fac1[i++]);
    else if (fac1[i].p > fac2[j].p)
      fac.emplace_back(fac2[j++]);
    else
    {
      using U = larger_int_t<P>;
      fac.emplace_back(fac1[i].p, fac1[i].e + fac2[j].e,
                       U(fac1[i].pe) * U(fac2[j].pe));
      i++, j++;
    }
  }
  fac.insert(fac.end(), fac1.begin() + i, fac1.end());
  fac.insert(fac.end(), fac2.begin() + j, fac2.end());
  return fac;
}
// https://github.com/miscalculation53/library/tree/wip/math/prime/large/primality_test.hpp

/*!
 * Portions corresponding to AtCoder Library (CC0-1.0): Miller-Rabin termination logic.
 * Upstream: atcoder/internal_math.hpp
 * https://github.com/atcoder/ac-library/tree/864245a00b00dd008d1abfdc239618fdb7d139da
 * License: https://github.com/atcoder/ac-library/blob/864245a00b00dd008d1abfdc239618fdb7d139da/LICENSE
 */

namespace internal
{

template <class mint, class Array>
bool is_prime_impl(ll n, const Array &bases)
{
  if (n <= 1)
    return false;
  if (n == 2 || n == 7 || n == 61)
    return true;
  if (n % 2 == 0)
    return false;
  ll d = (n - 1) >> countr_zero(n - 1);
  mint::set_mod(n);
  for (ll a : bases)
  {
    ll t = d;
    mint y = mint(a).pow(t);
    while (t != n - 1 && y != 1 && y != n - 1)
    {
      y *= y;
      t <<= 1;
    }
    if (y != n - 1 && t % 2 == 0)
      return false;
  }
  return true;
}

}; 

bool is_prime(ll n)
{
  static constexpr array<ll, 3> bases32 = {2, 7, 61};
  static constexpr array<ll, 7> bases64 = {2, 325, 9375, 28178, 450775, 9780504, 1795265022};
  if (n <= INT_MAX)
  {
    using mint = dynamic_modint32<INT_MIN>;
    return internal::is_prime_impl<mint>(n, bases32);
  }
  else
  {
    using mint = dynamic_modint64_odd<INT_MIN>;
    return internal::is_prime_impl<mint>(n, bases64);
  }
}
// https://github.com/miscalculation53/library/tree/wip/math/prime/large/factorize.hpp

namespace internal
{

template <class mint>
ll get_prime_factor_impl(ll n)
{
  mint::set_mod(n);
  int m = pow(n, .125);
  while (true)
  {
    int c = randrange(1, 100);
    mint x = 2, y = 2, prod = 1;
    ll g = 1;
    while (g == 1)
    {
      repi(i, m)
      {
        x = x * x + c;
        y = y * y + c, y = y * y + c;
        prod *= x - y;
      }
      g = gcd(prod.val(), n);
    }
    if (g == n)
      continue;
    if (is_prime(g))
      return g;
    else if (is_prime(n / g))
      return n / g;
    else
      return get_prime_factor_impl<mint>(g);
  }
}

ll get_prime_factor(ll n)
{
  if (n <= INT_MAX)
  {
    using mint = dynamic_modint32<INT_MIN>;
    return get_prime_factor_impl<mint>(n);
  }
  else
  {
    using mint = dynamic_modint64_odd<INT_MIN>;
    return get_prime_factor_impl<mint>(n);
  }
}

}; 

vc<PrimePower<ll>> factorize(ll n)
{
  vc<PrimePower<ll>> res;
  repi(p, 2, 100)
  {
    if (n % p == 0)
    {
      auto [e, pe, nn] = ord_pow_div(n, p);
      res.emplace_back(PrimePower<ll>(p, e, pe));
      n = nn;
    }
  }
  while (n > 1)
  {
    if (is_prime(n))
    {
      res.emplace_back(n);
      break;
    }
    ll p = internal::get_prime_factor(n);
    auto [e, pe, nn] = ord_pow_div(n, p);
    res.emplace_back(PrimePower<ll>(p, e, pe));
    n = nn;
  }
  sort(ALL(res), [&](const PrimePower<ll> &pp1, const PrimePower<ll> &pp2)
       { return pp1.p < pp2.p; });
  return res;
}
// https://github.com/miscalculation53/library/tree/wip/math/prime/large/euler_phi_carmichael.hpp

//! https://37zigen.com/primitive-root/#i-4
//! https://x.com/noshi91/status/1317025404700405760

namespace internal
{

template <class mint, class I>
vc<mint> lagrange_basis(const vc<I> &a, mint x)
{
  const int n = a.size();
  vc<mint> res(n);
  if (n == 0)
    return res;
  auto dfs = [&](auto dfs, mint v, int l, int r)
  {
    if (r - l == 1)
    {
      res[l] = v;
      return;
    }
    int m = (l + r) / 2;
    mint v_l = v, v_r = v;
    repi(i, l, m) v_r = v_r.pow(a[i]);
    repi(i, m, r) v_l = v_l.pow(a[i]);
    dfs(dfs, v_l, l, m);
    dfs(dfs, v_r, m, r);
  };
  dfs(dfs, x, 0, n);
  return res;
}

template <class mint, class P>
bool internal_is_primitive_root(ll x, ll p, const vc<PrimePower<P>> &fac)
{
  const int k = fac.size();
  mint::set_mod(p);
  vc<ll> a(k);
  repi(i, k) a[i] = fac[i].pe;
  auto b = lagrange_basis(a, mint(x));
  repi(i, k)
  {
    if (b[i].pow(fac[i].pe / fac[i].p) == 1)
      return false;
  }
  return true;
}

template <class mint, class P>
ll internal_primitive_root(ll p, const vc<PrimePower<P>> &fac)
{
  if (p == 2)
    return 1;
  while (true)
  {
    ll g = randrange(1, p);
    if (internal_is_primitive_root<mint>(g, p, fac))
      return g;
  }
}

}; 

template <class P>
ll primitive_root(ll p, const vc<PrimePower<P>> &fac)
{
  if (p <= INT_MAX)
  {
    using mint = dynamic_modint32<INT_MIN>;
    return internal::internal_primitive_root<mint>(p, fac);
  }
  else
  {
    using mint = dynamic_modint64_odd<INT_MIN>;
    return internal::internal_primitive_root<mint>(p, fac);
  }
}

// https://github.com/miscalculation53/library/tree/wip/math/convolution/convolution.hpp

/*!
 * Portions corresponding to AtCoder Library (CC0-1.0): primitive-root search and NTT butterflies.
 * Upstream: atcoder/convolution.hpp and atcoder/internal_math.hpp
 * https://github.com/atcoder/ac-library/tree/864245a00b00dd008d1abfdc239618fdb7d139da
 * License: https://github.com/atcoder/ac-library/blob/864245a00b00dd008d1abfdc239618fdb7d139da/LICENSE
 */

// https://github.com/miscalculation53/library/tree/wip/math/crt.hpp

/*!
 * Portions corresponding to AtCoder Library (CC0-1.0): two-congruence merge.
 * Upstream: atcoder/math.hpp
 * https://github.com/atcoder/ac-library/tree/864245a00b00dd008d1abfdc239618fdb7d139da
 * License: https://github.com/atcoder/ac-library/blob/864245a00b00dd008d1abfdc239618fdb7d139da/LICENSE
 */

namespace crt_internal
{
namespace crt_word_detail
{

}

namespace crt_word_dispatch
{
  constexpr bool constant_evaluation() { return __builtin_is_constant_evaluated(); }

}
}

namespace pre_crt_v2_adl
{
  void gcd() = delete;
  template <class T>
  auto call(const T &a, const T &b) -> decltype(gcd(a, b));
}

namespace pre_crt_v2_detail
{
  template <class T, bool = is_integral_ext<T>> struct work_type { using type = T; };
  template <class T> struct work_type<T, true> { using type = make_unsigned_ext_t<T>; };
  template <class T, class = void> struct has_adl_gcd : false_type {};
  template <class T> struct has_adl_gcd<T, void_t<decltype(pre_crt_v2_adl::call(declval<T>(), declval<T>()))>> : true_type {};
  template <class T>
  constexpr bool unbounded_integer = numeric_limits<T>::is_specialized &&
                                     numeric_limits<T>::is_integer && !numeric_limits<T>::is_bounded;

}

// https://github.com/miscalculation53/library/tree/wip/math/dot_product.hpp

// https://github.com/miscalculation53/library/tree/wip/algebra/algebra_basic_ops.hpp

// https://github.com/miscalculation53/library/tree/wip/algebra/algebra_base.hpp

// https://github.com/miscalculation53/library/tree/wip/algebra/internal_power.hpp

//! SPDX-License-Identifier: CC0-1.0

namespace internal_monoid_power
{

  template <class T, class I, class = void>
  struct HasMemberScalarProduct : false_type {};
  template <class T, class I>
  struct HasMemberScalarProduct<T, I, void_t<decltype(T(declval<const T &>().operator*({declval<I>()})))>> : true_type {};
  template <class T, class I, class = void>
  struct HasFreeScalarProduct : false_type {};
  template <class T, class I>
  struct HasFreeScalarProduct<T, I, void_t<decltype(T(operator*(declval<const T &>(), {declval<I>()})))>> : true_type {};
  template <class T, class I>
  struct HasScalarProduct : bool_constant<is_arithmetic_v<T> || is_same_v<T, f128> ||
                                         HasMemberScalarProduct<T, I>::value || HasFreeScalarProduct<T, I>::value> {};

  template <class M, class I>
  using ConstRefPower = typename M::S (*)(const typename M::S &, I);
  template <class M, class I>
  using ValuePower = typename M::S (*)(typename M::S, I);
  template <class M, class I, class = void>
  struct HasConstRefPower : false_type {};
  template <class M, class I>
  struct HasConstRefPower<M, I, void_t<decltype(static_cast<ConstRefPower<M, I>>(M::pow))>> : true_type {};
  template <class M, class I, class = void>
  struct HasValuePower : false_type {};
  template <class M, class I>
  struct HasValuePower<M, I, void_t<decltype(static_cast<ValuePower<M, I>>(M::pow))>> : true_type {};

}

template <class S_, auto op_, auto e_>
struct Monoid
{
  using S = S_;
  static constexpr auto op = op_;
  static constexpr auto e = e_;
};

template <class S_, auto op_, auto e_, auto inv_>
struct Group
{
  using S = S_;
  static constexpr auto op = op_;
  static constexpr auto e = e_;
  static constexpr auto inv = inv_;
};

template <class S_, auto add_, auto e0_, auto mul_, auto e1_>
struct SemiRing
{
  using S = S_;
  static constexpr auto add = add_;
  static constexpr auto e0 = e0_;
  static constexpr auto mul = mul_;
  static constexpr auto e1 = e1_;
};

template <class S_, auto add_, auto e0_, auto minus_, auto mul_, auto e1_>
struct Ring
{
  using S = S_;
  static constexpr auto add = add_;
  static constexpr auto e0 = e0_;
  static constexpr auto minus = minus_;
  static constexpr auto mul = mul_;
  static constexpr auto e1 = e1_;
};

template <class S_, auto add_, auto e0_, auto minus_, auto mul_, auto e1_, auto inv_>
struct Field
{
  using S = S_;
  static constexpr auto add = add_;
  static constexpr auto e0 = e0_;
  static constexpr auto minus = minus_;
  static constexpr auto mul = mul_;
  static constexpr auto e1 = e1_;
  static constexpr auto inv = inv_;
};

template <class M, class = void>
struct HasInverse : false_type {};
template <class M>
struct HasInverse<M, void_t<decltype(M::inv(declval<const typename M::S &>()))>> : true_type {};

template <class M>
struct OppositeMonoid
{
  using S = typename M::S;
  static constexpr auto e = M::e;
};
template <class G>
struct OppositeGroup
{
  using S = typename G::S;
  static constexpr auto e = G::e;
  static constexpr auto inv = G::inv;
};
template <class M>
struct NormalAndOppositeMonoid
{
  struct S
  {
    typename M::S normal;
    typename M::S opposite;
    S();
    template <class... Args,
              std::enable_if_t<std::is_constructible_v<typename M::S, Args...>, std::nullptr_t> = nullptr>
    S(Args &&...args);
    S(const typename M::S &normal, const typename M::S &opposite);
  };
  static constexpr S e();
};
template <class G>
struct NormalAndOppositeGroup
{
  struct S
  {
    typename G::S normal;
    typename G::S opposite;
    S();
    template <class... Args,
              std::enable_if_t<std::is_constructible_v<typename G::S, Args...>, std::nullptr_t> = nullptr>
    S(Args &&...args);
    S(const typename G::S &normal, const typename G::S &opposite);
  };
  static constexpr S e();
  static constexpr S inv(const S &a) { return {G::inv(a.normal), G::inv(a.opposite)}; }
};

template <class SR>
using MonoidOfSemiRingAdd = Monoid<typename SR::S, SR::add, SR::e0>;
template <class SR>
using MonoidOfSemiRingMul = Monoid<typename SR::S, SR::mul, SR::e1>;
template <class R>
using GroupOfRingAdd = Group<typename R::S, R::add, R::e0, R::minus>;
template <class K>
using GroupOfFieldMul = Group<typename K::S, K::mul, K::e1, K::inv>;

template <class Madd, class Mmul>
struct SemiRingFromMonoidMonoid
{
  static_assert(is_same_v<typename Madd::S, typename Mmul::S>, "Madd::S and Mmul::S must be identical");
  using S = typename Madd::S;
  static constexpr auto add = Madd::op;
  static constexpr auto e0 = Madd::e;
  static constexpr auto mul = Mmul::op;
  static constexpr auto e1 = Mmul::e;
};

template <class Gadd, class Mmul>
struct RingFromGroupMonoid
{
  static_assert(is_same_v<typename Gadd::S, typename Mmul::S>, "Gadd::S and Mmul::S must be identical");
  using S = typename Gadd::S;
  static constexpr auto add = Gadd::op;
  static constexpr auto e0 = Gadd::e;
  static constexpr auto minus = Gadd::inv;
  static constexpr auto mul = Mmul::op;
  static constexpr auto e1 = Mmul::e;
};

template <class Gadd, class Gmul>
struct FieldFromGroupGroup
{
  static_assert(is_same_v<typename Gadd::S, typename Gmul::S>, "Gadd::S and Gmul::S must be identical");
  using S = typename Gadd::S;
  static constexpr auto add = Gadd::op;
  static constexpr auto e0 = Gadd::e;
  static constexpr auto minus = Gadd::inv;
  static constexpr auto mul = Gmul::op;
  static constexpr auto e1 = Gmul::e;
  static constexpr auto inv = Gmul::inv;
};

template <class T, class = void>
struct has_e1 : false_type {};
template <class T>
struct has_e1<T, void_t<decltype(T::e1())>> : true_type {};
template <class T>
inline constexpr bool has_e1_v = has_e1<T>::value;

template <class T>
struct MonoidMul
{
  using S = T;
  static constexpr S e();
};

template <class T>
struct GroupAddSub
{
  using S = T;
  static constexpr S e();
  static constexpr S inv(S a) { return -a; }
};
template <class T>
struct GroupMulDiv
{
  using S = T;
  static constexpr S e();
  static constexpr S inv(S a) { return e() / a; }
};

template <class T, auto infty = nullptr>
using SemiRingMinPlus = SemiRingFromMonoidMonoid<MonoidMin<T, infty>, MonoidAdd<T>>;
template <class T, auto infty = nullptr>
using SemiRingMaxPlus = SemiRingFromMonoidMonoid<MonoidMax<T, infty>, MonoidAdd<T>>;
template <class T>
using RingAddSubMul = RingFromGroupMonoid<GroupAddSub<T>, MonoidMul<T>>;
template <class T>
using FieldAddSubMulDiv = FieldFromGroupGroup<GroupAddSub<T>, GroupMulDiv<T>>;
// https://github.com/miscalculation53/library/tree/wip/math/modint/internal_mod32.hpp

namespace internal
{

template <class S>
struct ordinary_mod32 : false_type {};
template <int mod, class PowInv>
struct ordinary_mod32<modint_impl<policy_static<mod>, PowInv>> : true_type {};
template <int id>
struct ordinary_mod32<dynamic_modint32<id>> : true_type {};

template <class G>
struct ordinary_mod32_add_group : false_type {};
template <class S>
struct ordinary_mod32_add_group<GroupAddSub<S>> : ordinary_mod32<S> {};
template <class S>
struct ordinary_mod32_add_group<Group<S, GroupAddSub<S>::op, GroupAddSub<S>::e, GroupAddSub<S>::inv>>
    : ordinary_mod32<S> {};

template <class F>
struct ordinary_mod32_field : false_type {};
template <class S>
struct ordinary_mod32_field<FieldAddSubMulDiv<S>> : ordinary_mod32<S> {};
}

namespace internal
{
template <class S>
using dot_product_mod32_value = ordinary_mod32<S>;

template <class SR>
struct dot_product_mod32 : false_type {};
template <class S>
struct dot_product_mod32<FieldAddSubMulDiv<S>> : dot_product_mod32_value<S> {};
template <class S>
struct dot_product_mod32<RingAddSubMul<S>> : dot_product_mod32_value<S> {};
template <class S>
struct dot_product_mod32<SemiRingFromMonoidMonoid<MonoidAdd<S>, MonoidMul<S>>>
    : dot_product_mod32_value<S> {};

} 

namespace internal
{

template <class T>
constexpr T primitive_root_constexpr(T 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;
  if (m == 1107296257)
    return 10;
  if (m == 1711276033)
    return 29;
  if (m == 1811939329)
    return 13;
  if (m == 2013265921)
    return 31;
  if (m == 2113929217)
    return 5;
  T divs[20] = {};
  divs[0] = 2;
  int cnt = 1;
  T x = (m - 1) / 2;
  while (x % 2 == 0)
    x /= 2;
  for (T i = 3; i <= x / i; i += 2)
  {
    if (x % i == 0)
    {
      divs[cnt++] = i;
      while (x % i == 0)
      {
        x /= i;
      }
    }
  }
  if (x > 1)
  {
    divs[cnt++] = x;
  }
  for (T g = 2;; g++)
  {
    bool ok = true;
    for (int i = 0; i < cnt; i++)
    {
      if (powmod_constexpr(g, (m - 1) / divs[i], m) == 1)
      {
        ok = false;
        break;
      }
    }
    if (ok)
      return g;
  }
}
template <auto m>
constexpr auto primitive_root_for_convolution = primitive_root_constexpr(m);

template <class T>
constexpr T ntt_generator_constexpr(T m)
{
  if (m <= INT_MAX) return primitive_root_constexpr(m);
  if ((m - 1) % 4 == 2) return m - 1;
  for (T g = 2; ; ++g)
    if (powmod_constexpr(g, (m - 1) / 2, m) == m - 1) return g;
}
template <auto m>
constexpr auto ntt_generator_for_convolution = ntt_generator_constexpr(m);

template <class mint, auto g = internal::ntt_generator_for_convolution<mint::mod()>>
struct fft_info
{
  static constexpr int rank2 = countr_zero(mint::mod() - 1);
  std::array<mint, std::max(3, rank2 + 1)> root;  
  std::array<mint, std::max(3, rank2 + 1)> iroot; 

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

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

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

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

}  

template <class mint>
bool ntt_ok(int n)
{
  if (n <= 0)
    return false;
  if constexpr (is_static_modint_v<mint>)
  {
    if constexpr (!internal::isprime<mint::mod()>)
      return false;
    static constexpr int rank2 = countr_zero(mint::mod() - 1);
    return ull(n) <= (1ULL << rank2);
  }
  else
    return false;
}

template <int id>
void ntt(vc<dynamic_modint32<id>> &) { assert(false); }

template <auto mod, class PowInv>
void ntt(vc<internal::modint_impl<internal::policy_static<mod>, PowInv>> &a)
{
  using mint = internal::modint_impl<internal::policy_static<mod>, PowInv>;
  int n = int(a.size());
  assert(n > 0);
  int h = countr_zero((unsigned int)n);
  assert(n == (1 << h));
  assert(ntt_ok<mint>(n));

  using wide = conditional_t<(mint::mod() <= INT_MAX), ull, u128>;
  static const internal::fft_info<mint> info;

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

template <auto mod, class PowInv>
void intt(vc<internal::modint_impl<internal::policy_static<mod>, PowInv>> &a)
{
  using mint = internal::modint_impl<internal::policy_static<mod>, PowInv>;
  int n = int(a.size());
  assert(n > 0);
  int h = countr_zero((unsigned int)n);
  assert(n == (1 << h));
  assert(ntt_ok<mint>(n));

  using wide = conditional_t<(mint::mod() <= INT_MAX), ull, u128>;
  static const internal::fft_info<mint> info;

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

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

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

namespace internal
{

}  

void main2()
{
  const ll P = mint103::mod();
  const ll g = primitive_root(P, factorize(P - 1));

  LL(N, Q);
  vc<mint103> A(N);
  rep(i, N) A[i] = xor128();
  dump(A);

  vc<mint103> pw(P - 1);
  pw[0] = 1;
  rep(i, 1, P - 1) pw[i] = pw[i - 1] * g;
  vl lg(P);
  rep(i, P - 1) lg[pw[i].val()] = i;

  vl B;
  rep(i, N) if (A[i] != 0) B.eb(lg[A[i].val()]);

  vl ipw(P);
  rep(i, P - 1) ipw[pw[i].val()] = i;

  const ll K = 2048;
  vc<mint998> f(P - 1);
  fe(b : B) f[safemod(-b, P - 1)] = 1;
  auto nf = f;
  nf.resize(2 * bit_ceil(SZ(nf)));
  ntt(nf);
  vvc<mint998> hs;
  rep(block, divceil(P, K))
  {
    vc<mint998> g(P - 1);
    rep(ans, block * K, min(P, (block + 1) * K))
    {
      ll z = ipw[ans];
      g[z] = 1;
    }
    g.resize(2 * bit_ceil(SZ(g)));
    ntt(g);
    rep(y, SZ(g)) g[y] *= nf[y];
    intt(g);
    const mint998 iv = mint998(g.size()).inv();
    rep(y, SZ(g)) g[y] *= iv;
    rep(y, P - 1, SZ(g)) g[y % (P - 1)] += g[y];
    g.resize(P - 1);
    hs.eb(g);
  }

  rep(_, Q)
  {
    LL(x);
    if (x == 0)
      PRINT(0);
    else
    {
      ll y = lg[x];
      // ll ans = 0;
      // fe(b : B) chmax(ans, pw[(y + b) % (P - 1)].val());
      // PRINT(ans);
      rep(block, divceil(P, K) - 1, -1, -1)
      {
        if (hs[block][y] != 0)
        {
          rep(ans, min(P, (block + 1) * K) - 1, block * K - 1, -1)
          {
            ll z = ipw[ans];
            ll b = safemod(z - y, P - 1);
            if (f[safemod(-b, P - 1)] != 0)
            {
              PRINT(ans);
              break;
            }
          }
          break;
        }
      }
    }
  }
}

void test()
{

}

// https://github.com/miscalculation53/library/tree/wip/template/template_main.hpp

template <auto init, auto main2, auto test>
struct Main
{
  Main()
  {
    cauto CERR = [](string val, string color)
    {
      string s = "\033[" + color + "m" + val + "\033[m";
      
    };
  
    CERR("\n[FAST_IO]\n\n", "32");
    
    cout << fixed << setprecision(20);
  
    init();
    
    CERR("\n[SINGLE_TESTCASE]\n\n", "36");
    main2();
  }
};
Main<init, main2, test> main_dummy;
int main() {}


// submit-code: hash=09a7acc3bf18 source=09a7acc3bf1830af566dbeac1ed8b9da1799f36d5f0f02d20760e0b8a372cb4e input=df2908542d42c6758df5264f214259cfce58289af866b226b93ea0bf698c522b body=53164d901de5611b4a26a6ac9339c8f5e96891333c59101e7e70fc6e17a0f0ed