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

#line 2 "Library\\lib\\alias"

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

#line 10 "Library\\lib\\alias"

#line 2 "Library\\lib\\bit"

#if __cplusplus > 201703L

#include <bit>

#elif __cplusplus > 201402L

#ifndef _GLIBCXX_BIT
#define _GLIBCXX_BIT 1

#include <limits>
#include <type_traits>

namespace std {

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 __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 __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 _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 workspace {

template <class _Tp> struct numeric_limits : std::numeric_limits<_Tp> {};

#ifdef __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 workspace
#line 13 "Library\\lib\\alias"

namespace workspace {

constexpr static 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 128-bit integer is not available.
#endif

template <class _T1, class _T2,
          typename = decltype(std::declval<const _T2 &>() <
                              std::declval<const _T1 &>())>
constexpr
    typename std::conditional<std::is_same<_T1, _T2>::value, const _T1 &,
                              typename std::common_type<_T1, _T2>::type>::type
    min(const _T1 &__x, const _T2 &__y) noexcept {
  return __y < __x ? __y : __x;
}

template <class _T1, class _T2, class _Compare,
          typename = decltype(std::declval<_Compare>()(
              std::declval<const _T2 &>(), std::declval<const _T1 &>()))>
constexpr
    typename std::conditional<std::is_same<_T1, _T2>::value, const _T1 &,
                              typename std::common_type<_T1, _T2>::type>::type
    min(const _T1 &__x, const _T2 &__y, _Compare __comp) noexcept {
  return __comp(__y, __x) ? __y : __x;
}

template <class _Tp, typename = decltype(std::declval<const _Tp &>() <
                                         std::declval<const _Tp &>())>
constexpr _Tp min(std::initializer_list<_Tp> __x) noexcept {
  return *std::min_element(__x.begin(), __x.end());
}

template <class _Tp, class _Compare,
          typename = decltype(std::declval<_Compare>()(
              std::declval<const _Tp &>(), std::declval<const _Tp &>()))>
constexpr _Tp min(std::initializer_list<_Tp> __x, _Compare __comp) noexcept {
  return *std::min_element(__x.begin(), __x.end(), __comp);
}

template <class _T1, class _T2,
          typename = decltype(std::declval<const _T1 &>() <
                              std::declval<const _T2 &>())>

constexpr
    typename std::conditional<std::is_same<_T1, _T2>::value, const _T1 &,
                              typename std::common_type<_T1, _T2>::type>::type
    max(const _T1 &__x, const _T2 &__y) noexcept {
  return __x < __y ? __y : __x;
}

template <class _T1, class _T2, class _Compare,
          typename = decltype(std::declval<_Compare>()(
              std::declval<const _T1 &>(), std::declval<const _T2 &>()))>
constexpr
    typename std::conditional<std::is_same<_T1, _T2>::value, const _T1 &,
                              typename std::common_type<_T1, _T2>::type>::type
    max(const _T1 &__x, const _T2 &__y, _Compare __comp) noexcept {
  return __comp(__x, __y) ? __y : __x;
}

template <class _Tp, typename = decltype(std::declval<const _Tp &>() <
                                         std::declval<const _Tp &>())>
constexpr _Tp max(std::initializer_list<_Tp> __x) noexcept {
  return *std::max_element(__x.begin(), __x.end());
}

template <class _Tp, class _Compare,
          typename = decltype(std::declval<_Compare>()(
              std::declval<const _Tp &>(), std::declval<const _Tp &>()))>
constexpr _Tp max(std::initializer_list<_Tp> __x, _Compare __comp) noexcept {
  return *std::max_element(__x.begin(), __x.end(), __comp);
}

#ifdef _GLIBCXX_BIT

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;
}

#endif

}  // namespace workspace
#line 6 "other-workspace\\y.cc"
// #include "lib/cxx20"
#line 1 "Library\\lib\\direct"
#ifdef ONLINE_JUDGE
#pragma GCC optimize("O3")
#pragma GCC target("avx,avx2")
#pragma GCC optimize("unroll-loops")
#endif
#line 8 "other-workspace\\y.cc"
// #include "lib/opt"
#line 2 "Library\\src\\sys\\call_once.hpp"

/**
 * @file call_once.hpp
 * @brief Call Once
 */

#line 9 "Library\\src\\sys\\call_once.hpp"

namespace workspace {

/**
 * @brief Call once.
 */
template <class _F> void call_once(_F &&__f) {
  static std::unordered_set<void *> __called;
  if (__called.count(std::addressof(__f))) return;
  __called.emplace(std::addressof(__f));
  __f();
}

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

/**
 * @file 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

/**
 * @return Elapsed time of the program.
 */
decltype(auto) elapsed() noexcept {
  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\\cached.hpp"

/**
 * @file cached.hpp
 * @brief Cached
 */

#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 {
 public:
  // Construct a new fixed-point object.
  fixed_point(_F __x) noexcept : __fn(__x) {}

  // Function call.
  template <class... _Args> decltype(auto) operator()(_Args... __args) {
    return _wrapper{__fn}(std::forward<_Args>(__args)...);
  }

 private:
  _F __fn;

  struct _wrapper {
    _F &__ref;

    template <class... _Args> decltype(auto) operator()(_Args... __args) {
      return __ref(*this, std::forward<_Args>(__args)...);
    }
  };
};

}  // namespace workspace
#line 2 "Library\\lib\\cxx17"

#ifndef _CXX17_CONSTEXPR
#if __cplusplus >= 201703L
#define _CXX17_CONSTEXPR constexpr
#else
#define _CXX17_CONSTEXPR
#endif
#endif

#if __cplusplus < 201703L

namespace std {

/**
 *  @brief  Return the size of a container.
 *  @param  __cont  Container.
 */
template <typename _Container>
constexpr auto size(const _Container& __cont) noexcept(noexcept(__cont.size()))
    -> decltype(__cont.size()) {
  return __cont.size();
}

/**
 *  @brief  Return the size of an array.
 */
template <typename _Tp, size_t _Nm>
constexpr size_t size(const _Tp (&)[_Nm]) noexcept {
  return _Nm;
}

}  // namespace std

#endif
#line 10 "Library\\src\\utils\\cached.hpp"

namespace workspace {

namespace _cached_impl {

// Convert keys to tuple.
template <class... _Args> struct as_tuple {
  using type = decltype(std::tuple_cat(
      std::declval<std::tuple<std::conditional_t<
          std::is_convertible<std::decay_t<_Args>, _Args>::value,
          std::decay_t<_Args>, _Args>>>()...));
};

// Associative array.
template <class _Value, class... _Keys>
struct assoc
    : std::integral_constant<int, !std::is_void<_Value>::value>,
      std::conditional_t<std::is_void<_Value>::value,
                         std::set<typename as_tuple<_Keys...>::type>,
                         std::map<typename as_tuple<_Keys...>::type, _Value>> {
};

// Non-resursive lambda type.
template <class _F, class = void> struct is_recursive : std::false_type {};

// Resursive lambda type.
template <class _F>
struct is_recursive<
    _F, std::__void_t<decltype(&_F::template operator()<fixed_point<_F> &>)>>
    : std::true_type {};

// Recursive ver.
template <class _F> class _recursive {
  template <class...> struct _cache;

  template <class _G, class _R, class _H, class... _Args>
  struct _cache<_R (_G::*)(_H, _Args...)> : assoc<_R, _Args...> {};

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

 public:
  using cache_type =
      _cache<decltype(&_F::template operator()<_recursive<_F> &>)>;

  _recursive(_F &&__x) noexcept : __fn(__x) {}

  // Function call.
  template <class... _Args> decltype(auto) operator()(_Args &&...__args) {
    return _wrapper{__fn, __c}(std::forward<_Args>(__args)...);
  }

 private:
  _F __fn;
  cache_type __c;

  struct _wrapper {
    _F &__fn;
    cache_type &__c;

    template <class... _Args> decltype(auto) operator()(_Args &&...__args) {
      typename cache_type::key_type __key{__args...};
      auto __i = __c.lower_bound(__key);

      if _CXX17_CONSTEXPR (cache_type::value) {
        if (__i != __c.end() && __i->first == __key) return __i->second;

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

      else if (__i == __c.end() || *__i != __key)
        __c.emplace_hint(__i, std::move(__key)),
            __fn(*this, std::forward<_Args>(__args)...);
    }
  };
};

// Non-recursive ver.
template <class _F> class _non_recursive {
  template <class _T, class = void> struct _get_func { using type = _T; };

  template <class _T>
  struct _get_func<_T, std::__void_t<decltype(&_T::operator())>> {
    using type = decltype(&_T::operator());
  };

  template <class...> struct _cache;

  template <class _R, class... _Args>
  struct _cache<_R(_Args...)> : assoc<_R, _Args...> {};

  template <class _R, class... _Args>
  struct _cache<_R (*)(_Args...)> : assoc<_R, _Args...> {};

  template <class _G, class _R, class... _Args>
  struct _cache<_R (_G::*)(_Args...)> : assoc<_R, _Args...> {};

  template <class _G, class _R, class... _Args>
  struct _cache<_R (_G::*)(_Args...) const> : assoc<_R, _Args...> {};

 public:
  using cache_type = _cache<typename _get_func<_F>::type>;

  _non_recursive(_F &&__x) noexcept : __fn(__x) {}

  // Function call.
  template <class... _Args> decltype(auto) operator()(_Args &&...__args) {
    typename cache_type::key_type __key{__args...};
    auto __i = __c.lower_bound(__key);

    if _CXX17_CONSTEXPR (cache_type::value) {
      if (__i != __c.end() && __i->first == __key) return __i->second;

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

    else if (__i == __c.end() || *__i != __key)
      __c.emplace_hint(__i, std::move(__key)),
          __fn(std::forward<_Args>(__args)...);
  }

 private:
  _F __fn;
  cache_type __c;
};

template <class _F>
using _cached = std::conditional_t<is_recursive<_F>::value, _recursive<_F>,
                                   _non_recursive<_F>>;

}  // namespace _cached_impl

/**
 * @brief Cached caller of function
 */
template <class _F> class cached : public _cached_impl::_cached<_F> {
 public:
  // Construct a new cached object.
  cached() noexcept : _cached_impl::_cached<_F>(_F{}) {}

  // Construct a new cached object.
  cached(_F __x) noexcept : _cached_impl::_cached<_F>(std::move(__x)) {}
};

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

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

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

#line 11 "Library\\src\\utils\\cat.hpp"

namespace workspace {

/**
 * @brief Concatenate two sequences.
 *
 * @param __c1
 * @param __c2
 * @return Concatenated sequence.
 */
template <class _C1, class _C2>
constexpr decltype(auto) cat(_C1 &&__c1, _C2 &&__c2) noexcept {
  auto __c = std::forward<_C1>(__c1);

  if _CXX17_CONSTEXPR (std::is_rvalue_reference<decltype(__c2)>::value)
    __c.insert(std::end(__c), std::make_move_iterator(std::begin(__c2)),
               std::make_move_iterator(std::end(__c2)));
  else
    __c.insert(std::end(__c), std::cbegin(__c2), std::cend(__c2));

  return __c;
}

/**
 * @return Concatenated sequence.
 */
template <class _C1, class _C2, class... _Args>
constexpr decltype(auto) cat(_C1 &&__c1, _C2 &&__c2,
                             _Args &&...__args) noexcept {
  return cat(cat(std::forward<_C1>(__c1), std::forward<_C2>(__c2)),
             std::forward<_Args>(__args)...);
}

}  // 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 {

/**
 * @brief Substitute __y for __x if __y < __x.
 * @param __x Reference
 * @param __y Comparison target
 * @return Whether or not __x is updated.
 */
template <class _T1, class _T2,
          typename = decltype(std::declval<_T2>() < std::declval<_T1 &>())>
typename std::enable_if<std::is_assignable<_T1 &, _T2>::value, bool>::type chle(
    _T1 &__x, _T2 &&__y) noexcept {
  return __y < __x ? __x = std::forward<_T2>(__y), true : false;
}

/**
 * @brief Substitute __y for __x if __x < __y.
 * @param __x Reference
 * @param __y Comparison target
 * @return Whether or not __x is updated.
 */
template <class _T1, class _T2,
          typename = decltype(std::declval<_T1 &>() < std::declval<_T2>())>
typename std::enable_if<std::is_assignable<_T1 &, _T2>::value, bool>::type chgr(
    _T1 &__x, _T2 &&__y) noexcept {
  return __x < __y ? __x = std::forward<_T2>(__y), true : false;
}

/**
 * @brief Substitute __y for __x if __comp(__y, __x) is true.
 * @param __x Reference
 * @param __y Comparison target
 * @param __comp Compare function object
 * @return Whether or not __x is updated.
 */
template <class _T1, class _T2, class _Compare,
          typename = decltype(std::declval<_Compare>()(std::declval<_T2>(),
                                                       std::declval<_T1 &>()))>
typename std::enable_if<std::is_assignable<_T1 &, _T2>::value, bool>::type chle(
    _T1 &__x, _T2 &&__y, _Compare __comp) noexcept {
  return __comp(__y, __x) ? __x = std::forward<_T2>(__y), true : false;
}

/**
 * @brief Substitute __y for __x if __comp(__x, __y) is true.
 * @param __x Reference
 * @param __y Comparison target
 * @param __comp Compare function object
 * @return Whether or not __x is updated.
 */
template <class _T1, class _T2, class _Compare,
          typename = decltype(std::declval<_Compare>()(std::declval<_T1 &>(),
                                                       std::declval<_T2>()))>
typename std::enable_if<std::is_assignable<_T1 &, _T2>::value, bool>::type chgr(
    _T1 &__x, _T2 &&__y, _Compare __comp) noexcept {
  return __comp(__x, __y) ? __x = std::forward<_T2>(__y), true : false;
}

}  // namespace workspace
#line 1 "Library\\src\\utils\\compare.hpp"
/**
 * @file compare.hpp
 * @brief Compare
 */

#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; };

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

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

#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;

/**
 * @brief Return type of subscripting ( @c [] ) access.
 */
template <class _Tp>
using subscripted_type =
    typename std::decay<decltype(std::declval<_Tp&>()[0])>::type;

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

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

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

template <class _Tp, class = void> struct has_mod : std::false_type {};

template <class _Tp>
struct has_mod<_Tp, std::__void_t<decltype(_Tp::mod)>> : std::true_type {};

template <class _Tp, class = void> struct is_integral_ext : std::false_type {};
template <class _Tp>
struct is_integral_ext<
    _Tp, typename std::enable_if<std::is_integral<_Tp>::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 _Tp>
constexpr static bool is_integral_ext_v = is_integral_ext<_Tp>::value;

#endif

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

#if __INT128_DEFINED__

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

#endif

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

template <typename _Tp> struct multiplicable {
  using type = std::conditional_t<
      is_integral_ext<_Tp>::value,
      std::conditional_t<std::is_signed<_Tp>::value,
                         typename multiplicable_int<_Tp>::type,
                         typename multiplicable_uint<_Tp>::type>,
      _Tp>;
};

template <class> struct first_arg { using type = void; };

template <class _R, class _Tp, class... _Args>
struct first_arg<_R(_Tp, _Args...)> {
  using type = _Tp;
};

template <class _R, class _Tp, class... _Args>
struct first_arg<_R (*)(_Tp, _Args...)> {
  using type = _Tp;
};

template <class _G, class _R, class _Tp, class... _Args>
struct first_arg<_R (_G::*)(_Tp, _Args...)> {
  using type = _Tp;
};

template <class _G, class _R, class _Tp, class... _Args>
struct first_arg<_R (_G::*)(_Tp, _Args...) const> {
  using type = _Tp;
};

template <class _Tp, class = void> struct parse_compare : first_arg<_Tp> {};

template <class _Tp>
struct parse_compare<_Tp, std::__void_t<decltype(&_Tp::operator())>>
    : first_arg<decltype(&_Tp::operator())> {};

}  // namespace workspace
#line 7 "Library\\src\\utils\\compare.hpp"

#if __cplusplus >= 201703L

namespace workspace {

/**
 * @brief Compare 2 points by their value of `atan2`.
 *
 * @return
 */
template <class _Tp>
bool compare_arg(const _Tp& __p1, const _Tp& __p2) noexcept {
  const auto& [__x1, __y1] = __p1;
  const auto& [__x2, __y2] = __p2;

  using value_type = std::decay_t<decltype(__x1)>;
  using mul_type = typename multiplicable<value_type>::type;

  if (__y1 == value_type(0))
    return value_type(0) <= __x1 &&
           (value_type(0) < __y2 ||
            (__y2 == value_type(0) && __x2 < value_type(0)));

  return value_type(0) < __y1
             ? value_type(0) <= __y2 &&
                   mul_type(__y1) * __x2 < mul_type(__x1) * __y2
             : value_type(0) <= __y2 ||
                   mul_type(__y1) * __x2 < mul_type(__x1) * __y2;
}

}  // namespace workspace

#endif
#line 7 "Library\\lib\\utils"
// #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 16 "Library\\src\\utils\\io\\istream.hpp"

namespace workspace {

namespace _istream_impl {

template <class _Tp, typename = void> struct helper {
  helper(std::istream &__is, _Tp &__x) {
    if _CXX17_CONSTEXPR (has_begin<_Tp &>::value)
      for (auto &&__e : __x) helper<std::decay_t<decltype(__e)>>(__is, __e);
    else
      static_assert(has_begin<_Tp>::value, "istream unsupported type.");
  }
};

template <class _Tp>
struct helper<_Tp, std::__void_t<decltype(std::declval<std::istream &>() >>
                                          std::declval<_Tp &>())>> {
  helper(std::istream &__is, _Tp &__x) { __is >> __x; }
};

#ifdef __SIZEOF_INT128__

template <> struct helper<__uint128_t, void> {
  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 helper<__int128_t, void> {
  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 helper<std::pair<_T1, _T2>> {
  helper(std::istream &__is, std::pair<_T1, _T2> &__x) {
    helper<_T1>(__is, __x.first), helper<_T2>(__is, __x.second);
  }
};

template <class... _Tp> struct helper<std::tuple<_Tp...>> {
  helper(std::istream &__is, std::tuple<_Tp...> &__x) { iterate(__is, __x); }

 private:
  template <class _Tuple, size_t _Nm = 0>
  void iterate(std::istream &__is, _Tuple &__x) {
    if _CXX17_CONSTEXPR (_Nm != std::tuple_size<_Tuple>::value) {
      helper<typename std::tuple_element<_Nm, _Tuple>::type>(
          __is, std::get<_Nm>(__x)),
          iterate<_Tuple, _Nm + 1>(__is, __x);
    }
  }
};

}  // namespace _istream_impl

/**
 * @brief A wrapper class for std::istream.
 */
class istream : public std::istream {
 public:
  /**
   * @brief Wrapped operator.
   */
  template <typename _Tp> istream &operator>>(_Tp &__x) {
    _istream_impl::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 9 "Library\\lib\\utils"
// #include "src/utils/io/ostream.hpp"
// #include "src/utils/io/read.hpp"
#line 2 "Library\\src\\utils\\grid\\motion.hpp"

/**
 * @file motion.hpp
 * @brief Motion
 */

#line 9 "Library\\src\\utils\\grid\\motion.hpp"

#line 11 "Library\\src\\utils\\grid\\motion.hpp"

namespace workspace {

/**
 * @brief Transpose.
 * @param __grid
 */
template <class _Grid,
          typename = decltype(std::declval<std::decay_t<_Grid>>()[0].resize(0))>
constexpr decltype(auto) transpose(_Grid &&__grid) noexcept {
#if __cplusplus < 201703L
  auto __h = __grid.size(), __w = __grid[0].size();
#else
  auto __h = std::size(__grid), __w = std::size(__grid[0]);
#endif
  std::decay_t<_Grid> __t(__w);
  for (auto &&__r : __t) __r.resize(__h);
  for (size_t __i = 0; __i != __h; ++__i)
    for (size_t __j = 0; __j != __w; ++__j)
      if _CXX17_CONSTEXPR (std::is_rvalue_reference<decltype(__grid)>::value)
        __t[__j][__i] = std::move(__grid[__i][__j]);
      else
        __t[__j][__i] = __grid[__i][__j];
  return __t;
}

/**
 * @brief Transpose.
 * @param __grid
 */
template <class _Tp, size_t _Rows, size_t _Cols>
constexpr decltype(auto) transpose(const _Tp (&__grid)[_Rows][_Cols]) noexcept {
  std::array<std::array<_Tp, _Rows>, _Cols> __t;
  for (size_t __i = 0; __i != _Rows; ++__i)
    for (size_t __j = 0; __j != _Cols; ++__j) __t[__j][__i] = __grid[__i][__j];
  return __t;
}

/**
 * @brief Transpose.
 * @param __grid
 */
template <class _Tp, size_t _Rows, size_t _Cols>
constexpr decltype(auto) transpose(_Tp(&&__grid)[_Rows][_Cols]) noexcept {
  std::array<std::array<_Tp, _Rows>, _Cols> __t;
  for (size_t __i = 0; __i != _Rows; ++__i)
    for (size_t __j = 0; __j != _Cols; ++__j)
      __t[__j][__i] = std::move(__grid[__i][__j]);
  return __t;
}

/**
 * @brief Transpose.
 * @param __grid
 */
template <class _Tp, size_t _Rows, size_t _Cols>
constexpr decltype(auto) transpose(
    const std::array<std::array<_Tp, _Cols>, _Rows> &__grid) noexcept {
  std::array<std::array<_Tp, _Rows>, _Cols> __t;
  for (size_t __i = 0; __i != _Rows; ++__i)
    for (size_t __j = 0; __j != _Cols; ++__j) __t[__j][__i] = __grid[__i][__j];
  return __t;
}

/**
 * @brief Transpose.
 * @param __grid
 */
template <class _Tp, size_t _Rows, size_t _Cols>
constexpr decltype(auto) transpose(
    std::array<std::array<_Tp, _Cols>, _Rows> &&__grid) noexcept {
  std::array<std::array<_Tp, _Rows>, _Cols> __t;
  for (size_t __i = 0; __i != _Rows; ++__i)
    for (size_t __j = 0; __j != _Cols; ++__j)
      __t[__j][__i] = std::move(__grid[__i][__j]);
  return __t;
}

/**
 * @brief Roll the grid counter-clockwise.
 * @param __grid
 */
template <class _Grid> decltype(auto) roll_ccw(_Grid &&__grid) noexcept {
  if _CXX17_CONSTEXPR (std::is_rvalue_reference<decltype(__grid)>::value) {
    auto __t = transpose(std::move(__grid));
    std::reverse(std::begin(__t), std::end(__t));
    return __t;
  } else {
    auto __t = transpose(__grid);
    std::reverse(std::begin(__t), std::end(__t));
    return __t;
  }
}

/**
 * @brief Roll the grid clockwise.
 * @param __grid
 */
template <class _Grid> decltype(auto) roll_cw(_Grid &&__grid) noexcept {
  if _CXX17_CONSTEXPR (std::is_rvalue_reference<decltype(__grid)>::value) {
    std::reverse(std::begin(__grid), std::end(__grid));
    return transpose(std::move(__grid));
  } else {
    auto __t = transpose(__grid);
    for (auto &&__r : __t) std::reverse(std::begin(__r), std::end(__r));
    return __t;
  }
}

}  // namespace workspace
#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 {

/**
 * @brief Setup I/O.
 * @param __n Standard output precision
 */
void io_setup(int __n) {
  std::cin.tie(0)->sync_with_stdio(0);
  std::cout << std::fixed << std::setprecision(__n);

#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.
 *
 * @param __dim Dimension
 * @param __x Initial value
 */
template <typename _Tp, class _Dim, size_t _Nm>
constexpr decltype(auto) make_vector([[maybe_unused]] _Dim* __dim,
                                     const _Tp& __x = _Tp()) {
  static_assert(std::is_convertible<_Dim, size_t>::value);

  if constexpr (_Nm)
    return std::vector(*__dim,
                       make_vector<_Tp, _Dim, _Nm - 1>(std::next(__dim), __x));
  else
    return __x;
}

/**
 * @brief Make a multi-dimensional vector.
 *
 * @param __dim Dimension
 * @param __x Initial value
 */
template <typename _Tp, class _Dim, size_t _Nm>
constexpr decltype(auto) make_vector(const _Dim (&__dim)[_Nm],
                                     const _Tp& __x = _Tp()) {
  return make_vector<_Tp, _Dim, _Nm>((_Dim*)__dim, __x);
}

/**
 * @brief Make a multi-dimensional vector.
 *
 * @param __dim Dimension
 * @param __x Initial value
 */
template <typename _Tp, class _Dim, size_t _Nm = 0>
constexpr decltype(auto) make_vector([[maybe_unused]] const _Dim& __dim,
                                     const _Tp& __x = _Tp()) {
  if constexpr (_Nm == std::tuple_size<_Dim>::value)
    return __x;

  else {
    static_assert(
        std::is_convertible<std::tuple_element_t<_Nm, _Dim>, size_t>::value);

    return std::vector(std::get<_Nm>(__dim),
                       make_vector<_Tp, _Dim, _Nm + 1>(__dim, __x));
  }
}

}  // namespace workspace

#endif
#line 16 "Library\\lib\\utils"
// #include "src/utils/py-like/enumerate.hpp"
#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 _reversed_impl {

template <class _Container> class reversed {
  _Container __cont;

 public:
  constexpr reversed(_Container &&__cont) noexcept : __cont(__cont) {}

  constexpr decltype(auto) begin() noexcept { return std::rbegin(__cont); }
  constexpr decltype(auto) begin() const noexcept {
    return std::rbegin(__cont);
  }

  constexpr decltype(auto) end() noexcept { return std::rend(__cont); }
  constexpr decltype(auto) end() const noexcept { return std::rend(__cont); }

  constexpr decltype(auto) size() const noexcept {
    return
#if __cplusplus < 201703L
        __cont.size();
#else
        std::size(__cont);
#endif
  }
};

}  // namespace _reversed_impl

template <class _Container>
constexpr decltype(auto) reversed(_Container &&__cont) noexcept {
  return _reversed_impl::reversed<_Container>{std::forward<_Container>(__cont)};
}

template <class _Tp>
constexpr decltype(auto) reversed(
    std::initializer_list<_Tp> &&__cont) noexcept {
  return _reversed_impl::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::random_access_iterator_tag;

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

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

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

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

    constexpr iterator &operator++() noexcept {
      ++current;
      return *this;
    }
    constexpr iterator &operator++(int) noexcept {
      auto __tmp = *this;
      ++current;
      return __tmp;
    }

    constexpr iterator &operator--() noexcept {
      --current;
      return *this;
    }
    constexpr iterator &operator--(int) noexcept {
      auto __tmp = *this;
      --current;
      return __tmp;
    }

    constexpr difference_type operator-(const iterator &__x) const noexcept {
      return current - __x.current;
    }

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

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

    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());
  }

  constexpr size_t size() const noexcept {
    return std::distance(__first, __last);
  }
};

template <class... _Args>
constexpr decltype(auto) rrange(_Args &&...__args) noexcept {
  return reversed(range(std::forward<_Args>(__args)...));
}

template <class _Container>
constexpr decltype(auto) iterate(_Container &&__cont) noexcept {
  return range(std::begin(__cont), std::end(__cont));
}

template <class _Container>
constexpr decltype(auto) riterate(_Container &&__cont) noexcept {
  return range(std::rbegin(__cont), std::rend(__cont));
}

}  // namespace workspace

#endif
#line 18 "Library\\lib\\utils"
// #include "src/utils/py-like/reversed.hpp"
// #include "src/utils/py-like/zip.hpp"
#line 2 "Library\\src\\utils\\rand\\rng.hpp"

/**
 * @file rng.hpp
 * @brief Random Number Generator
 */

#line 9 "Library\\src\\utils\\rand\\rng.hpp"

namespace workspace {

template <typename _Arithmetic>
using uniform_distribution = typename std::conditional<
    std::is_integral<_Arithmetic>::value,
    std::uniform_int_distribution<_Arithmetic>,
    std::uniform_real_distribution<_Arithmetic>>::type;

template <typename _Arithmetic, class _Engine = std::mt19937>
class random_number_generator : uniform_distribution<_Arithmetic> {
  using base = uniform_distribution<_Arithmetic>;

  _Engine __engine;

 public:
  random_number_generator(_Arithmetic __min, _Arithmetic __max)
      : base(__min, __max), __engine(std::random_device{}()) {}

  random_number_generator(_Arithmetic __max = 1)
      : random_number_generator(0, __max) {}

  random_number_generator(typename base::param_type const& __param)
      : base(__param), __engine(std::random_device{}()) {}

  decltype(auto) operator()() noexcept { return base::operator()(__engine); }
};

}  // namespace workspace
#line 2 "Library\\src\\utils\\rand\\shuffle.hpp"

/**
 * @file shuffle.hpp
 * @brief Shuffle
 */

#line 10 "Library\\src\\utils\\rand\\shuffle.hpp"

namespace workspace {

template <class _RAIter, class _Engine = std::mt19937>
void shuffle(_RAIter __first, _RAIter __last) {
  static _Engine __engine(std::random_device{}());
  std::shuffle(__first, __last, __engine);
}

}  // namespace workspace
#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 23 "Library\\lib\\utils"
// #include "src\utils\rand\tree.hpp"
// #include "src\utils\reference_list.hpp"
#line 2 "Library\\src\\utils\\io\\input.hpp"

/**
 * @file input.hpp
 * @brief Input
 */

#line 9 "Library\\src\\utils\\io\\input.hpp"

namespace workspace {

namespace _input_impl {

template <class _Tp, bool _Is_class = false> class input {
  _Tp __value;

  template <class _Arg, class... _Args> struct is_same : std::false_type {};
  template <class _Arg> struct is_same<_Arg, _Arg> : std::true_type {};

 public:
  operator _Tp &() noexcept { return __value; }
  operator const _Tp &() const noexcept { return __value; }

  template <class... _Args>
  input(_Args &&...__args) noexcept : __value(std::forward<_Args>(__args)...) {
    if _CXX17_CONSTEXPR (!is_same<decltype(*this), _Args...>::value &&
                         !is_same<_Tp, _Args...>::value)
      cin >> __value;
  }

  input &operator=(const _Tp &__x) noexcept { return __value = __x, *this; }
};

template <class _Tp> class input<_Tp, true> : public _Tp {
  template <class _Arg, class... _Args> struct is_same : std::false_type {};
  template <class _Arg> struct is_same<_Arg, _Arg> : std::true_type {};

 public:
  operator _Tp &() noexcept { return *this; }
  operator const _Tp &() const noexcept { return *this; }

  template <class... _Args>
  input(_Args &&...__args) noexcept : _Tp(std::forward<_Args>(__args)...) {
    if _CXX17_CONSTEXPR (!is_same<decltype(*this), _Args...>::value &&
                         !is_same<_Tp, _Args...>::value)
      cin >> *this;
  }

  input &operator=(const _Tp &__x) noexcept {
    _Tp::operator=(__x);
    return *this;
  }
};

}  // namespace _input_impl

template <class _Tp>
using input = _input_impl::input<_Tp, std::is_class<_Tp>::value>;

}  // namespace workspace
#line 11 "other-workspace\\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\\algebra\\rational.hpp"

/**
 * @file rational.hpp
 * @brief Rational
 */

#line 9 "Library\\src\\algebra\\rational.hpp"

namespace workspace {

/**
 * @brief Rational
 *
 * @tparam _Tp Ring structure
 */
template <class _Tp> class rational {
  constexpr void normalize() noexcept;

 public:
  _Tp __num{0}, __den{1};

  constexpr rational() noexcept = default;

  constexpr rational(const _Tp &__x) noexcept : __num(__x) {}

  constexpr rational(const _Tp &__x, const _Tp __y) noexcept
      : __num(__x), __den(__y) {}

  constexpr operator bool() const noexcept { return operator!=(0); }

  constexpr rational operator+() const noexcept { return *this; }

  constexpr rational operator-() const noexcept { return {-__num, __den}; }

  constexpr rational operator+(const rational &__x) const noexcept {
    return {__num * __x.__den + __x.__num * __den, __den * __x.__den};
  }

  constexpr rational operator-(const rational &__x) const noexcept {
    return {__num * __x.__den - __x.__num * __den, __den * __x.__den};
  }

  constexpr rational operator*(const rational &__x) noexcept {
    return {__num * __x.__num, __den * __x.__den};
  }

  constexpr rational operator/(const rational &__x) noexcept {
    assert(__x.__num != _Tp(0));
    return {__num * __x.__den, __den * __x.__num};
  }

  constexpr rational &operator+=(const rational &__x) noexcept {
    (__num *= __x.__den) += __den * __x.__num, __den *= __x.__den;
    return *this;
  }

  constexpr rational &operator-=(const rational &__x) noexcept {
    (__num *= __x.__den) -= __den * __x.__num, __den *= __x.__den;
    return *this;
  }

  constexpr rational &operator*=(const rational &__x) noexcept {
    __num *= __x.__num, __den *= __x.__den;
    return *this;
  }

  constexpr rational &operator/=(const rational &__x) noexcept {
    assert(__x.__num != _Tp(0));
    __num *= __x.__den, __den *= __x.__num;
    return *this;
  }

  constexpr bool operator==(const rational &__x) const noexcept {
    return __num == __x.__num && __den == __x.den;
  }

  constexpr bool operator!=(const rational &__x) const noexcept {
    return __num != __x.__num || __den != __x.den;
  }

  constexpr bool operator<(const rational &__x) const noexcept {
    return __num * __x.__den < __den * __x.__num;
  }
};

}  // namespace workspace
#line 2 "Library\\src\\modular\\modint.hpp"

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

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

#line 2 "Library\\src\\number_theory\\sqrt_mod.hpp"

/**
 * @file sqrt_mod.hpp
 * @brief Tonelli-Shanks Algorithm
 */

#line 2 "Library\\src\\number_theory\\pow_mod.hpp"

/**
 * @file mod_pow.hpp
 * @brief Modular Exponentiation
 */

#line 9 "Library\\src\\number_theory\\pow_mod.hpp"

#line 11 "Library\\src\\number_theory\\pow_mod.hpp"

namespace workspace {

/**
 * @brief Compile time modular exponentiation.
 *
 * @param __x
 * @param __n Exponent
 * @param __mod Modulus
 * @return
 */
template <class _Tp>
constexpr std::enable_if_t<(is_integral_ext<_Tp>::value), _Tp> pow_mod(
    _Tp __x, _Tp __n, _Tp __mod) noexcept {
  assert(__mod > 0);

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

  if ((__x %= __mod) < 0) __x += __mod;

  mul_type __y{1};

  while (__n) {
    if (__n & 1) (__y *= __x) %= __mod;
    __x = (mul_type)__x * __x % __mod;
    __n >>= 1;
  }

  return __y;
};

}  // namespace workspace
#line 10 "Library\\src\\number_theory\\sqrt_mod.hpp"

namespace workspace {

/**
 * @brief Compile time modular square root.
 *
 * @param __x
 * @param __mod Modulus
 * @return One if it exists. Otherwise -1.
 */
template <class _Tp>
constexpr std::enable_if_t<(is_integral_ext<_Tp>::value), _Tp> sqrt_mod(
    _Tp __x, _Tp __mod) noexcept {
  assert(__mod > 0);

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

  if ((__x %= __mod) < 0) __x += __mod;

  if (!__x) return 0;

  if (__mod == 2) return __x;

  if (pow_mod(__x, __mod >> 1, __mod) != 1) return -1;

  _Tp __z = __builtin_ctz(__mod - 1), __q = __mod >> __z;

  mul_type __a = pow_mod(__x, (__q + 1) >> 1, __mod), __b = 2;
  while (pow_mod<_Tp>(__b, __mod >> 1, __mod) == 1) ++__b;
  __b = pow_mod<_Tp>(__b, __q, __mod);

  _Tp __shift = 0;

  for (auto __r = __a * __a % __mod * pow_mod(__x, __mod - 2, __mod) % __mod;
       __r != 1; (__r *= (__b *= __b) %= __mod) %= __mod) {
    auto __bsf = __z;

    for (auto __e = __r; __e != 1; --__bsf) (__e *= __e) %= __mod;

    while (++__shift != __bsf) (__b *= __b) %= __mod;

    (__a *= __b) %= __mod;
  }

  return __a;
};

}  // namespace workspace
#line 15 "Library\\src\\modular\\modint.hpp"

namespace workspace {

namespace _modint_impl {

template <auto _Mod, unsigned _Storage> struct modint {
  static_assert(is_integral_ext<decltype(_Mod)>::value,
                "_Mod must be integral type.");

  using mod_type = std::make_signed_t<typename std::conditional<
      0 < _Mod, std::add_const_t<decltype(_Mod)>, decltype(_Mod)>::type>;

  using value_type = std::decay_t<mod_type>;

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

  // Modulus
  static mod_type mod;

  static unsigned storage;

 private:
  value_type value = 0;

  struct direct_ctor_t {};
  constexpr static direct_ctor_t direct_ctor_tag{};

  // Direct constructor
  template <class _Tp>
  constexpr modint(_Tp __n, direct_ctor_t) noexcept : value(__n) {}

 public:
  constexpr modint() noexcept = default;

  template <class _Tp, typename = std::enable_if_t<is_integral_ext<_Tp>::value>>
  constexpr modint(_Tp __n) noexcept
      : value((__n %= mod) < 0 ? __n += mod : __n) {}

  constexpr modint(bool __n) noexcept : value(__n) {}

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

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

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

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

  constexpr modint &operator--() noexcept {
    if (!value)
      value = mod - 1;
    else
      --value;
    return *this;
  }

  constexpr modint operator+() const noexcept { return *this; }

  constexpr modint operator-() const noexcept {
    return {value ? mod - value : 0, direct_ctor_tag};
  }

  // }} unary operators

  // operator+= {{

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

  template <class _Tp>
  constexpr std::enable_if_t<is_integral_ext<_Tp>::value, modint> &operator+=(
      _Tp const &__x) noexcept {
    if (((value += __x) %= mod) < 0) value += mod;
    return *this;
  }

  // }} operator+=

  // operator+ {{

  template <class _Tp>
  constexpr std::enable_if_t<is_integral_ext<_Tp>::value, modint> operator+(
      _Tp const &__x) const noexcept {
    return modint{*this} += __x;
  }

  constexpr modint operator+(modint __x) const noexcept { return __x += *this; }

  template <class _Tp>
  constexpr friend std::enable_if_t<is_integral_ext<_Tp>::value, modint>
  operator+(_Tp const &__x, modint __y) noexcept {
    return __y += __x;
  }

  // }} operator+

  // operator-= {{

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

  template <class _Tp>
  constexpr std::enable_if_t<is_integral_ext<_Tp>::value, modint> &operator-=(
      _Tp __x) noexcept {
    if (((value -= __x) %= mod) < 0) value += mod;
    return *this;
  }

  // }} operator-=

  // operator- {{

  template <class _Tp>
  constexpr std::enable_if_t<is_integral_ext<_Tp>::value, modint> operator-(
      _Tp const &__x) const noexcept {
    return modint{*this} -= __x;
  }

  constexpr modint operator-(const modint &__x) const noexcept {
    return modint{*this} -= __x;
  }

  template <class _Tp>
  constexpr friend std::enable_if_t<is_integral_ext<_Tp>::value, modint>
  operator-(_Tp __x, const modint &__y) noexcept {
    if (((__x -= __y.value) %= mod) < 0) __x += mod;
    return {__x, direct_ctor_tag};
  }

  // }} operator-

  // operator*= {{

  constexpr modint &operator*=(const modint &__x) noexcept {
    value =
        static_cast<value_type>(value * static_cast<mul_type>(__x.value) % mod);
    return *this;
  }

  template <class _Tp>
  constexpr std::enable_if_t<is_integral_ext<_Tp>::value, modint> &operator*=(
      _Tp __x) noexcept {
    value = static_cast<value_type>(
        value * mul_type((__x %= mod) < 0 ? __x + mod : __x) % mod);
    return *this;
  }

  // }} operator*=

  // operator* {{

  constexpr modint operator*(const modint &__x) const noexcept {
    return {static_cast<mul_type>(value) * __x.value % mod, direct_ctor_tag};
  }

  template <class _Tp>
  constexpr std::enable_if_t<is_integral_ext<_Tp>::value, modint> operator*(
      _Tp __x) const noexcept {
    __x %= mod;
    if (__x < 0) __x += mod;
    return {static_cast<mul_type>(value) * __x % mod, direct_ctor_tag};
  }

  template <class _Tp>
  constexpr friend std::enable_if_t<is_integral_ext<_Tp>::value, modint>
  operator*(_Tp __x, const modint &__y) noexcept {
    __x %= mod;
    if (__x < 0) __x += mod;
    return {static_cast<mul_type>(__x) * __y.value % mod, direct_ctor_tag};
  }

  // }} 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];
  }

  static value_type _div(mul_type __r, value_type __x) noexcept {
    assert(__x != value_type(0));
    if (!__r) return 0;

    std::make_signed_t<value_type> __v{};
    bool __neg = __x < 0 ? __x = -__x, true : false;

    if (static_cast<decltype(storage)>(__x) < storage)
      __v = _mem(__x);
    else {
      value_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;

    return __r == mul_type(1) ? static_cast<value_type>(__v)
                              : static_cast<value_type>(__r * __v % mod);
  }

 public:
  // operator/= {{

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

  template <class _Tp>
  constexpr std::enable_if_t<is_integral_ext<_Tp>::value, modint> &operator/=(
      _Tp __x) noexcept {
    if (value) value = _div(value, __x %= mod);
    return *this;
  }

  // }} operator/=

  // operator/ {{

  constexpr modint operator/(const modint &__x) const noexcept {
    if (!value) return {};
    return {_div(value, __x.value), direct_ctor_tag};
  }

  template <class _Tp>
  constexpr std::enable_if_t<is_integral_ext<_Tp>::value, modint> operator/(
      _Tp __x) const noexcept {
    if (!value) return {};
    return {_div(value, __x %= mod), direct_ctor_tag};
  }

  template <class _Tp>
  constexpr friend std::enable_if_t<is_integral_ext<_Tp>::value, modint>
  operator/(_Tp __x, const modint &__y) noexcept {
    if (!__x) return {};
    if ((__x %= mod) < 0) __x += mod;
    return {_div(__x, __y.value), direct_ctor_tag};
  }

  // }} operator/

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

  template <class _Tp>
  constexpr std::enable_if_t<is_integral_ext<_Tp>::value, modint> pow(
      _Tp __e) const noexcept {
    modint __r{1, direct_ctor_tag};

    for (modint __b{__e < 0 ? __e = -__e, _div(1, value) : value,
                              direct_ctor_tag};
         __e; __e >>= 1, __b *= __b)
      if (__e & 1) __r *= __b;

    return __r;
  }

  template <class _Tp>
  constexpr friend std::enable_if_t<is_integral_ext<_Tp>::value, modint> pow(
      modint __b, _Tp __e) noexcept {
    if (__e < 0) {
      __e = -__e;
      __b.value = _div(1, __b.value);
    }

    modint __r{1, direct_ctor_tag};

    for (; __e; __e >>= 1, __b *= __b)
      if (__e & 1) __r *= __b;

    return __r;
  }

  constexpr modint sqrt() const noexcept {
    return {sqrt_mod(value, mod), direct_ctor_tag};
  }

  friend constexpr modint sqrt(const modint &__x) noexcept {
    return {sqrt_mod(__x.value, mod), direct_ctor_tag};
  }

  template <class _Os>
  friend _Os &operator<<(_Os &__os, const modint &__x) noexcept {
    return __os << __x.value;
  }

  friend std::istream &operator>>(std::istream &__is, modint &__x) noexcept {
    std::string __s;
    __is >> __s;
    bool __neg = false;
    if (__s.front() == '-') {
      __neg = true;
      __s.erase(__s.begin());
    }
    __x = 0;
    for (char __c : __s) __x = __x * 10 + (__c - '0');
    if (__neg) __x = -__x;
    return __is;
  }
};

template <auto _Mod, unsigned _Storage>
typename modint<_Mod, _Storage>::mod_type modint<_Mod, _Storage>::mod =
    _Mod > 0 ? _Mod : 0;

template <auto _Mod, unsigned _Storage>
unsigned modint<_Mod, _Storage>::storage = _Storage;

}  // namespace _modint_impl

template <auto _Mod, unsigned _Storage = 0,
          typename = std::enable_if_t<(_Mod > 0)>>
using modint = _modint_impl::modint<_Mod, _Storage>;

template <unsigned _Id = 0>
using modint_runtime = _modint_impl::modint<-(signed)_Id, 0>;

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

namespace workspace {

using mint = modint<998244353>;

template <class T, typename = decltype(declval<T>() + 1)> void f(T) {}

void main() {
  // start here!

  input<int> n, k;
  priority_queue<rational<int>> Q;
  mint sum;
  while (n--) {
    input<int> a;
    Q.emplace(a, 1);
    sum += (int)a;
  }
  mint ans{1};
  for (auto i : range((int)k)) {
    ans *= i + 1;
    ans /= sum;
  }
  while (k--) {
    auto [x, y] = Q.top();
    Q.pop();
    ans *= x;
    ans /= y++;
    Q.emplace(x, y);
  }
  cout << ans << "\n";
}

}  // namespace workspace