#line 1 "other-workspace\\tmp.cpp"
#if defined(ONLINE_JUDGE)  // && 0
#pragma GCC optimize("Ofast,unroll-loops")
#pragma GCC target("sse,sse2,sse3,ssse3,sse4,popcnt,mmx,avx,avx2")
#endif

// #undef _GLIBCXX_DEBUG
#include <bits/extc++.h>

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

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

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

// #include "bit"
#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);
}

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

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

#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

namespace workspace {

#ifdef _GLIBCXX_BIT

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

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

template <class _Tp> constexpr _Tp popcount(_Tp __x) noexcept {
  return std::__popcount(__x);
}

#endif

template <class _Tp>
constexpr _Tp test_bit(const _Tp& __b, std::size_t __n) noexcept {
  return __b >> __n & 1;
}

}  // namespace workspace
#line 9 "Library\\lib\\cxx20"

#if __cplusplus <= 201703L

#include <algorithm>
#include <vector>

namespace std {

/**
 * @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();
}

/**
 * @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\\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 {

// Fixed Point Combinator.
template <class _F> class fixed_point {
  struct _wrapper {
    _F &__ref;

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

  _F __fn;

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

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

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

#line 8 "Library\\src\\utils\\hash.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; };

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 = void> struct has_begin : std::false_type {};

template <class _Tp>
struct has_begin<
    _Tp, std::__void_t<decltype(std::begin(std::declval<const _Tp&>()))>>
    : std::true_type {
  using type = decltype(std::begin(std::declval<const _Tp&>()));
};

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

template <class _Tp>
struct has_size<_Tp, std::__void_t<decltype(std::size(std::declval<_Tp>()))>>
    : std::true_type {};

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

template <class _Tp>
struct has_resize<_Tp, std::__void_t<decltype(std::declval<_Tp>().resize(
                           std::declval<size_t>()))>> : 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())> {};

template <class _Container, class = void> struct get_dimension {
  static constexpr size_t value = 0;
};

template <class _Container>
struct get_dimension<_Container,
                     std::enable_if_t<has_begin<_Container>::value>> {
  static constexpr size_t value =
      1 + get_dimension<typename std::iterator_traits<
              typename has_begin<_Container>::type>::value_type>::value;
};

}  // namespace workspace
#line 10 "Library\\src\\utils\\hash.hpp"

namespace workspace {

template <class T, class = void> struct hash : std::hash<T> {};

template <class _Tp> struct hash<_Tp *> : std::hash<_Tp *> {};

#if __cplusplus >= 201703L
template <class Unique_bits_type>
struct hash<Unique_bits_type,
            enable_if_trait_type<Unique_bits_type,
                                 std::has_unique_object_representations>> {
  size_t operator()(uint64_t x) const {
    static const uint64_t m = std::random_device{}();
    x ^= x >> 23;
    x ^= m;
    x ^= x >> 47;
    return x - (x >> 32);
  }
};
#endif

template <class Key> size_t hash_combine(const size_t &seed, const Key &key) {
  return seed ^
         (hash<Key>()(key) + 0x9e3779b9 /* + (seed << 6) + (seed >> 2) */);
}
template <class T1, class T2> struct hash<std::pair<T1, T2>> {
  size_t operator()(const std::pair<T1, T2> &pair) const {
    return hash_combine(hash<T1>()(pair.first), pair.second);
  }
};
template <class... T> class hash<std::tuple<T...>> {
  template <class Tuple, size_t index = std::tuple_size<Tuple>::value - 1>
  struct tuple_hash {
    static uint64_t apply(const Tuple &t) {
      return hash_combine(tuple_hash<Tuple, index - 1>::apply(t),
                          std::get<index>(t));
    }
  };
  template <class Tuple> struct tuple_hash<Tuple, size_t(-1)> {
    static uint64_t apply([[maybe_unused]] const Tuple &t) { return 0; }
  };

 public:
  uint64_t operator()(const std::tuple<T...> &t) const {
    return tuple_hash<std::tuple<T...>>::apply(t);
  }
};
template <class hash_table> struct hash_table_wrapper : hash_table {
  using key_type = typename hash_table::key_type;
  size_t count(const key_type &key) const {
    return hash_table::find(key) != hash_table::end();
  }
  template <class... Args> auto emplace(Args &&...args) {
    return hash_table::insert(typename hash_table::value_type(args...));
  }
};
template <class Key, class Mapped = __gnu_pbds::null_type>
using cc_hash_table =
    hash_table_wrapper<__gnu_pbds::cc_hash_table<Key, Mapped, hash<Key>>>;
template <class Key, class Mapped = __gnu_pbds::null_type>
using gp_hash_table =
    hash_table_wrapper<__gnu_pbds::gp_hash_table<Key, Mapped, hash<Key>>>;
template <class Key, class Mapped>
using unordered_map = std::unordered_map<Key, Mapped, hash<Key>>;
template <class Key> using unordered_set = std::unordered_set<Key, hash<Key>>;

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

template <class _Tp>
struct is_hashable<_Tp, std::__void_t<decltype(std::declval<hash<_Tp>>()(
                            std::declval<const _Tp &>()))>> : std::true_type {};

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

#line 2 "Library\\lib\\cxx14"

#ifndef _CXX14_CONSTEXPR
#if __cplusplus >= 201402L
#define _CXX14_CONSTEXPR constexpr
#else
#define _CXX14_CONSTEXPR
#endif
#endif
#line 4 "Library\\lib\\cxx17"

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

#ifndef _CXX17_STATIC_ASSERT
#if __cplusplus >= 201703L
#define _CXX17_STATIC_ASSERT static_assert
#else
#define _CXX17_STATIC_ASSERT assert
#endif
#endif

#line 22 "Library\\lib\\cxx17"

#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;
}

/**
 *  @brief  Return whether a container is empty.
 *  @param  __cont  Container.
 */
template <typename _Container>
[[nodiscard]] constexpr auto empty(const _Container& __cont) noexcept(
    noexcept(__cont.empty())) -> decltype(__cont.empty()) {
  return __cont.empty();
}

/**
 *  @brief  Return whether an array is empty (always false).
 */
template <typename _Tp, size_t _Nm>
[[nodiscard]] constexpr bool empty(const _Tp (&)[_Nm]) noexcept {
  return false;
}

/**
 *  @brief  Return whether an initializer_list is empty.
 *  @param  __il  Initializer list.
 */
template <typename _Tp>
[[nodiscard]] constexpr bool empty(initializer_list<_Tp> __il) noexcept {
  return __il.size() == 0;
}

struct monostate {};

}  // namespace std

#else

#include <variant>

#endif
#line 11 "Library\\src\\utils\\cached.hpp"

namespace workspace {

namespace _cached_impl {

// Convert keys to tuple.
template <class... _Args>
using as_tuple = 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>>>()...));

template <class _Tp, class _V> struct hash_map : unordered_map<_Tp, _V> {
  template <class... _Args>
  decltype(auto) lower_bound(_Args &&...__args) noexcept {
    return unordered_map<_Tp, _V>::find(std::forward<_Args>(__args)...);
  }
};

template <class _Tp> struct hash_set : unordered_set<_Tp> {
  template <class... _Args>
  decltype(auto) lower_bound(_Args &&...__args) noexcept {
    return unordered_set<_Tp>::find(std::forward<_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::conditional_t<is_hashable<as_tuple<_Keys...>>::value,
                                      hash_set<as_tuple<_Keys...>>,
                                      std::set<as_tuple<_Keys...>>>,
                   std::conditional_t<
                       //  is_hashable<as_tuple<_Keys...>>::value,
                       false, hash_map<as_tuple<_Keys...>, _Value>,
                       std::map<as_tuple<_Keys...>, _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...> {};

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

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

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

 private:
  _F __fn;
  cache_type __c;

  struct _wrapper {
    _F &__fn;
    cache_type &__c;

    template <class... _Args>
    decltype(auto) operator()(_Args &&...__args) noexcept(
        noexcept(__fn(*this, std::forward<_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)...);
    }
  };

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

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

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

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

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

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

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

 private:
  _F __fn;
  cache_type __c;

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

  // Function call.
  template <class... _Args>
  decltype(auto) operator()(_Args &&...__args) noexcept(
      noexcept(__fn(std::forward<_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)...);
  }
};

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\\lib\\utils"
// #include "src/utils/cat.hpp"
#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 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 2 "Library\\src\\utils\\io\\ostream.hpp"

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

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

#line 11 "Library\\src\\utils\\io\\ostream.hpp"

namespace workspace {

template <class _Os> struct is_ostream {
  template <typename... _Args>
  static std::true_type __test(std::basic_ostream<_Args...> *);
  static std::false_type __test(void *);
  constexpr static bool value = decltype(__test(std::declval<_Os *>()))::value;
};

template <class _Os>
using ostream_ref =
    typename std::enable_if<is_ostream<_Os>::value, _Os &>::type;

/**
 * @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), ostream_ref<_Os>>::type
operator<<(_Os &__os, const _Tp (&__a)[_Nm]) {
  if _CXX17_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::array.
 *
 * @param __os Output stream
 * @param __a Array
 * @return Reference to __os.
 */
template <class _Os, class _Tp, size_t _Nm>
ostream_ref<_Os> operator<<(_Os &__os, const std::array<_Tp, _Nm> &__a) {
  if _CXX17_CONSTEXPR (_Nm) {
    __os << __a[0];
    for (size_t __i = 1; __i != _Nm; ++__i) __os << ' ' << __a[__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>
ostream_ref<_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),
                        ostream_ref<_Os>>::type
operator<<(_Os &__os, const _Tp &__t) {
  if _CXX17_CONSTEXPR (_Nm != std::tuple_size<_Tp>::value) {
    if _CXX17_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_convertible<std::decay_t<_Container>, std::string>::value &&
        !std::is_convertible<std::decay_t<_Container>, char *>::value,
    ostream_ref<_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> ostream_ref<_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> ostream_ref<_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 8 "Library\\lib\\utils"
// #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 11 "Library\\lib\\utils"
// #include "src/utils/iterator/category.hpp"
// #include "src/utils/iterator/reverse.hpp"
// #include "src/utils/make_vector.hpp"
#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 2 "Library\\src\\utils\\py-like\\reversed.hpp"

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

#line 9 "Library\\src\\utils\\py-like\\reversed.hpp"

#line 11 "Library\\src\\utils\\py-like\\reversed.hpp"

namespace workspace {

// Reversed container.
template <class _Container> class reversed {
  _Container __c;

 public:
  template <class _Tp>
  constexpr reversed(_Tp &&__x) noexcept : __c(std::forward<_Container>(__x)) {}

  template <class _Tp>
  constexpr reversed(std::initializer_list<_Tp> __x) noexcept : __c(__x) {}

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

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

  constexpr bool empty() const noexcept { return std::empty(__c); }

  constexpr decltype(auto) size() const noexcept { return std::size(__c); }

  using iterator = decltype(std::rbegin(__c));
  using const_iterator = decltype(std::crbegin(__c));

  using size_type = decltype(std::size(__c));
  using difference_type =
      typename std::iterator_traits<iterator>::difference_type;
  using value_type = typename std::iterator_traits<iterator>::value_type;

  using reference = typename std::iterator_traits<iterator>::reference;
  using const_reference =
      typename std::iterator_traits<const_iterator>::reference;
};

#if __cpp_deduction_guides >= 201606L

template <class _Tp> reversed(_Tp &&) -> reversed<_Tp>;

template <class _Tp>
reversed(std::initializer_list<_Tp>) -> reversed<std::initializer_list<_Tp>>;

#endif

}  // namespace workspace
#line 9 "Library\\src\\utils\\py-like\\range.hpp"

namespace workspace {

template <class _Index> class range {
  _Index __first, __last;

 public:
  class iterator {
    _Index __i;

   public:
    using difference_type = std::ptrdiff_t;
    using value_type = _Index;
    using pointer = void;
    using reference = value_type;
    using iterator_category = std::random_access_iterator_tag;

    constexpr iterator() = default;
    constexpr iterator(const _Index &__x) noexcept : __i(__x) {}

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

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

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

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

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

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

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

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

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

  using value_type = _Index;
  using reference = value_type;

  using difference_type = std::ptrdiff_t;
  using size_type = std::size_t;

  using const_iterator = iterator;

  using reverse_iterator = std::reverse_iterator<iterator>;
  using const_reverse_iterator = reverse_iterator;

  template <class _Tp1, class _Tp2>
  constexpr range(const _Tp1 &__first, const _Tp2 &__last) noexcept
      : __first(__first), __last(__last) {}

  template <class _Tp>
  constexpr range(const _Tp &__last) noexcept : __first(), __last(__last) {}

  constexpr iterator begin() const noexcept { return {__first}; }
  constexpr const_iterator cbegin() const noexcept { return begin(); }

  constexpr iterator end() const noexcept { return {__last}; }
  constexpr const_iterator cend() const noexcept { return end(); }

  constexpr reverse_iterator rbegin() const noexcept {
    return reverse_iterator{end()};
  }
  constexpr const_reverse_iterator crbegin() const noexcept { return rbegin(); }

  constexpr reverse_iterator rend() const noexcept {
    return reverse_iterator{begin()};
  }
  constexpr const_reverse_iterator crend() const noexcept { return rend(); }

  constexpr size_type size() const noexcept { return __last - __first; }
};

#if __cpp_deduction_guides >= 201606L

template <class _Tp1, class _Tp2>
range(const _Tp1 &, const _Tp2 &)
    -> range<std::decay_t<decltype(++std::declval<_Tp1 &>())>>;

template <class _Tp>
range(const _Tp &) -> range<std::decay_t<decltype(++std::declval<_Tp &>())>>;

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

#endif

}  // namespace workspace
#line 2 "Library\\src\\utils\\py-like\\zip.hpp"

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

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

#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 15 "Library\\src\\utils\\py-like\\zip.hpp"

namespace workspace {

template <class _Tuple> class zipped : protected _Tuple {
 public:
  zipped(const _Tuple &__x) : _Tuple(__x) {}

  template <std::size_t _Nm> constexpr decltype(auto) get() const &noexcept {
    return *std::get<_Nm>(*this);
  }

  template <std::size_t _Nm> constexpr decltype(auto) get() &&noexcept {
    auto __tmp = *std::get<_Nm>(*this);
    return __tmp;
  }
};

template <class... _Containers> class zip {
  std::tuple<_Containers...> __c;

  template <std::size_t _Nm = 0>
  constexpr decltype(auto) begin_cat() const noexcept {
    if _CXX17_CONSTEXPR (_Nm != sizeof...(_Containers)) {
      return std::tuple_cat(std::tuple(std::begin(std::get<_Nm>(__c))),
                            begin_cat<_Nm + 1>());
    } else
      return std::tuple<>();
  }

  template <std::size_t _Nm = 0>
  constexpr decltype(auto) end_cat() const noexcept {
    if _CXX17_CONSTEXPR (_Nm != sizeof...(_Containers)) {
      return std::tuple_cat(std::tuple(std::end(std::get<_Nm>(__c))),
                            end_cat<_Nm + 1>());
    } else
      return std::tuple<>();
  }

  using _Iterator_tuple =
      std::tuple<decltype(std::begin(std::declval<_Containers>()))...>;

 public:
  using size_type = std::size_t;

  class iterator : public zipped<_Iterator_tuple> {
    using _Base = zipped<_Iterator_tuple>;

   public:
    using difference_type = std::ptrdiff_t;
    using value_type = _Base;
    using pointer = void;
    using reference = value_type &;
    using iterator_category =
        typename common_iterator_category<_Iterator_tuple>::type;

   protected:
    template <std::size_t _Nm = 0>
    constexpr bool equal(const iterator &__x) const noexcept {
      if _CXX17_CONSTEXPR (_Nm != sizeof...(_Containers)) {
        return std::get<_Nm>(*this) == std::get<_Nm>(__x) ||
               equal<_Nm + 1>(__x);
      } else
        return false;
    }

    template <std::size_t _Nm = 0> constexpr void increment() noexcept {
      if _CXX17_CONSTEXPR (_Nm != sizeof...(_Containers))
        ++std::get<_Nm>(*this), increment<_Nm + 1>();
    }

    template <std::size_t _Nm = 0> constexpr void decrement() noexcept {
      if _CXX17_CONSTEXPR (_Nm != sizeof...(_Containers))
        --std::get<_Nm>(*this), decrement<_Nm + 1>();
    }

    template <std::size_t _Nm = 0>
    constexpr void advance(difference_type __d) noexcept {
      if _CXX17_CONSTEXPR (_Nm != sizeof...(_Containers))
        std::get<_Nm>(*this) += __d, advance<_Nm + 1>(__d);
    }

   public:
    iterator(const _Iterator_tuple &__x) : _Base(__x) {}

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

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

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

    constexpr bool operator<(const iterator &__x) const noexcept {
      return std::get<0>(*this) < std::get<0>(__x);
    }
    constexpr bool operator<=(const iterator &__x) const noexcept {
      return std::get<0>(*this) <= std::get<0>(__x);
    }
    constexpr bool operator>(const iterator &__x) const noexcept {
      return !operator<=(__x);
    }
    constexpr bool operator>=(const iterator &__x) const noexcept {
      return !operator>(__x);
    }

    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 &__x) const noexcept {
      return std::get<0>(*this) - std::get<0>(__x);
    }

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

  using const_iterator = iterator;

  template <class... _Args>
  constexpr zip(_Args &&...__args) noexcept
      : __c(std::forward<_Args>(__args)...) {}

  template <class... _Args>
  constexpr zip(std::initializer_list<_Args>... __args) noexcept
      : __c(std::forward<_Args>(__args)...) {}

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

  constexpr iterator end() const noexcept { return {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()};
  }

#if __cplusplus >= 201703L
  constexpr size_type size() const noexcept {
    size_type __n = -1;
    std::apply(
        [&__n](auto &&__x) {
          if (__n < __x.size()) __n = __x.size();
        },
        __c);
    return __n;
  }
#endif
};

#if __cpp_deduction_guides >= 201606L

template <class... _Args> zip(_Args &&...) -> zip<_Args...>;

// For gcc version < 10
// https://gcc.gnu.org/bugzilla/show_bug.cgi?id=80438
template <class _Tp, class... _Args>
zip(_Tp &&, _Args &&...) -> zip<_Tp, _Args...>;

template <class... _Args>
zip(std::initializer_list<_Args>...) -> zip<std::initializer_list<_Args>...>;

#endif

}  // namespace workspace

namespace std {

template <size_t _Nm, class _Tuple>
struct tuple_element<_Nm, workspace::zipped<_Tuple>> {
  using type = remove_reference_t<
      typename iterator_traits<tuple_element_t<_Nm, _Tuple>>::reference>;
};

template <class _Tuple>
struct tuple_size<workspace::zipped<_Tuple>> : tuple_size<_Tuple> {};

}  // namespace std
#line 10 "Library\\src\\utils\\py-like\\enumerate.hpp"

#if __cplusplus >= 201703L

namespace workspace {

namespace _enumerate_impl {

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)...));
}

}  // namespace _enumerate_impl

template <class... _Args>
constexpr decltype(auto) enumerate(_Args &&... __args) noexcept {
  return zip(range(_enumerate_impl::min_size(__args...)),
             std::forward<_Args>(__args)...);
}

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

}  // namespace workspace

#endif
#line 17 "Library\\lib\\utils"
// #include "src/utils/py-like/zip.hpp"
// #include "src/utils/rand/rng.hpp"
#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 21 "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 10 "Library\\src\\utils\\io\\input.hpp"

namespace workspace {

template <class _Tp = int_least64_t, bool _Is_class = std::is_class<_Tp>::value>
class input {
  _Tp __value;

  template <class... _Args> struct is_convertible : std::false_type {};
  template <class _Arg>
  struct is_convertible<_Arg> : std::is_convertible<_Arg, _Tp> {};

 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 (not is_convertible<_Args...>::value) cin >> __value;
  }
};

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

 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 (not is_convertible<_Args...>::value) cin >> *this;
  }

  template <class _E>
  input(std::initializer_list<_E> __l) noexcept : _Tp(__l) {}
};

// Integrality.
template <class _Tp>
struct is_integral_ext<input<_Tp>> : is_integral_ext<_Tp> {};

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

/**
 * @file print.hpp
 * @brief Print
 */

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

namespace workspace {

/**
 * @brief Print
 * @tparam _Sep
 * @tparam _End
 */
template <char _Sep = ' ', char _End = '\n', class _Tp, class... _Args>
void print(_Tp &&__x, _Args &&...__args) noexcept {
  if _CXX17_CONSTEXPR (sizeof...(_Args))
    cout << __x << _Sep, print(std::forward<_Args>(__args)...);
  else
    cout << __x << _End;
}

void flush() noexcept { cout << std::flush; }

}  // namespace workspace
#line 13 "other-workspace\\tmp.cpp"

signed main() {
  using namespace workspace;

  io_setup(15);

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

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

  return case_info.iterate();
}

namespace workspace {

void main() {
  // start here!

  input N, M;
  input<vector<i64>> V(N);

  vector<i64> zeta(1 << M);
  for (auto &&v : V) {
    auto contained = 0;
    for (auto i : range(M)) {
      if (input<char>() == 'o') {
        contained ^= 1 << i;
      }
    }
    zeta[contained] += v;
  }

  for (auto i : range(M)) {
    for (auto s : range(1 << M)) {
      if (s >> i & 1) {
        zeta[s] += zeta[s ^ 1 << i];
      }
    }
  }

  vector<i64> dp(1 << M, -10000000000000);
  dp.back() = 0;

  for (auto s : rrange(dp.size())) {
    for (auto i : range(M)) {
      if (s >> i & 1) {
        auto a = zeta[s] - zeta[s ^ 1 << i];
        chgr(dp[s ^ 1 << i], dp[s] + a * a);
      }
    }
  }

  print(dp.front());
}

}  // namespace workspace

/**
 * @note
 */