#line 1 "other/y.cc"
// #undef _GLIBCXX_DEBUG
// #define NDEBUG
#include <bits/extc++.h>

#line 2 "Library/lib/alias"

/**
 * @file alias
 * @brief Alias
 */

#line 13 "Library/lib/alias"

#line 2 "Library/lib/bit"

#if __cplusplus > 201703L

#include <bit>

#else

#ifndef _GLIBCXX_BIT
#define _GLIBCXX_BIT 1

#include <limits>
#include <type_traits>

namespace std {

template <typename _Tp> constexpr _Tp __rotl(_Tp __x, int __s) noexcept {
  constexpr auto _Nd = numeric_limits<_Tp>::digits;
  const int __r = __s % _Nd;
  if (__r == 0)
    return __x;
  else if (__r > 0)
    return (__x << __r) | (__x >> ((_Nd - __r) % _Nd));
  else
    return (__x >> -__r) | (__x << ((_Nd + __r) % _Nd));  // rotr(x, -r)
}

template <typename _Tp> constexpr _Tp __rotr(_Tp __x, int __s) noexcept {
  constexpr auto _Nd = numeric_limits<_Tp>::digits;
  const int __r = __s % _Nd;
  if (__r == 0)
    return __x;
  else if (__r > 0)
    return (__x >> __r) | (__x << ((_Nd - __r) % _Nd));
  else
    return (__x << -__r) | (__x >> ((_Nd + __r) % _Nd));  // rotl(x, -r)
}

template <typename _Tp> constexpr int __countl_zero(_Tp __x) noexcept {
  constexpr auto _Nd = numeric_limits<_Tp>::digits;

  if (__x == 0) return _Nd;

  constexpr auto _Nd_ull = numeric_limits<unsigned long long>::digits;
  constexpr auto _Nd_ul = numeric_limits<unsigned long>::digits;
  constexpr auto _Nd_u = numeric_limits<unsigned>::digits;

  if _GLIBCXX17_CONSTEXPR (_Nd <= _Nd_u) {
    constexpr int __diff = _Nd_u - _Nd;
    return __builtin_clz(__x) - __diff;
  } else if _GLIBCXX17_CONSTEXPR (_Nd <= _Nd_ul) {
    constexpr int __diff = _Nd_ul - _Nd;
    return __builtin_clzl(__x) - __diff;
  } else if _GLIBCXX17_CONSTEXPR (_Nd <= _Nd_ull) {
    constexpr int __diff = _Nd_ull - _Nd;
    return __builtin_clzll(__x) - __diff;
  } else  // (_Nd > _Nd_ull)
  {
    static_assert(_Nd <= (2 * _Nd_ull),
                  "Maximum supported integer size is 128-bit");

    unsigned long long __high = __x >> _Nd_ull;
    if (__high != 0) {
      constexpr int __diff = (2 * _Nd_ull) - _Nd;
      return __builtin_clzll(__high) - __diff;
    }
    constexpr auto __max_ull = numeric_limits<unsigned long long>::max();
    unsigned long long __low = __x & __max_ull;
    return (_Nd - _Nd_ull) + __builtin_clzll(__low);
  }
}

template <typename _Tp> constexpr int __countl_one(_Tp __x) noexcept {
  if (__x == numeric_limits<_Tp>::max()) return numeric_limits<_Tp>::digits;
  return __countl_zero<_Tp>((_Tp)~__x);
}

template <typename _Tp> constexpr int __countr_zero(_Tp __x) noexcept {
  constexpr auto _Nd = numeric_limits<_Tp>::digits;

  if (__x == 0) return _Nd;

  constexpr auto _Nd_ull = numeric_limits<unsigned long long>::digits;
  constexpr auto _Nd_ul = numeric_limits<unsigned long>::digits;
  constexpr auto _Nd_u = numeric_limits<unsigned>::digits;

  if _GLIBCXX17_CONSTEXPR (_Nd <= _Nd_u)
    return __builtin_ctz(__x);
  else if _GLIBCXX17_CONSTEXPR (_Nd <= _Nd_ul)
    return __builtin_ctzl(__x);
  else if _GLIBCXX17_CONSTEXPR (_Nd <= _Nd_ull)
    return __builtin_ctzll(__x);
  else  // (_Nd > _Nd_ull)
  {
    static_assert(_Nd <= (2 * _Nd_ull),
                  "Maximum supported integer size is 128-bit");

    constexpr auto __max_ull = numeric_limits<unsigned long long>::max();
    unsigned long long __low = __x & __max_ull;
    if (__low != 0) return __builtin_ctzll(__low);
    unsigned long long __high = __x >> _Nd_ull;
    return __builtin_ctzll(__high) + _Nd_ull;
  }
}

template <typename _Tp> constexpr int __countr_one(_Tp __x) noexcept {
  if (__x == numeric_limits<_Tp>::max()) return numeric_limits<_Tp>::digits;
  return __countr_zero((_Tp)~__x);
}

template <typename _Tp> constexpr int __popcount(_Tp __x) noexcept {
  constexpr auto _Nd = numeric_limits<_Tp>::digits;

  if (__x == 0) return 0;

  constexpr auto _Nd_ull = numeric_limits<unsigned long long>::digits;
  constexpr auto _Nd_ul = numeric_limits<unsigned long>::digits;
  constexpr auto _Nd_u = numeric_limits<unsigned>::digits;

  if _GLIBCXX17_CONSTEXPR (_Nd <= _Nd_u)
    return __builtin_popcount(__x);
  else if _GLIBCXX17_CONSTEXPR (_Nd <= _Nd_ul)
    return __builtin_popcountl(__x);
  else if _GLIBCXX17_CONSTEXPR (_Nd <= _Nd_ull)
    return __builtin_popcountll(__x);
  else  // (_Nd > _Nd_ull)
  {
    static_assert(_Nd <= (2 * _Nd_ull),
                  "Maximum supported integer size is 128-bit");

    constexpr auto __max_ull = numeric_limits<unsigned long long>::max();
    unsigned long long __low = __x & __max_ull;
    unsigned long long __high = __x >> _Nd_ull;
    return __builtin_popcountll(__low) + __builtin_popcountll(__high);
  }
}

template <typename _Tp> constexpr bool __has_single_bit(_Tp __x) noexcept {
  return __popcount(__x) == 1;
}

template <typename _Tp> constexpr _Tp __bit_ceil(_Tp __x) noexcept {
  constexpr auto _Nd = numeric_limits<_Tp>::digits;
  if (__x == 0 || __x == 1) return 1;
  auto __shift_exponent = _Nd - __countl_zero((_Tp)(__x - 1u));
#ifdef _GLIBCXX_HAVE_BUILTIN_IS_CONSTANT_EVALUATED
  if (!__builtin_is_constant_evaluated()) {
    __glibcxx_assert(__shift_exponent != numeric_limits<_Tp>::digits);
  }
#endif
  using __promoted_type = decltype(__x << 1);
  if _GLIBCXX17_CONSTEXPR (!is_same<__promoted_type, _Tp>::value) {
    const int __extra_exp = sizeof(__promoted_type) / sizeof(_Tp) / 2;
    __shift_exponent |= (__shift_exponent & _Nd) << __extra_exp;
  }
  return (_Tp)1u << __shift_exponent;
}

template <typename _Tp> constexpr _Tp __bit_floor(_Tp __x) noexcept {
  constexpr auto _Nd = numeric_limits<_Tp>::digits;
  if (__x == 0) return 0;
  return (_Tp)1u << (_Nd - __countl_zero((_Tp)(__x >> 1)));
}

template <typename _Tp> constexpr _Tp __bit_width(_Tp __x) noexcept {
  constexpr auto _Nd = numeric_limits<_Tp>::digits;
  return _Nd - __countl_zero(__x);
}

}  // namespace std

#endif

#endif
#line 2 "Library/lib/limits"

#line 4 "Library/lib/limits"

namespace std {

#if defined(__STRICT_ANSI__) && defined(__SIZEOF_INT128__)

template <> struct numeric_limits<__uint128_t> {
  constexpr static __uint128_t max() { return ~__uint128_t(0); }
  constexpr static __uint128_t min() { return 0; }
};

template <> struct numeric_limits<__int128_t> {
  constexpr static __int128_t max() {
    return numeric_limits<__uint128_t>::max() >> 1;
  }
  constexpr static __int128_t min() { return -max() - 1; }
};

#endif

}  // namespace std
#line 16 "Library/lib/alias"

namespace workspace {

constexpr char eol = '\n';

using namespace std;

using i32 = int_least32_t;
using u32 = uint_least32_t;
using i64 = int_least64_t;
using u64 = uint_least64_t;

#ifdef __SIZEOF_INT128__
using i128 = __int128_t;
using u128 = __uint128_t;
#else
#warning 128bit integer is not available.
#endif

template <class T, class Comp = less<T>>
using priority_queue = std::priority_queue<T, vector<T>, Comp>;

template <class T> using stack = std::stack<T, vector<T>>;

template <typename _Tp> constexpr _Tp __bsf(_Tp __x) noexcept {
  return std::__countr_zero(__x);
}

template <typename _Tp> constexpr _Tp __bsr(_Tp __x) noexcept {
  return std::__bit_width(__x) - 1;
}

}  // namespace workspace
#line 2 "Library/lib/cxx20"

/*
 * @file cxx20
 * @brief C++20 Features
 */

#line 9 "Library/lib/cxx20"

#if __cplusplus <= 201703L

#include <algorithm>
#include <vector>

namespace std {

/*
 * @fn erase_if
 * @brief Erase the elements of a container that do not satisfy the condition.
 * @param __cont Container.
 * @param __pred Predicate.
 * @return Number of the erased elements.
 */
template <typename _Tp, typename _Alloc, typename _Predicate>
inline typename vector<_Tp, _Alloc>::size_type erase_if(
    vector<_Tp, _Alloc>& __cont, _Predicate __pred) {
  const auto __osz = __cont.size();
  __cont.erase(remove_if(__cont.begin(), __cont.end(), __pred), __cont.end());
  return __osz - __cont.size();
}

/*
 * @fn erase
 * @brief Erase the elements of a container that are equal to the given value.
 * @param __cont Container.
 * @param __value Value.
 * @return Number of the erased elements.
 */
template <typename _Tp, typename _Alloc, typename _Up>
inline typename vector<_Tp, _Alloc>::size_type erase(
    vector<_Tp, _Alloc>& __cont, const _Up& __value) {
  const auto __osz = __cont.size();
  __cont.erase(remove(__cont.begin(), __cont.end(), __value), __cont.end());
  return __osz - __cont.size();
}

}  // namespace std

#endif
#line 2 "Library/lib/direct"

/*
 * @file direct
 * @brief Pragma Directive
 */

#ifdef ONLINE_JUDGE

#pragma GCC optimize("O3")
#pragma GCC target("avx,avx2")
#pragma GCC optimize("unroll-loops")

#endif
#line 8 "other/y.cc"
// #include "lib/opt"
#line 2 "Library/src/sys/clock.hpp"

/*
 * @fn clock.hpp
 * @brief Clock
 */

#line 9 "Library/src/sys/clock.hpp"

namespace workspace {

using namespace std::chrono;

namespace internal {
// The start time of the program.
const auto start_time{system_clock::now()};
}  // namespace internal

/*
 * @fn elapsed
 * @return elapsed time of the program
 */
int64_t elapsed() {
  const auto end_time{system_clock::now()};
  return duration_cast<milliseconds>(end_time - internal::start_time).count();
}

}  // namespace workspace
#line 2 "Library/src/sys/ejection.hpp"

/**
 * @file ejection.hpp
 * @brief Ejection
 */

#line 9 "Library/src/sys/ejection.hpp"

namespace workspace {

namespace internal {

struct ejection {
  bool exit = 0;
};

}  // namespace internal

/**
 * @brief eject from a try block, throw nullptr
 * @param arg output
 */
template <class Tp> void eject(Tp const &arg) {
  std::cout << arg << "\n";
  throw internal::ejection{};
}

void exit() { throw internal::ejection{true}; }

}  // namespace workspace
#line 2 "Library/src/sys/iteration.hpp"

/**
 * @file iteration.hpp
 * @brief Case Iteration
 */

#line 9 "Library/src/sys/iteration.hpp"

#line 11 "Library/src/sys/iteration.hpp"

namespace workspace {

void main();

struct {
  // 1-indexed
  unsigned current{0};
  unsigned total{1};

  void read() { (std::cin >> total).ignore(); }

  int iterate() {
    static bool once = false;
    assert(!once);
    once = true;
    while (current++ < total) {
      try {
        main();
      } catch (internal::ejection const& status) {
        if (status.exit) break;
      }
    }
    return 0;
  }
} case_info;

}  // namespace workspace
#line 2 "Library/src/utils/cat.hpp"

/**
 * @file cat.hpp
 * @brief Cat
 */

#line 9 "Library/src/utils/cat.hpp"

namespace workspace {

template <class C1, class C2>
constexpr C1 &&cat(C1 &&__c1, C2 const &__c2) noexcept {
  __c1.insert(__c1.end(), std::begin(__c2), std::end(__c2));
  return __c1;
}

}  // namespace workspace
#line 2 "Library/src/utils/chval.hpp"

/*
 * @file chval.hpp
 * @brief Change Less/Greater
 */

#line 9 "Library/src/utils/chval.hpp"

namespace workspace {

/*
 * @fn chle
 * @brief Substitute y for x if comp(y, x) is true.
 * @param x Reference
 * @param y Const reference
 * @param comp Compare function
 * @return Whether or not x is updated
 */
template <class Tp, class Comp = std::less<Tp>>
bool chle(Tp &x, const Tp &y, Comp comp = Comp()) {
  return comp(y, x) ? x = y, true : false;
}

/*
 * @fn chge
 * @brief Substitute y for x if comp(x, y) is true.
 * @param x Reference
 * @param y Const reference
 * @param comp Compare function
 * @return Whether or not x is updated
 */
template <class Tp, class Comp = std::less<Tp>>
bool chge(Tp &x, const Tp &y, Comp comp = Comp()) {
  return comp(x, y) ? x = y, true : false;
}

}  // namespace workspace
#line 2 "Library/src/utils/fixed_point.hpp"

/**
 * @file fixed_point.hpp
 * @brief Fixed Point Combinator
 */

#line 9 "Library/src/utils/fixed_point.hpp"

namespace workspace {

/**
 * @brief Fixed Point Combinator
 */
template <class _F> class fixed_point {
  _F __fn;

 public:
  /**
   * @param __fn 1st argument callable with the rest of its arguments, and the
   * return type specified.
   */
  fixed_point(_F &&__fn) noexcept : __fn(std::forward<_F>(__fn)) {}

  /**
   * @brief Apply *this to 1st argument.
   * @param __args Rest of arguments.
   */
  template <class... _Args>
  decltype(auto) operator()(_Args &&... __args) const noexcept {
    return __fn(*this, std::forward<_Args>(__args)...);
  }
};

/**
 * @brief Cached version of Fixed Point Combinator
 */
template <class _F> class cached_fixed_point {
  template <class...> struct _cache;

  template <class _G, class _R, class _H, class... _Args>
  struct _cache<_R (_G::*)(_H, _Args...) const>
      : std::map<std::tuple<_Args...>, _R> {};

  using cache =
      _cache<decltype(&_F::template operator()<cached_fixed_point<_F> &>)>;

  _F __fn;
  static cache __ca;

 public:
  /**
   * @param __fn 1st argument callable with the rest of its arguments, and the
   * return type specified.
   */
  cached_fixed_point(_F &&__fn) noexcept : __fn(std::forward<_F>(__fn)) {}

  /**
   * @brief Apply *this to 1st argument.
   * @param __args Rest of arguments.
   */
  template <class... _Args>
  decltype(auto) operator()(_Args &&... __args) noexcept {
    typename cache::key_type __key(__args...);

    if (auto __i = __ca.lower_bound(__key);
        __i != __ca.end() && __i->first == __key)
      return __i->second;

    else
      return __ca
          .emplace_hint(__i, std::move(__key),
                        __fn(*this, std::forward<_Args>(__args)...))
          ->second;
  }
};

template <class _F>
typename cached_fixed_point<_F>::cache cached_fixed_point<_F>::__ca;

}  // namespace workspace
#line 5 "Library/lib/utils"
// #include "src/utils/grid.hpp"
// #include "src/utils/hash.hpp"
#line 2 "Library/src/utils/io/istream.hpp"

/**
 * @file istream.hpp
 * @brief Input Stream
 */

#include <cxxabi.h>

#line 13 "Library/src/utils/io/istream.hpp"

#line 2 "Library/src/utils/sfinae.hpp"

/**
 * @file sfinae.hpp
 * @brief SFINAE
 */

#line 10 "Library/src/utils/sfinae.hpp"
#include <type_traits>

#ifndef __INT128_DEFINED__

#ifdef __SIZEOF_INT128__
#define __INT128_DEFINED__ 1
#else
#define __INT128_DEFINED__ 0
#endif

#endif

namespace std {

#if __INT128_DEFINED__

template <> struct make_signed<__uint128_t> { using type = __int128_t; };
template <> struct make_signed<__int128_t> { using type = __int128_t; };

template <> struct make_unsigned<__uint128_t> { using type = __uint128_t; };
template <> struct make_unsigned<__int128_t> { using type = __uint128_t; };

#endif

}  // namespace std

namespace workspace {

template <class Tp, class... Args> struct variadic_front { using type = Tp; };

template <class... Args> struct variadic_back;

template <class Tp> struct variadic_back<Tp> { using type = Tp; };

template <class Tp, class... Args> struct variadic_back<Tp, Args...> {
  using type = typename variadic_back<Args...>::type;
};

template <class type, template <class> class trait>
using enable_if_trait_type = typename std::enable_if<trait<type>::value>::type;

template <class Container>
using element_type = typename std::decay<decltype(
    *std::begin(std::declval<Container&>()))>::type;

template <class T, class = std::nullptr_t>
struct has_begin : std::false_type {};

template <class T>
struct has_begin<T, decltype(std::begin(std::declval<T>()), nullptr)>
    : std::true_type {};

template <class T, class = int> struct mapped_of {
  using type = element_type<T>;
};
template <class T>
struct mapped_of<T,
                 typename std::pair<int, typename T::mapped_type>::first_type> {
  using type = typename T::mapped_type;
};
template <class T> using mapped_type = typename mapped_of<T>::type;

template <class T, class = void> struct is_integral_ext : std::false_type {};
template <class T>
struct is_integral_ext<
    T, typename std::enable_if<std::is_integral<T>::value>::type>
    : std::true_type {};

#if __INT128_DEFINED__

template <> struct is_integral_ext<__int128_t> : std::true_type {};
template <> struct is_integral_ext<__uint128_t> : std::true_type {};

#endif

#if __cplusplus >= 201402

template <class T>
constexpr static bool is_integral_ext_v = is_integral_ext<T>::value;

#endif

template <typename T, typename = void> struct multiplicable_uint {
  using type = uint_least32_t;
};
template <typename T>
struct multiplicable_uint<
    T, typename std::enable_if<(2 < sizeof(T)) &&
                               (!__INT128_DEFINED__ || sizeof(T) <= 4)>::type> {
  using type = uint_least64_t;
};

#if __INT128_DEFINED__

template <typename T>
struct multiplicable_uint<T, typename std::enable_if<(4 < sizeof(T))>::type> {
  using type = __uint128_t;
};

#endif

template <typename T> struct multiplicable_int {
  using type =
      typename std::make_signed<typename multiplicable_uint<T>::type>::type;
};

}  // namespace workspace
#line 15 "Library/src/utils/io/istream.hpp"

namespace workspace {

namespace internal {

template <class Tp, typename = std::nullptr_t> struct istream_helper {
  istream_helper(std::istream &is, Tp &x) {
    if constexpr (has_begin<Tp>::value)
      for (auto &&e : x)
        istream_helper<typename std::decay<decltype(e)>::type>(is, e);
    else
      static_assert(has_begin<Tp>::value, "istream unsupported type.");
  }
};

template <class Tp>
struct istream_helper<
    Tp,
    decltype(std::declval<std::decay<decltype(std::declval<std::istream &>() >>
                                              std::declval<Tp &>())>>(),
             nullptr)> {
  istream_helper(std::istream &is, Tp &x) { is >> x; }
};

#ifdef __SIZEOF_INT128__

template <> struct istream_helper<__uint128_t, std::nullptr_t> {
  istream_helper(std::istream &__is, __uint128_t &__x) {
    std::string __s;
    __is >> __s;
    bool __neg = false;
    if (__s.front() == '-') __neg = true, __s.erase(__s.begin());
    __x = 0;
    for (char __d : __s) {
      __x *= 10;
      __d -= '0';
      if (__neg)
        __x -= __d;
      else
        __x += __d;
    }
  }
};

template <> struct istream_helper<__int128_t, std::nullptr_t> {
  istream_helper(std::istream &__is, __int128_t &__x) {
    std::string __s;
    __is >> __s;
    bool __neg = false;
    if (__s.front() == '-') __neg = true, __s.erase(__s.begin());
    __x = 0;
    for (char __d : __s) {
      __x *= 10;
      __d -= '0';
      if (__neg)
        __x -= __d;
      else
        __x += __d;
    }
  }
};

#endif  // INT128

template <class T1, class T2> struct istream_helper<std::pair<T1, T2>> {
  istream_helper(std::istream &is, std::pair<T1, T2> &x) {
    istream_helper<T1>(is, x.first), istream_helper<T2>(is, x.second);
  }
};

template <class... Tps> struct istream_helper<std::tuple<Tps...>> {
  istream_helper(std::istream &is, std::tuple<Tps...> &x) { iterate(is, x); }

 private:
  template <class Tp, size_t N = 0> void iterate(std::istream &is, Tp &x) {
    if constexpr (N == std::tuple_size<Tp>::value)
      return;
    else
      istream_helper<typename std::tuple_element<N, Tp>::type>(is,
                                                               std::get<N>(x)),
          iterate<Tp, N + 1>(is, x);
  }
};

}  // namespace internal

/**
 * @brief A wrapper class for std::istream.
 */
class istream : public std::istream {
 public:
  /**
   * @brief Wrapped operator.
   */
  template <typename Tp> istream &operator>>(Tp &x) {
    internal::istream_helper<Tp>(*this, x);
    if (std::istream::fail()) {
      static auto once = atexit([] {
        std::cerr << "\n\033[43m\033[30mwarning: failed to read \'"
                  << abi::__cxa_demangle(typeid(Tp).name(), 0, 0, 0)
                  << "\'.\033[0m\n\n";
      });
      assert(!once);
    }
    return *this;
  }
};

decltype(auto) cin = static_cast<istream &>(std::cin);

}  // namespace workspace
#line 2 "Library/src/utils/io/ostream.hpp"

/**
 * @file ostream.hpp
 * @brief Output Stream
 */

#line 9 "Library/src/utils/io/ostream.hpp"

namespace workspace {

/**
 * @brief Stream insertion operator for C-style array.
 *
 * @param __os Output stream
 * @param __a Array
 * @return Reference to __os.
 */
template <class _Os, class _Tp, size_t _Nm>
typename std::enable_if<bool(sizeof(_Tp) > 2), _Os &>::type operator<<(
    _Os &__os, const _Tp (&__a)[_Nm]) {
  if constexpr (_Nm) {
    __os << *__a;
    for (auto __i = __a + 1, __e = __a + _Nm; __i != __e; ++__i)
      __os << ' ' << *__i;
  }
  return __os;
}

/**
 * @brief Stream insertion operator for std::pair.
 *
 * @param __os Output stream
 * @param __p Pair
 * @return Reference to __os.
 */
template <class _Os, class _T1, class _T2>
_Os &operator<<(_Os &__os, const std::pair<_T1, _T2> &__p) {
  return __os << __p.first << ' ' << __p.second;
}

/**
 * @brief Stream insertion operator for std::tuple.
 *
 * @param __os Output stream
 * @param __t Tuple
 * @return Reference to __os.
 */
template <class _Os, class _Tp, size_t _Nm = 0>
typename std::enable_if<bool(std::tuple_size<_Tp>::value + 1), _Os &>::type
operator<<(_Os &__os, const _Tp &__t) {
  if constexpr (_Nm != std::tuple_size<_Tp>::value) {
    if constexpr (_Nm) __os << ' ';
    __os << std::get<_Nm>(__t);
    operator<<<_Os, _Tp, _Nm + 1>(__os, __t);
  }
  return __os;
}

template <class _Os, class _Container,
          typename = decltype(std::begin(std::declval<_Container>()))>
typename std::enable_if<
    !std::is_same<typename std::decay<_Container>::type, std::string>::value &&
        !std::is_same<typename std::decay<_Container>::type, char *>::value,
    _Os &>::type
operator<<(_Os &__os, const _Container &__cont) {
  bool __h = true;
  for (auto &&__e : __cont) __h ? __h = 0 : (__os << ' ', 0), __os << __e;
  return __os;
}

#ifdef __SIZEOF_INT128__

/**
 * @brief Stream insertion operator for __int128_t.
 *
 * @param __os Output Stream
 * @param __x 128-bit integer
 * @return Reference to __os.
 */
template <class _Os> _Os &operator<<(_Os &__os, __int128_t __x) {
  if (!__x) return __os << '0';
  if (__x < 0) __os << '-';
  char __s[40], *__p = __s;
  while (__x) {
    auto __d = __x % 10;
    *__p++ = '0' + (__x < 0 ? -__d : __d);
    __x /= 10;
  }
  *__p = 0;
  for (char *__t = __s; __t < --__p; ++__t) *__t ^= *__p ^= *__t ^= *__p;
  return __os << __s;
}

/**
 * @brief Stream insertion operator for __uint128_t.
 *
 * @param __os Output Stream
 * @param __x 128-bit unsigned integer
 * @return Reference to __os.
 */
template <class _Os> _Os &operator<<(_Os &__os, __uint128_t __x) {
  if (!__x) return __os << '0';
  char __s[40], *__p = __s;
  while (__x) *__p++ = '0' + __x % 10, __x /= 10;
  *__p = 0;
  for (char *__t = __s; __t < --__p; ++__t) *__t ^= *__p ^= *__t ^= *__p;
  return __os << __s;
}

#endif

}  // namespace workspace
#line 9 "Library/lib/utils"
// #include "src/utils/io/read.hpp"
#line 2 "Library/src/utils/io/setup.hpp"

/*
 * @file setup.hpp
 * @brief I/O Setup
 */

#line 10 "Library/src/utils/io/setup.hpp"

namespace workspace {

/*
 * @fn io_setup
 * @brief Setup I/O.
 * @param precision Standard output precision
 */
void io_setup(int precision) {
  std::ios::sync_with_stdio(false);
  std::cin.tie(nullptr);
  std::cout << std::fixed << std::setprecision(precision);

#ifdef _buffer_check
  atexit([] {
    char bufc;
    if (std::cin >> bufc)
      std::cerr << "\n\033[43m\033[30mwarning: buffer not empty.\033[0m\n\n";
  });
#endif
}

}  // namespace workspace
#line 2 "Library/src/utils/iterator/category.hpp"

/*
 * @file category.hpp
 * @brief Iterator Category
 */

#line 10 "Library/src/utils/iterator/category.hpp"

namespace workspace {

/*
 * @tparam Tuple Tuple of iterator types
 */
template <class Tuple, size_t N = std::tuple_size<Tuple>::value - 1>
struct common_iterator_category {
  using type = typename std::common_type<
      typename common_iterator_category<Tuple, N - 1>::type,
      typename std::iterator_traits<typename std::tuple_element<
          N, Tuple>::type>::iterator_category>::type;
};

template <class Tuple> struct common_iterator_category<Tuple, 0> {
  using type = typename std::iterator_traits<
      typename std::tuple_element<0, Tuple>::type>::iterator_category;
};

}  // namespace workspace
#line 2 "Library/src/utils/iterator/reverse.hpp"

/*
 * @file reverse_iterator.hpp
 * @brief Reverse Iterator
 */

#if __cplusplus >= 201703L

#include <iterator>
#include <optional>

namespace workspace {

/*
 * @class reverse_iterator
 * @brief Wrapper class for `std::reverse_iterator`.
 * @see http://gcc.gnu.org/PR51823
 */
template <class Iterator>
class reverse_iterator : public std::reverse_iterator<Iterator> {
  using base_std = std::reverse_iterator<Iterator>;
  std::optional<typename base_std::value_type> deref;

 public:
  using base_std::reverse_iterator;

  constexpr typename base_std::reference operator*() noexcept {
    if (!deref) {
      Iterator tmp = base_std::current;
      deref = *--tmp;
    }
    return deref.value();
  }

  constexpr reverse_iterator &operator++() noexcept {
    base_std::operator++();
    deref.reset();
    return *this;
  }
  constexpr reverse_iterator &operator--() noexcept {
    base_std::operator++();
    deref.reset();
    return *this;
  }
  constexpr reverse_iterator operator++(int) noexcept {
    base_std::operator++();
    deref.reset();
    return *this;
  }
  constexpr reverse_iterator operator--(int) noexcept {
    base_std::operator++();
    deref.reset();
    return *this;
  }
};

}  // namespace workspace

#endif
#line 2 "Library/src/utils/make_vector.hpp"

/*
 * @file make_vector.hpp
 * @brief Multi-dimensional Vector
 */

#if __cplusplus >= 201703L

#include <tuple>
#include <vector>

namespace workspace {

/*
 * @brief Make a multi-dimensional vector.
 * @tparam Tp type of the elements
 * @tparam N dimension
 * @tparam S integer type
 * @param sizes The size of each dimension
 * @param init The initial value
 */
template <typename Tp, size_t N, typename S>
constexpr auto make_vector([[maybe_unused]] S* sizes, Tp const& init = Tp()) {
  static_assert(std::is_convertible_v<S, size_t>);
  if constexpr (N)
    return std::vector(*sizes,
                       make_vector<Tp, N - 1, S>(std::next(sizes), init));
  else
    return init;
}

/*
 * @brief Make a multi-dimensional vector.
 * @param sizes The size of each dimension
 * @param init The initial value
 */
template <typename Tp, size_t N, typename S>
constexpr auto make_vector(const S (&sizes)[N], Tp const& init = Tp()) {
  return make_vector<Tp, N, S>((S*)sizes, init);
}

/*
 * @brief Make a multi-dimensional vector.
 * @param sizes The size of each dimension
 * @param init The initial value
 */
template <typename Tp, size_t N, typename S, size_t I = 0>
constexpr auto make_vector([[maybe_unused]] std::array<S, N> const& sizes,
                           Tp const& init = Tp()) {
  static_assert(std::is_convertible_v<S, size_t>);
  if constexpr (I == N)
    return init;
  else
    return std::vector(sizes[I], make_vector<Tp, N, S, I + 1>(sizes, init));
}

/*
 * @brief Make a multi-dimensional vector.
 * @param sizes The size of each dimension
 * @param init The initial value
 */
template <typename Tp, size_t N = SIZE_MAX, size_t I = 0, class... Args>
constexpr auto make_vector([[maybe_unused]] std::tuple<Args...> const& sizes,
                           Tp const& init = Tp()) {
  using tuple_type = std::tuple<Args...>;
  if constexpr (I == std::tuple_size_v<tuple_type> || I == N)
    return init;
  else {
    static_assert(
        std::is_convertible_v<std::tuple_element_t<I, tuple_type>, size_t>);
    return std::vector(std::get<I>(sizes),
                       make_vector<Tp, N, I + 1>(sizes, init));
  }
}

/*
 * @brief Make a multi-dimensional vector.
 * @param sizes The size of each dimension
 * @param init The initial value
 */
template <typename Tp, class Fst, class Snd>
constexpr auto make_vector(std::pair<Fst, Snd> const& sizes,
                           Tp const& init = Tp()) {
  static_assert(std::is_convertible_v<Fst, size_t>);
  static_assert(std::is_convertible_v<Snd, size_t>);
  return make_vector({(size_t)sizes.first, (size_t)sizes.second}, init);
}

}  // namespace workspace

#endif
#line 2 "Library/src/utils/py-like/enumerate.hpp"

/*
 * @file enumerate.hpp
 * @brief Enumerate
 */

#line 2 "Library/src/utils/py-like/range.hpp"

/**
 * @file range.hpp
 * @brief Range
 */

#line 9 "Library/src/utils/py-like/range.hpp"

#line 2 "Library/src/utils/py-like/reversed.hpp"

/**
 * @file reversed.hpp
 * @brief Reversed
 */

#include <initializer_list>
#line 10 "Library/src/utils/py-like/reversed.hpp"

namespace workspace {

namespace internal {

template <class Container> class reversed {
  Container cont;

 public:
  constexpr reversed(Container &&cont) : cont(cont) {}

  constexpr auto begin() { return std::rbegin(cont); }
  constexpr auto end() { return std::rend(cont); }
};

}  // namespace internal

template <class Container> constexpr auto reversed(Container &&cont) noexcept {
  return internal::reversed<Container>{std::forward<Container>(cont)};
}

template <class Tp>
constexpr auto reversed(std::initializer_list<Tp> &&cont) noexcept {
  return internal::reversed<std::initializer_list<Tp>>{
      std::forward<std::initializer_list<Tp>>(cont)};
}

}  // namespace workspace
#line 12 "Library/src/utils/py-like/range.hpp"

#if __cplusplus >= 201703L

namespace workspace {

template <class Index> class range {
  Index first, last;

 public:
  class iterator {
    Index current;

   public:
    using difference_type = std::ptrdiff_t;
    using value_type = Index;
    using reference = typename std::add_const<Index>::type &;
    using pointer = iterator;
    using iterator_category = std::bidirectional_iterator_tag;

    constexpr iterator(Index const &__i = Index()) noexcept : current(__i) {}

    constexpr bool operator==(iterator const &rhs) const noexcept {
      return current == rhs.current;
    }
    constexpr bool operator!=(iterator const &rhs) const noexcept {
      return current != rhs.current;
    }

    constexpr iterator &operator++() noexcept {
      ++current;
      return *this;
    }
    constexpr iterator &operator--() noexcept {
      --current;
      return *this;
    }

    constexpr reference operator*() const noexcept { return current; }
  };

  constexpr range(Index first, Index last) noexcept
      : first(first), last(last) {}
  constexpr range(Index last) noexcept : first(), last(last) {}

  constexpr iterator begin() const noexcept { return iterator{first}; }
  constexpr iterator end() const noexcept { return iterator{last}; }

  constexpr reverse_iterator<iterator> rbegin() const noexcept {
    return reverse_iterator<iterator>(end());
  }
  constexpr reverse_iterator<iterator> rend() const noexcept {
    return reverse_iterator<iterator>(begin());
  }
};

template <class... Args> constexpr auto rrange(Args &&... args) noexcept {
  return internal::reversed(range(std::forward<Args>(args)...));
}

}  // namespace workspace

#endif
#line 2 "Library/src/utils/py-like/zip.hpp"

/**
 * @file zip.hpp
 * @brief Zip
 */

#line 11 "Library/src/utils/py-like/zip.hpp"

#line 14 "Library/src/utils/py-like/zip.hpp"

#if __cplusplus >= 201703L

namespace workspace {

namespace internal {

template <class> struct zipped_iterator;

template <class...> struct zipped_iterator_tuple;

template <class... Args> class zipped {
  using ref_tuple = std::tuple<Args...>;
  ref_tuple args;

  template <size_t N = 0> constexpr auto begin_cat() const noexcept {
    if constexpr (N != std::tuple_size<ref_tuple>::value) {
      return std::tuple_cat(std::tuple(std::begin(std::get<N>(args))),
                            begin_cat<N + 1>());
    } else
      return std::tuple<>();
  }

  template <size_t N = 0> constexpr auto end_cat() const noexcept {
    if constexpr (N != std::tuple_size<ref_tuple>::value) {
      return std::tuple_cat(std::tuple(std::end(std::get<N>(args))),
                            end_cat<N + 1>());
    } else
      return std::tuple<>();
  }

 public:
  constexpr zipped(Args &&... args) noexcept : args(args...) {}

  class iterator {
    using base_tuple = typename zipped_iterator_tuple<Args...>::type;

   public:
    using iterator_category =
        typename common_iterator_category<base_tuple>::type;
    using difference_type = std::ptrdiff_t;
    using value_type = zipped_iterator<base_tuple>;
    using reference = zipped_iterator<base_tuple> &;
    using pointer = iterator;

   protected:
    value_type current;

    template <size_t N = 0>
    constexpr bool equal(const iterator &rhs) const noexcept {
      if constexpr (N != std::tuple_size<base_tuple>::value) {
        return std::get<N>(current) == std::get<N>(rhs.current) ||
               equal<N + 1>(rhs);
      } else
        return false;
    }

    template <size_t N = 0> constexpr void increment() noexcept {
      if constexpr (N != std::tuple_size<base_tuple>::value) {
        ++std::get<N>(current);
        increment<N + 1>();
      }
    }

    template <size_t N = 0> constexpr void decrement() noexcept {
      if constexpr (N != std::tuple_size<base_tuple>::value) {
        --std::get<N>(current);
        decrement<N + 1>();
      }
    }

    template <size_t N = 0>
    constexpr void advance(difference_type __d) noexcept {
      if constexpr (N != std::tuple_size<base_tuple>::value) {
        std::get<N>(current) += __d;
        advance<N + 1>(__d);
      }
    }

   public:
    constexpr iterator() noexcept = default;
    constexpr iterator(base_tuple const &current) noexcept : current(current) {}

    constexpr bool operator==(const iterator &rhs) const noexcept {
      return equal(rhs);
    }
    constexpr bool operator!=(const iterator &rhs) const noexcept {
      return !equal(rhs);
    }

    constexpr iterator &operator++() noexcept {
      increment();
      return *this;
    }
    constexpr iterator &operator--() noexcept {
      decrement();
      return *this;
    }

    constexpr bool operator<(const iterator &rhs) const noexcept {
      return std::get<0>(current) < std::get<0>(rhs.current);
    }

    constexpr bool operator<=(const iterator &rhs) const noexcept {
      return std::get<0>(current) <= std::get<0>(rhs.current);
    }

    constexpr iterator &operator+=(difference_type __d) noexcept {
      advance(__d);
      return *this;
    }

    constexpr iterator &operator-=(difference_type __d) noexcept {
      advance(-__d);
      return *this;
    }

    constexpr iterator operator+(difference_type __d) const noexcept {
      return iterator{*this} += __d;
    }

    constexpr iterator operator-(difference_type __d) const noexcept {
      return iterator{*this} -= __d;
    }

    constexpr difference_type operator-(const iterator &rhs) const noexcept {
      return std::get<0>(current) - std::get<0>(rhs.current);
    }

    constexpr reference operator*() noexcept { return current; }
  };

  constexpr iterator begin() const noexcept { return iterator{begin_cat()}; }
  constexpr iterator end() const noexcept { return iterator{end_cat()}; }

  constexpr reverse_iterator<iterator> rbegin() const noexcept {
    return reverse_iterator<iterator>{end()};
  }
  constexpr reverse_iterator<iterator> rend() const noexcept {
    return reverse_iterator<iterator>{begin()};
  }
};

template <class Tp, class... Args> struct zipped_iterator_tuple<Tp, Args...> {
  using type = decltype(std::tuple_cat(
      std::declval<std::tuple<decltype(std::begin(std::declval<Tp>()))>>(),
      std::declval<typename zipped_iterator_tuple<Args...>::type>()));
};

template <> struct zipped_iterator_tuple<> { using type = std::tuple<>; };

template <class Iter_tuple> struct zipped_iterator : Iter_tuple {
  constexpr zipped_iterator(Iter_tuple const &__t) noexcept
      : Iter_tuple::tuple(__t) {}

  constexpr zipped_iterator(zipped_iterator const &__t) = default;

  constexpr zipped_iterator &operator=(zipped_iterator const &__t) = default;

  // Avoid move initialization.
  constexpr zipped_iterator(zipped_iterator &&__t)
      : zipped_iterator(static_cast<zipped_iterator const &>(__t)) {}

  // Avoid move assignment.
  zipped_iterator &operator=(zipped_iterator &&__t) {
    return operator=(static_cast<zipped_iterator const &>(__t));
  }

  template <size_t N>
  friend constexpr auto &get(zipped_iterator<Iter_tuple> const &__z) noexcept {
    return *std::get<N>(__z);
  }

  template <size_t N>
  friend constexpr auto get(zipped_iterator<Iter_tuple> &&__z) noexcept {
    return *std::get<N>(__z);
  }
};

}  // namespace internal

}  // namespace workspace

namespace std {

template <size_t N, class Iter_tuple>
struct tuple_element<N, workspace::internal::zipped_iterator<Iter_tuple>> {
  using type = typename remove_reference<typename iterator_traits<
      typename tuple_element<N, Iter_tuple>::type>::reference>::type;
};

template <class Iter_tuple>
struct tuple_size<workspace::internal::zipped_iterator<Iter_tuple>>
    : tuple_size<Iter_tuple> {};

}  // namespace std

namespace workspace {

template <class... Args> constexpr auto zip(Args &&... args) noexcept {
  return internal::zipped<Args...>(std::forward<Args>(args)...);
}

template <class... Args>
constexpr auto zip(std::initializer_list<Args> const &... args) noexcept {
  return internal::zipped<const std::initializer_list<Args>...>(args...);
}

}  // namespace workspace

#endif
#line 10 "Library/src/utils/py-like/enumerate.hpp"

#if __cplusplus >= 201703L

namespace workspace {

constexpr size_t min_size() noexcept { return SIZE_MAX; }

template <class Container, class... Args>
constexpr size_t min_size(Container const &cont, Args &&... args) noexcept {
  return std::min(std::size(cont), min_size(std::forward<Args>(args)...));
}

template <class... Args> constexpr auto enumerate(Args &&... args) noexcept {
  return zip(range(min_size(args...)), std::forward<Args>(args)...);
}

template <class... Args>
constexpr auto enumerate(std::initializer_list<Args> const &... args) noexcept {
  return zip(range(min_size(args...)), std::vector(args)...);
}

}  // namespace workspace

#endif
#line 16 "Library/lib/utils"
// #include "src/utils/py-like/reversed.hpp"
// #include "src/utils/py-like/zip.hpp"
// #include "src/utils/rand/rng.hpp"
// #include "src/utils/rand/shuffle.hpp"
#line 2 "Library/src/utils/round_div.hpp"

/*
 * @file round_div.hpp
 * @brief Round Integer Division
 */

#line 9 "Library/src/utils/round_div.hpp"

#line 11 "Library/src/utils/round_div.hpp"

namespace workspace {

/*
 * @fn floor_div
 * @brief floor of fraction.
 * @param x the numerator
 * @param y the denominator
 * @return maximum integer z s.t. z <= x / y
 * @note y must be nonzero.
 */
template <typename T1, typename T2>
constexpr typename std::enable_if<(is_integral_ext<T1>::value &&
                                   is_integral_ext<T2>::value),
                                  typename std::common_type<T1, T2>::type>::type
floor_div(T1 x, T2 y) {
  assert(y != 0);
  if (y < 0) x = -x, y = -y;
  return x < 0 ? (x - y + 1) / y : x / y;
}

/*
 * @fn ceil_div
 * @brief ceil of fraction.
 * @param x the numerator
 * @param y the denominator
 * @return minimum integer z s.t. z >= x / y
 * @note y must be nonzero.
 */
template <typename T1, typename T2>
constexpr typename std::enable_if<(is_integral_ext<T1>::value &&
                                   is_integral_ext<T2>::value),
                                  typename std::common_type<T1, T2>::type>::type
ceil_div(T1 x, T2 y) {
  assert(y != 0);
  if (y < 0) x = -x, y = -y;
  return x < 0 ? x / y : (x + y - 1) / y;
}

}  // namespace workspace
#line 21 "Library/lib/utils"
// #include "src/utils/sfinae.hpp"
#line 11 "other/y.cc"

signed main() {
  using namespace workspace;

  io_setup(15);

  /* given
    case_info.read();  //*/

  /* unspecified
    case_info.total = -1;  //*/

  return case_info.iterate();
}

#line 2 "Library/src/combinatorics/binomial.hpp"

/**
 * @file binomial.hpp
 * @brief Binomial Coefficient
 * @date 2021-01-15
 *
 *
 */

#line 2 "Library/src/modular/modint.hpp"

/**
 * @file modint.hpp
 *
 * @brief Modular Arithmetic
 */

#line 12 "Library/src/modular/modint.hpp"

#line 14 "Library/src/modular/modint.hpp"

namespace workspace {

namespace internal {

/**
 * @brief Base of modular arithmetic.
 *
 * @tparam Mod identifier, which represents modulus if positive
 * @tparam Storage Reserved size for inverse calculation
 */
template <auto Mod, unsigned Storage> struct modint_base {
  static_assert(is_integral_ext<decltype(Mod)>::value,
                "Mod must be integral type.");

  using mod_type = typename std::make_signed<typename std::conditional<
      0 < Mod, typename std::add_const<decltype(Mod)>::type,
      decltype(Mod)>::type>::type;

  using value_type = typename std::decay<mod_type>::type;

  using mul_type = typename multiplicable_uint<value_type>::type;

  static mod_type mod;

  static value_type storage;

  constexpr static void reserve(unsigned __n) noexcept { storage = __n; }

 protected:
  value_type value = 0;

 public:
  constexpr modint_base() noexcept = default;

  template <class int_type,
            typename std::enable_if<is_integral_ext<int_type>::value>::type * =
                nullptr>
  constexpr modint_base(int_type n) noexcept
      : value((n %= mod) < 0 ? n += mod : n) {}

  constexpr modint_base(bool n) noexcept : value(n) {}

  constexpr operator value_type() const noexcept { return value; }

  constexpr static modint_base one() noexcept { return 1; }

  // unary operators {{
  constexpr modint_base operator++(int) noexcept {
    modint_base __t{*this};
    operator++();
    return __t;
  }

  constexpr modint_base operator--(int) noexcept {
    modint_base __t{*this};
    operator--();
    return __t;
  }

  constexpr modint_base &operator++() noexcept {
    if (++value == mod) value = 0;
    return *this;
  }

  constexpr modint_base &operator--() noexcept {
    if (!value) value = mod;
    --value;
    return *this;
  }

  constexpr modint_base operator-() const noexcept {
    modint_base __t;
    __t.value = value ? mod - value : 0;
    return __t;
  }

  // }} unary operators

  // operator+= {{

  constexpr modint_base &operator+=(modint_base const &rhs) noexcept {
    if ((value += rhs.value) >= mod) value -= mod;
    return *this;
  }

  template <class int_type>
  constexpr typename std::enable_if<is_integral_ext<int_type>::value,
                                    modint_base>::type &
  operator+=(int_type const &rhs) noexcept {
    if (((value += rhs) %= mod) < 0) value += mod;
    return *this;
  }

  // }} operator+=

  // operator+ {{

  template <class int_type>
  constexpr typename std::enable_if<is_integral_ext<int_type>::value,
                                    modint_base>::type
  operator+(int_type const &rhs) const noexcept {
    return modint_base{*this} += rhs;
  }

  constexpr modint_base operator+(modint_base rhs) const noexcept {
    return rhs += *this;
  }

  template <class int_type>
  constexpr friend typename std::enable_if<is_integral_ext<int_type>::value,
                                           modint_base>::type
  operator+(int_type const &lhs, modint_base rhs) noexcept {
    return rhs += lhs;
  }

  // }} operator+

  // operator-= {{

  constexpr modint_base &operator-=(modint_base const &rhs) noexcept {
    if ((value -= rhs.value) < 0) value += mod;
    return *this;
  }

  template <class int_type>
  constexpr typename std::enable_if<is_integral_ext<int_type>::value,
                                    modint_base>::type &
  operator-=(int_type rhs) noexcept {
    if (((value -= rhs) %= mod) < 0) value += mod;
    return *this;
  }

  // }} operator-=

  // operator- {{

  template <class int_type>
  constexpr typename std::enable_if<is_integral_ext<int_type>::value,
                                    modint_base>::type
  operator-(int_type const &rhs) const noexcept {
    return modint_base{*this} -= rhs;
  }

  constexpr modint_base operator-(modint_base const &rhs) const noexcept {
    modint_base __t;
    if (((__t.value = value) -= rhs.value) < 0) __t.value += mod;
    return __t;
  }

  template <class int_type>
  constexpr friend typename std::enable_if<is_integral_ext<int_type>::value,
                                           modint_base>::type
  operator-(int_type lhs, modint_base const &rhs) noexcept {
    if (((lhs -= rhs.value) %= mod) < 0) lhs += mod;
    modint_base __t;
    __t.value = lhs;
    return __t;
  }

  // }} operator-

  // operator*= {{

  constexpr modint_base &operator*=(modint_base const &rhs) noexcept {
    if (!rhs.value)
      value = 0;
    else if (value) {
      mul_type __r(value);
      value = static_cast<value_type &&>((__r *= rhs.value) %= mod);
    }
    return *this;
  }

  template <class int_type>
  constexpr typename std::enable_if<is_integral_ext<int_type>::value,
                                    modint_base>::type &
  operator*=(int_type rhs) noexcept {
    if (!rhs)
      value = 0;
    else if (value) {
      if ((rhs %= mod) < 0) rhs += mod;
      mul_type __r(value);
      value = static_cast<value_type &&>((__r *= rhs) %= mod);
    }
    return *this;
  }

  // }} operator*=

  // operator* {{

  constexpr modint_base operator*(modint_base const &rhs) const noexcept {
    if (!value or !rhs.value) return {};
    mul_type __r(value);
    modint_base __t;
    __t.value = static_cast<value_type &&>((__r *= rhs.value) %= mod);
    return __t;
  }

  template <class int_type>
  constexpr typename std::enable_if<is_integral_ext<int_type>::value,
                                    modint_base>::type
  operator*(int_type rhs) const noexcept {
    if (!value or !rhs) return {};
    if ((rhs %= mod) < 0) rhs += mod;
    mul_type __r(value);
    modint_base __t;
    __t.value = static_cast<value_type &&>((__r *= rhs) %= mod);
    return __t;
  }

  template <class int_type>
  constexpr friend typename std::enable_if<is_integral_ext<int_type>::value,
                                           modint_base>::type
  operator*(int_type lhs, modint_base const &rhs) noexcept {
    if (!lhs or !rhs.value) return {};
    if ((lhs %= mod) < 0) lhs += mod;
    mul_type __r(lhs);
    modint_base __t;
    __t.value = static_cast<value_type &&>((__r *= rhs.value) %= mod);
    return __t;
  }

  // }} operator*

 protected:
  static value_type _mem(value_type __x) {
    static std::vector<value_type> __m{0, 1};
    static value_type __i = (__m.reserve(Storage), 1);
    while (__i < __x) {
      ++__i;
      __m.emplace_back(mod - mul_type(mod / __i) * __m[mod % __i] % mod);
    }
    return __m[__x];
  }

  template <class int_type>
  constexpr static typename std::enable_if<is_integral_ext<int_type>::value,
                                           value_type>::type
  _div(mul_type __r, int_type __x) noexcept {
    assert(__x);
    if (!__r) return 0;
    int_type __v{};
    bool __neg = __x < 0 ? __x = -__x, true : false;
    if (__x < storage)
      __v = _mem(__x);
    else {
      int_type __y{mod}, __u{1}, __t;
      while (__x)
        __t = __y / __x, __y ^= __x ^= (__y -= __t * __x) ^= __x,
        __v ^= __u ^= (__v -= __t * __u) ^= __u;
      if (__y < 0) __neg ^= 1;
    }
    if (__neg)
      __v = 0 < __v ? mod - __v : -__v;
    else if (__v < 0)
      __v += mod;
    if (__r == 1) return static_cast<value_type>(__v);
    return static_cast<value_type>((__r *= __v) %= mod);
  }

 public:
  // operator/= {{

  constexpr modint_base &operator/=(modint_base const &rhs) noexcept {
    if (value) value = _div(value, rhs.value);
    return *this;
  }

  template <class int_type>
  constexpr typename std::enable_if<is_integral_ext<int_type>::value,
                                    modint_base>::type &
  operator/=(int_type rhs) noexcept {
    if (value) value = _div(value, rhs %= mod);
    return *this;
  }

  // }} operator/=

  // operator/ {{

  constexpr modint_base operator/(modint_base const &rhs) const noexcept {
    if (!value) return {};
    modint_base __t;
    __t.value = _div(value, rhs.value);
    return __t;
  }

  template <class int_type>
  constexpr typename std::enable_if<is_integral_ext<int_type>::value,
                                    modint_base>::type
  operator/(int_type rhs) const noexcept {
    if (!value) return {};
    modint_base __t;
    __t.value = _div(value, rhs %= mod);
    return __t;
  }

  template <class int_type>
  constexpr friend typename std::enable_if<is_integral_ext<int_type>::value,
                                           modint_base>::type
  operator/(int_type lhs, modint_base const &rhs) noexcept {
    if (!lhs) return {};
    if ((lhs %= mod) < 0) lhs += mod;
    modint_base __t;
    __t.value = _div(lhs, rhs.value);
    return __t;
  }

  // }} operator/

  constexpr modint_base inv() const noexcept { return _div(1, value); }

  template <class int_type>
  friend constexpr typename std::enable_if<is_integral_ext<int_type>::value,
                                           modint_base>::type
  pow(modint_base b, int_type e) noexcept {
    if (e < 0) {
      e = -e;
      b.value = _div(1, b.value);
    }
    modint_base __r;
    for (__r.value = 1; e; e >>= 1, b *= b)
      if (e & 1) __r *= b;
    return __r;
  }

  template <class int_type>
  constexpr typename std::enable_if<is_integral_ext<int_type>::value,
                                    modint_base>::type
  pow(int_type e) const noexcept {
    modint_base __r, b;
    __r.value = 1;
    for (b.value = e < 0 ? e = -e, _div(1, value) : value; e; e >>= 1, b *= b)
      if (e & 1) __r *= b;
    return __r;
  }

  friend std::ostream &operator<<(std::ostream &os,
                                  const modint_base &rhs) noexcept {
    return os << rhs.value;
  }

  friend std::istream &operator>>(std::istream &is, modint_base &rhs) noexcept {
    intmax_t value;
    rhs = (is >> value, value);
    return is;
  }
};

template <auto Mod, unsigned Storage>
typename modint_base<Mod, Storage>::mod_type modint_base<Mod, Storage>::mod =
    Mod > 0 ? Mod : 0;

template <auto Mod, unsigned Storage>
typename modint_base<Mod, Storage>::value_type
    modint_base<Mod, Storage>::storage = Storage;

}  // namespace internal

/**
 * @brief Modular arithmetic.
 *
 * @tparam Mod modulus
 * @tparam Storage Reserved size for inverse calculation
 */
template <auto Mod, unsigned Storage = 0,
          typename std::enable_if<(Mod > 0)>::type * = nullptr>
using modint = internal::modint_base<Mod, Storage>;

/**
 * @brief Runtime modular arithmetic.
 *
 * @tparam type_id uniquely assigned
 * @tparam Storage Reserved size for inverse calculation
 */
template <unsigned type_id = 0, unsigned Storage = 0>
using modint_runtime = internal::modint_base<-(signed)type_id, Storage>;

// #define modint_newtype modint_runtime<__COUNTER__>

}  // namespace workspace
#line 2 "Library/src/combinatorics/factorial.hpp"

/**
 * @file factorial.hpp
 * @brief Factorial
 * @date 2021-01-15
 *
 *
 */

#line 14 "Library/src/combinatorics/factorial.hpp"

namespace workspace {

template <class Tp> Tp factorial(int_fast32_t __x) noexcept {
  if (__x < 0) return 0;
  static std::vector<Tp> __t{1};
  static int_fast32_t __i = (__t.reserve(0x1000000), 0);
  while (__i < __x) {
    ++__i;
    __t.emplace_back(__t.back() * __i);
  }
  return __t[__x];
}

template <class Tp> Tp factorial_inverse(int_fast32_t __x) noexcept {
  if (__x < 0) return 0;
  static std::vector<Tp> __t{1};
  static int_fast32_t __i = (__t.reserve(0x1000000), 0);
  while (__i < __x) {
    ++__i;
    __t.emplace_back(__t.back() / __i);
  }
  return __t[__x];
}

}  // namespace workspace
#line 13 "Library/src/combinatorics/binomial.hpp"

namespace workspace {

/**
 * @brief Binomial coefficient for integer args.
 */
template <class Tp> Tp binomial(int_fast32_t __x, int_fast32_t __y) {
  if (!__y) return 1;
  if (__y < 0 || __x < __y) return 0;
  return factorial<Tp>(__x) * factorial_inverse<Tp>(__y) *
         factorial_inverse<Tp>(__x - __y);
}

}  // namespace workspace
#line 2 "Library/src/algebra/polynomial.hpp"

/**
 * @file polynomial.hpp
 * @brief Polynomial
 * @date 2021-02-19
 *
 *
 */

#line 12 "Library/src/algebra/polynomial.hpp"

#line 2 "Library/src/algebra/convolution/ntt.hpp"

#line 2 "Library/src/number_theory/ext_gcd.hpp"

/*
 * @file ext_gcd
 * @brief Extended Euclidean Algorithm
 */

#line 9 "Library/src/number_theory/ext_gcd.hpp"

#line 11 "Library/src/number_theory/ext_gcd.hpp"

namespace workspace {

/**
 * @param a Integer
 * @param b Integer
 * @return Pair of integers (x, y) s.t. ax + by = g = gcd(a, b), |x| < |b/g|,
 * |y| < |a/g|.
 * @note return (0, 0) if (a, b) = (0, 0)
 */
template <typename T1, typename T2> constexpr auto ext_gcd(T1 a, T2 b) {
  static_assert(is_integral_ext<T1>::value);
  static_assert(is_integral_ext<T2>::value);
  using result_type =
      typename std::make_signed<typename std::common_type<T1, T2>::type>::type;
  result_type p{1}, q{}, r{}, s{1}, t{};
  while (b) {
    r ^= p ^= r ^= p -= (t = a / b) * r;
    s ^= q ^= s ^= q -= t * s;
    b ^= a ^= b ^= a %= b;
  }
  if (a < 0) p = -p, q = -q;
  return std::make_pair(p, q);
}

}  // namespace workspace
#line 2 "Library/src/number_theory/primitive_root.hpp"

/**
 * @file primitive_root.hpp
 * @brief Primitive Root
 * @date 2020-12-28
 */

#line 10 "Library/src/number_theory/primitive_root.hpp"

namespace workspace {

/**
 * @brief Compile time primitive root.
 *
 * @tparam __mod A positive integer
 * @return Minimum positive one if it exists. Otherwise 0.
 */
template <class Tp>
constexpr typename std::enable_if<(is_integral_ext<Tp>::value), Tp>::type
primitive_root(const Tp __mod) noexcept {
  assert(__mod > 0);
  using int_type = typename multiplicable_uint<Tp>::type;

  int_type __r = __mod, __p[16] = {}, *__q = __p;
  for (int_type __i = 2; __i <= __r / __i; ++__i) {
    if (__r % __i) continue;
    *__q++ = __i;
    while (!(__r % __i)) __r /= __i;
  }
  if (__r != 1) *__q++ = __r;

  int_type __tot = __mod;
  for (__q = __p; *__q; *__q++ = 0) (__tot /= *__q) *= *__q - 1;
  __r = __tot, __q = __p + 1, __p[0] = 1;
  for (int_type __i = 2; __i <= __r / __i; ++__i) {
    if (__r % __i) continue;
    *__q++ = __i;
    while (!(__r % __i)) __r /= __i;
  }
  if (__r != 1) *__q++ = __r;

  for (Tp __r = 1; __r != __mod; ++__r) {
    auto __cnt = 0;
    for (__q = __p; *__q; ++__q) {
      int_type __w = 1;
      for (int_type __e = __tot / *__q, __x = __r; __e;
           __e >>= 1, (__x *= __x) %= __mod)
        if (__e & 1) (__w *= __x) %= __mod;
      if (__w == 1 && ++__cnt > 1) break;
    }
    if (__cnt == 1) return __r;
  }

  return 0;
};

}  // namespace workspace
#line 6 "Library/src/algebra/convolution/ntt.hpp"

namespace workspace {

namespace ntt_impl {

template <class _Tp> struct __coef {
  _Tp sum_e[30];  // sum_e[i] = ies[0] * ... * ies[i - 1] * es[i]

  constexpr __coef() : sum_e{} {
    _Tp es[30] = {}, ies[30] = {};  // es[i]^(2^(2+i)) == 1
    int cnt2 = __builtin_ctz(_Tp::mod - 1);
    _Tp e = 1;
    {
      auto p = (_Tp::mod - 1) >> cnt2;
      _Tp w = primitive_root(_Tp::mod);
      while (p) {
        if (p & 1) e *= w;
        p >>= 1;
        w *= w;
      }
    }
    _Tp ie = ext_gcd(decltype(_Tp::mod)(e), _Tp::mod).first;
    for (int i = cnt2; i >= 2; i--) {
      // e^(2^i) == 1
      es[i - 2] = e;
      ies[i - 2] = ie;
      e *= e;
      ie *= ie;
    }
    _Tp now = 1;
    for (int i = 0; i <= cnt2 - 2; i++) {
      sum_e[i] = es[i] * now;
      now *= ies[i];
    }
  }
};

template <class _Tp> struct __icoef {
  _Tp sum_ie[30];  // sum_e[i] = ies[0] * ... * ies[i - 1] * es[i]

  constexpr __icoef() : sum_ie{} {
    _Tp es[30] = {}, ies[30] = {};  // es[i]^(2^(2+i)) == 1
    int cnt2 = __builtin_ctz(_Tp::mod - 1);
    _Tp e = 1;
    {
      auto p = (_Tp::mod - 1) >> cnt2;
      _Tp w = primitive_root(_Tp::mod);
      while (p) {
        if (p & 1) e *= w;
        p >>= 1;
        w *= w;
      }
    }
    _Tp ie = ext_gcd(decltype(_Tp::mod)(e), _Tp::mod).first;
    for (int i = cnt2; i >= 2; i--) {
      // e^(2^i) == 1
      es[i - 2] = e;
      ies[i - 2] = ie;
      e *= e;
      ie *= ie;
    }
    _Tp now = 1;
    for (int i = 0; i <= cnt2 - 2; i++) {
      sum_ie[i] = ies[i] * now;
      now *= es[i];
    }
  }
};

template <class _Tp> struct __ipow2 {
  _Tp __ip2[30];
  constexpr __ipow2() : __ip2{1, (1 + _Tp::mod) / 2} {
    for (size_t __i = 1; __i + 1 != std::size(__ip2); ++__i)
      __ip2[__i + 1] = __ip2[__i] * __ip2[1];
  }
};

template <class _RAIter> constexpr void ntt(_RAIter __first, _RAIter __last) {
  using modint = typename std::decay<decltype(*__first)>::type;
  constexpr __coef<modint> _;

  int h = __builtin_ctz(__last - __first);
  for (int ph = 1; ph <= h; ph++) {
    int w = 1 << (ph - 1), p = 1 << (h - ph);
    modint now = -1;
    for (int s = 0; s < w; s++) {
      auto __l = __first + (s << (h - ph + 1));
      auto __r = __first + p + (s << (h - ph + 1));
      for (int i = 0; i < p; ++i, ++__l, ++__r) {
        auto __tmp = *__l;
        *__r *= now;
        *__l -= *__r;
        *__r += __tmp;
      }
      now *= _.sum_e[__builtin_ctz(~(unsigned int)(s))];
    }
  }
}

template <class _RAIter> constexpr void intt(_RAIter __first, _RAIter __last) {
  using modint = typename std::decay<decltype(*__first)>::type;
  constexpr __icoef<modint> _;

  int h = __builtin_ctz(__last - __first);
  for (int ph = h; ph >= 1; ph--) {
    int w = 1 << (ph - 1), p = 1 << (h - ph);
    modint inow = 1;
    for (int s = 0; s < w; s++) {
      auto __l = __first + (s << (h - ph + 1));
      auto __r = __first + p + (s << (h - ph + 1));
      for (int i = 0; i < p; ++i, ++__l, ++__r) {
        auto __tmp = (*__l - *__r) * inow;
        *__l += *__r;
        *__r = __tmp;
      }
      inow *= _.sum_ie[__builtin_ctz(~(unsigned int)(s))];
    }
  }

  constexpr __ipow2<modint> __;
  while (__first != __last) *--__last *= __.__ip2[h];
}

template <class _A> constexpr void ntt(_A &a) {
  using modint = typename std::decay<decltype(a[0])>::type;
  constexpr __coef<modint> _;

  int h = __builtin_ctz(a.size());
  for (int ph = 1; ph <= h; ph++) {
    int w = 1 << (ph - 1), p = 1 << (h - ph);
    modint now = 1;
    for (int s = 0; s < w; s++) {
      int offset = s << (h - ph + 1);
      for (int i = 0; i < p; i++) {
        auto l = a[i + offset];
        auto r = a[i + offset + p] * now;
        a[i + offset] = l + r;
        a[i + offset + p] = l - r;
      }
      now *= _.sum_e[__builtin_ctz(~(unsigned int)(s))];
    }
  }
}

template <class _A> constexpr void intt(_A &a) {
  using modint = typename std::decay<decltype(a[0])>::type;
  constexpr __icoef<modint> _;

  int h = __builtin_ctz(a.size());
  for (int ph = h; ph >= 1; ph--) {
    int w = 1 << (ph - 1), p = 1 << (h - ph);
    modint inow = 1;
    for (int s = 0; s < w; s++) {
      int offset = s << (h - ph + 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] = (l - r) * inow;
      }
      inow *= _.sum_ie[__builtin_ctz(~(unsigned int)(s))];
    }
  }

  constexpr __ipow2<modint> __;
  for (auto &x : a) x *= __.__ip2[h];
}

}  // namespace ntt_impl

using ntt_impl::intt;
using ntt_impl::ntt;

}  // namespace workspace
#line 14 "Library/src/algebra/polynomial.hpp"

namespace workspace {

template <class _Tp, std::size_t _Conv_threshold = 60>
class polynomial : public std::vector<_Tp> {
 protected:
  using vector = std::vector<_Tp>;
  using poly = polynomial;

  void conv_naive(const poly& __x) {
    if (__x._M_impl._M_start == __x._M_impl._M_finish)
      vector::_M_erase_at_end(vector::_M_impl._M_start);

    else {
      vector::_M_default_append(__x._M_impl._M_finish - __x._M_impl._M_start -
                                1);

      for (auto __i = vector::_M_impl._M_finish;
           __i-- != vector::_M_impl._M_start;) {
        auto __j = __i, __k = __x._M_impl._M_start;
        *__i *= *__k++;

        while (__j != vector::_M_impl._M_start && __k != __x._M_impl._M_finish)
          *__i += *--__j * *__k++;
      }
    }
  }

  void conv_ntt(poly&& __x) {
    size_type __n = vector::size(), __m = __x.size();
    size_type __len = 1 << (32 - __builtin_clz(__n + __m - 1));

    vector::_M_default_append(__len - __n);
    __x._M_default_append(__len - __m);

    ntt(vector::_M_impl._M_start, vector::_M_impl._M_finish);
    ntt(__x._M_impl._M_start, __x._M_impl._M_finish);

    for (auto __i = vector::_M_impl._M_start, __j = __x._M_impl._M_start;
         __i != vector::_M_impl._M_finish; ++__i, ++__j)
      *__i *= *__j;

    intt(vector::_M_impl._M_start, vector::_M_impl._M_finish);

    vector::_M_erase_at_end(vector::_M_impl._M_start + __n + __m - 1);
  }

 public:
  using vector::vector;
  using size_type = typename vector::size_type;

  typename vector::reference operator[](size_type __i) {
    if (size_type __n = vector::_M_impl._M_finish - vector::_M_impl._M_start;
        __n <= __i) {
      vector::_M_default_append(__i + 1 - __n);
    }
    return *(vector::_M_impl._M_start + __i);
  }

  typename vector::const_reference operator[](size_type __i) const {
    return __i < size_type(vector::_M_impl._M_finish - vector::_M_impl._M_start)
               ? *(vector::_M_impl._M_start + __i)
               : 0;
  }

  poly operator+() const { return *this; }

  poly operator-() const {
    poly __x = *this;
    for (auto __i = __x._M_impl._M_start; __i != __x._M_impl._M_finish; ++__i)
      *__i = -*__i;
    return __x;
  }

  poly& operator+=(const poly& __x) {
    if (vector::size() < __x.size())
      vector::_M_default_append(__x.size() - vector::size());
    for (auto __i = vector::_M_impl._M_start, __j = __x._M_impl._M_start;
         __j != __x._M_impl._M_finish; ++__i, ++__j)
      *__i += *__j;
    return *this;
  }

  poly& operator+=(const _Tp& __x) {
    if (__x != static_cast<_Tp>(0)) {
      if (vector::_M_impl._M_start == vector::_M_impl._M_finish)
        vector::emplace_back(__x);
      else
        *vector::_M_impl._M_start = __x;
    }
    return *this;
  }

  poly& operator-=(const poly& __x) {
    if (vector::size() < __x.size())
      vector::_M_default_append(__x.size() - vector::size());
    for (auto __i = vector::_M_impl._M_start, __j = __x._M_impl._M_start;
         __j != __x._M_impl._M_finish; ++__i, ++__j)
      *__i -= *__j;
    return *this;
  }

  poly& operator-=(const _Tp& __x) {
    if (__x != static_cast<_Tp>(0)) {
      if (vector::_M_impl._M_start == vector::_M_impl._M_finish)
        vector::emplace_back(-__x);
      else
        *vector::_M_impl._M_start -= __x;
    }
    return *this;
  }

  typename std::enable_if<bool(_Tp::mod > 2), poly&>::type operator*=(
      const poly& __x) {
    std::min(vector::size(), __x.size()) > _Conv_threshold ? conv_ntt(poly(__x))
                                                           : conv_naive(__x);
    return *this;
  }

  typename std::enable_if<bool(_Tp::mod > 2), poly&>::type operator*=(
      poly&& __x) {
    std::min(vector::size(), __x.size()) > _Conv_threshold
        ? conv_ntt(std::move(__x))
        : conv_naive(__x);
    return *this;
  }

  poly& operator*=(const _Tp& __x) {
    if (__x == static_cast<_Tp>(0))
      vector::_M_erase_at_end(vector::_M_impl._M_start);
    else
      for (auto __i = vector::_M_impl._M_start;
           __i != vector::_M_impl._M_finish; ++__i)
        *__i *= __x;
    return *this;
  }

  poly& operator/=(const _Tp& __x) {
    assert(__x != static_cast<_Tp>(0));
    for (auto __i = vector::_M_impl._M_start; __i != vector::_M_impl._M_finish;
         ++__i)
      *__i /= __x;
    return *this;
  }

  poly& operator/=(const poly& __x);

  poly& operator%=(const poly& __x);

  template <class _T> poly operator+(_T&& __x) const {
    return poly(*this).operator+=(std::forward<_T>(__x));
  }

  template <class _T> poly operator-(_T&& __x) const {
    return poly(*this).operator-=(std::forward<_T>(__x));
  }

  template <class _T> poly operator*(_T&& __x) const {
    return poly(*this).operator*=(std::forward<_T>(__x));
  }

  template <class _T> poly operator/(_T&& __x) const {
    return poly(*this).operator/=(std::forward<_T>(__x));
  }

  template <class _T> poly operator%(_T&& __x) const {
    return poly(*this).operator%=(std::forward<_T>(__x));
  }
};

}  // namespace workspace
#line 28 "other/y.cc"

namespace workspace {

using mint = modint<998244353>;
using poly = polynomial<mint>;

auto fa = factorial<mint>;
auto fi = factorial_inverse<mint>;
auto b = binomial<mint>;

void main() {
  // start here!

  int h, w, k;
  cin >> h >> w >> k;
  --h, --w;
  k /= 2;
  ++k;
  const auto n = min(h, w);
  poly fv;
  for (auto r : range(k * 2)) {
    poly p;
    for (auto i : range(n / k + 1)) {
      p[i] = fi(k * 2 * i + r);
    }
    p *= p;
    for (auto i = r, j = 0; i <= n; i += k * 2, j += 2) {
      fv[i] += p[j];
    }
    for (auto i = r < k ? r + k : r - k; i <= n; i += k * 2) {
      if (i >= k) fv[i] -= p[(i - k) / k / 2 + (i + k) / k / 2];
    }
  }
  mint ans;
  for (auto i : range(n + 1)) {
    ans += fv[i] * fi(h - i) * fi(w - i);
  }
  ans *= fa(h + w);
  cout << ans << "\n";
}

}  // namespace workspace