#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 1 "Library\\lib\\utils"
// #include "src/utils/cached.hpp"
// #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\\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 5 "Library\\lib\\utils"
// #include "src/utils/hash.hpp"
// #include "src/utils/io/istream.hpp"
// #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 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\\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"
// #include "src/utils/py-like/enumerate.hpp"
#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 16 "Library\\lib\\utils"
// #include "src/utils/py-like/reversed.hpp"
// #include "src/utils/py-like/zip.hpp"
// #include "src/utils/rand/rng.hpp"
// #include "src/utils/rand/shuffle.hpp"
// #include "src/utils/round_div.hpp"
// #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 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 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 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 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 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();
}

#line 2 "Library\\src\\graph\\directed\\flow\\Dinic.hpp"

/**
 * @file Dinic.hpp
 * @brief Dinic's Algorithm
 */

#line 9 "Library\\src\\graph\\directed\\flow\\Dinic.hpp"

#line 2 "Library\\src\\graph\\directed\\flow\\base.hpp"

/**
 * @file base.hpp
 * @brief Flow Graph
 */

#line 12 "Library\\src\\graph\\directed\\flow\\base.hpp"

namespace workspace {

template <class _Cap, class _Cost = void> class flow_graph {
 protected:
  class adjacency_impl;

 public:
  using container_type = std::vector<adjacency_impl>;
  using size_type = typename container_type::size_type;

  class unweighted_edge {
   public:
    size_type tail;  // Source
    size_type head;  // Destination
    _Cap capacity;   // Capacity
    _Cap flow;       // Flow

    unweighted_edge(size_type __s, size_type __d, const _Cap &__u = 1)
        : tail(__s), head(__d), capacity(__u), flow(0) {
      assert(!(capacity < static_cast<_Cap>(0))),
          assert(!(flow < static_cast<_Cap>(0)));
    }

    // tail, head, capacity, flow
    template <class _Os>
    friend _Os &operator<<(_Os &__os, const unweighted_edge &__e) {
      return __os << __e.tail << ' ' << __e.head << ' ' << __e.capacity << ' '
                  << __e.flow;
    }

   protected:
    unweighted_edge() = default;

    unweighted_edge(size_type __s, size_type __d, const _Cap &__u,
                    const _Cap &__f)
        : tail(__s), head(__d), capacity(__u), flow(__f) {}

    unweighted_edge make_rev() const { return {head, tail, flow, capacity}; }
  };

  class weighted_edge : public unweighted_edge {
   public:
    _Cost cost;  // _Cost

    weighted_edge(const unweighted_edge &__e, const _Cost &__c = 0)
        : unweighted_edge(__e), cost(__c) {}

    weighted_edge(size_type __s, size_type __d, const _Cap &__u = 1,
                  const _Cost &__c = 0)
        : unweighted_edge(__s, __d, __u), cost(__c) {}

    // tail, head, capacity, flow, cost
    template <class _Os>
    friend _Os &operator<<(_Os &__os, const weighted_edge &__e) {
      return __os << static_cast<unweighted_edge>(__e) << ' ' << __e.cost;
    }

   protected:
    weighted_edge() = default;

    weighted_edge make_rev() const {
      return {unweighted_edge::make_rev(), -cost};
    }
  };

  using edge = std::conditional_t<std::is_void<_Cost>::value, unweighted_edge,
                                  weighted_edge>;

 protected:
  struct edge_impl : edge {
    bool aux = false;
    edge_impl *rev = nullptr;

    edge_impl() = default;

    edge_impl(const edge &__e) : edge(__e) {}
    edge_impl(edge &&__e) : edge(__e) {}

    void push(_Cap __f) {
      edge::capacity -= __f;
      edge::flow += __f;
      if (rev) {
        rev->capacity += __f;
        rev->flow -= __f;
      }
    }

    edge_impl make_rev() {
      edge_impl __e = edge::make_rev();
      __e.aux = true;
      __e.rev = this;
      return __e;
    }
  };

 public:
  class adjacency {
   public:
    using value_type = edge;
    using reference = edge &;
    using const_reference = edge const &;
    using pointer = edge *;
    using const_pointer = const edge *;

    class iterator {
      edge_impl *__p;

     public:
      iterator(edge_impl *__p = nullptr) : __p(__p) {}

      bool operator!=(const iterator &__x) const { return __p != __x.__p; }

      bool operator==(const iterator &__x) const { return __p == __x.__p; }

      iterator &operator++() {
        do ++__p;
        while (__p->rev && __p->aux);
        return *this;
      }

      iterator operator++(int) {
        auto __cp = *this;
        do ++__p;
        while (__p->rev && __p->aux);
        return __cp;
      }

      iterator &operator--() {
        do --__p;
        while (__p->aux);
        return *this;
      }

      iterator operator--(int) {
        auto __cp = *this;
        do --__p;
        while (__p->aux);
        return __cp;
      }

      pointer operator->() const { return __p; }

      reference operator*() const { return *__p; }
    };

    class const_iterator {
      const edge_impl *__p;

     public:
      const_iterator(const edge_impl *__p = nullptr) : __p(__p) {}

      bool operator!=(const const_iterator &__x) const {
        return __p != __x.__p;
      }

      bool operator==(const const_iterator &__x) const {
        return __p == __x.__p;
      }

      const_iterator &operator++() {
        do ++__p;
        while (__p->rev && __p->aux);
        return *this;
      }

      const_iterator operator++(int) {
        auto __cp = *this;
        do ++__p;
        while (__p->rev && __p->aux);
        return __cp;
      }

      const_iterator &operator--() {
        do --__p;
        while (__p->aux);
        return *this;
      }

      const_iterator operator--(int) {
        auto __cp = *this;
        do --__p;
        while (__p->aux);
        return __cp;
      }

      const_pointer operator->() const { return __p; }

      const_reference operator*() const { return *__p; }
    };

    adjacency()
        : first(new edge_impl[2]), last(first + 1), __s(first), __t(first) {}

    ~adjacency() { delete[] first; }

    const_reference operator[](size_type __i) const {
      assert(__i < size());
      return *(first + __i);
    }

    size_type size() const { return __t - first; }

    auto begin() { return iterator{__s}; }
    auto begin() const { return const_iterator{__s}; }

    auto end() { return iterator{__t}; }
    auto end() const { return const_iterator{__t}; }

    /**
     * @brief Construct a new adjacency object
     *
     * @param __x Rvalue reference to another object
     */
    adjacency(adjacency &&__x) : first(nullptr) { operator=(std::move(__x)); }

    /**
     * @brief Assignment operator.
     *
     * @param __x Rvalue reference to another object
     * @return Reference to this object.
     */
    adjacency &operator=(adjacency &&__x) {
      delete[] first;
      first = __x.first, __x.first = nullptr;
      last = __x.last, __s = __x.__s, __t = __x.__t;
      return *this;
    }

   protected:
    edge_impl *first, *last, *__s, *__t;
  };

  using value_type = adjacency;
  using reference = adjacency &;
  using const_reference = adjacency const &;

 protected:
  class adjacency_impl : public adjacency {
   public:
    using _Base = adjacency;
    using _Base::__s;
    using _Base::__t;
    using _Base::first;
    using _Base::last;

    using iterator = edge_impl *;

    iterator push(const edge_impl &__e) {
      realloc();
      *__t = __e;
      if (__s->aux) ++__s;
      return __t++;
    }

    iterator push(edge_impl &&__e) {
      realloc();
      *__t = std::move(__e);
      if (__s->aux) ++__s;
      return __t++;
    }

    iterator begin() const { return first; }

    iterator end() const { return __t; }

    void realloc() {
      if (__t == last) {
        size_type __n(last - first);
        iterator loc = new edge_impl[__n << 1 | 1];
        __s += loc - first;
        __t = loc;
        for (iterator __p{first}; __p != last; ++__p, ++__t) {
          *__t = *__p;
          if (__p->rev) __p->rev->rev = __t;
        }
        delete[] first;
        first = loc;
        last = __t + __n;
      }
    }
  };

  // Only member variable.
  container_type graph;

 public:
  /**
   * @brief Construct a new flow graph object
   *
   * @param __n Number of vertices
   */
  flow_graph(size_type __n = 0) : graph(__n) {}

  /**
   * @brief Construct a new flow graph object
   *
   * @param __x Const reference to another object
   */
  flow_graph(const flow_graph &__x) : graph(__x.size()) {
    for (auto &&__adj : __x)
      for (auto &&__e : __adj) add_edge(__e);
  }

  /**
   * @brief Construct a new flow graph object
   *
   * @param __x Rvalue reference to another object
   */
  flow_graph(flow_graph &&__x) : graph(std::move(__x.graph)) {}

  /**
   * @brief Assignment operator.
   *
   * @param __x Const reference to another object
   * @return Reference to this object.
   */
  flow_graph &operator=(const flow_graph &__x) {
    return operator=(std::move(flow_graph{__x}));
  }

  /**
   * @brief Assignment operator.
   *
   * @param __x Rvalue reference to another object
   * @return Reference to this object.
   */
  flow_graph &operator=(flow_graph &&__x) {
    graph = std::move(__x.graph);
    return *this;
  }

  /**
   * @return Whether the graph is empty.
   */
  bool empty() const { return graph.empty(); }

  /**
   * @return Number of nodes.
   */
  size_type size() const { return graph.size(); }

  /**
   * @param node Node
   * @return Referece to the adjacency list of the node.
   */
  reference operator[](size_type node) {
    assert(node < size());
    return graph[node];
  }

  /**
   * @param node Node
   * @return Const referece to the adjacency list of the node.
   */
  const_reference operator[](size_type node) const {
    assert(node < size());
    return graph[node];
  }

  class iterator : public container_type::iterator {
    using _Base = typename container_type::iterator;

   public:
    using reference = adjacency &;
    using pointer = adjacency *;

    iterator(const _Base &__i) : _Base(__i) {}

    pointer operator->() const { return _Base::operator->(); }

    reference operator*() const { return _Base::operator*(); }
  };

  class const_iterator : public container_type::const_iterator {
    using _Base = typename container_type::const_iterator;

   public:
    using const_reference = const adjacency &;
    using const_pointer = const adjacency *;

    const_iterator(const _Base &__i) : _Base(__i) {}

    const_pointer operator->() const { return _Base::operator->(); }

    const_reference operator*() const { return _Base::operator*(); }
  };

  auto begin() { return iterator{graph.begin()}; }
  auto begin() const { return const_iterator{graph.begin()}; }

  auto end() { return iterator{graph.end()}; }
  auto end() const { return const_iterator{graph.end()}; }

  /**
   * @brief Add a node to the graph.
   *
   * @return Index of the node.
   */
  size_type add_node() { return add_nodes(1).front(); }

  /**
   * @brief Add some nodes to the graph.
   *
   * @param __n Number of nodes added
   * @return List of indices of the nodes.
   */
  std::vector<size_type> add_nodes(size_type __n) noexcept {
    std::vector<size_type> __nodes(__n);
    std::iota(__nodes.begin(), __nodes.end(), graph.size());
    graph.resize(graph.size() + __n);
    return __nodes;
  }

  /**
   * @brief Add a directed edge to the graph.
   *
   * @return Reference to the edge.
   */
  template <class... _Args>
  typename std::enable_if<std::is_constructible<edge, _Args...>::value,
                          edge &>::type
  add_edge(_Args &&...__args) {
    edge_impl __e = edge(std::forward<_Args>(__args)...);
    assert(__e.tail < size()), assert(__e.head < size());
    edge_impl *__p = graph[__e.tail].push(std::move(__e));
    // Careful with a self loop.
    if (__p->tail != __p->head)
      __p->rev = graph[__p->head].push(__p->make_rev());
    return *__p;
  }

  /**
   * @brief Add an undirected edge to the graph. Its cost must be non-negative.
   *
   * @return Reference to the edge.
   */
  template <class... _Args> edge &add_undirected_edge(_Args &&...__args) {
    edge_impl __e = edge(std::forward<_Args>(__args)...);
    assert(__e.tail < size()), assert(__e.head < size());
    __e.flow += __e.capacity;
    edge_impl *__p = graph[__e.tail].push(std::move(__e));
    // Careful with a self loop.
    if (__p->tail != __p->head) {
      edge_impl __r = __p->make_rev();
      __r.aux = false;
      __p->rev = graph[__p->head].push(std::move(__r));
    }
    return *__p;
  }

  template <class _Os>
  friend _Os &operator<<(_Os &__os, flow_graph const &__g) {
    for (const auto &__adj : __g)
      for (const auto &__e : __adj) __os << __e << "\n";
    return __os;
  }
};

}  // namespace workspace
#line 11 "Library\\src\\graph\\directed\\flow\\Dinic.hpp"

namespace workspace {

/**
 * @brief Compute the maximum flow.
 * @tparam _Cap Capacity type
 */
template <class _Cap> class Dinic : public flow_graph<_Cap> {
  using _Base = flow_graph<_Cap>;

 public:
  using _Base::_Base;
  using typename _Base::size_type;

 protected:
  constexpr static size_type nil = -1;

  std::vector<size_type> __level;
  std::vector<typename _Base::container_type::value_type::iterator> __iter;

  _Cap dfs(size_type __src, size_type __dst, _Cap __limit) noexcept {
    if (__src == __dst) return __limit;

    _Cap __flow(0);

    for (auto &__e{__iter[__dst]}; __e != _Base::graph[__dst].end(); ++__e)
      if (static_cast<_Cap>(0) < __e->flow &&
          __level[__e->head] < __level[__dst])
        if (_Cap achv = dfs(__src, __e->head, std::min(__limit, __e->flow));
            static_cast<_Cap>(0) < achv) {
          __e->push(-achv);
          __flow += achv, __limit -= achv;
          if (__limit == static_cast<_Cap>(0)) break;
        }

    return __flow;
  }

 public:
  /**
   * @brief Run Dinic's algorithm.
   * @param __src Source
   * @param __dst Destination
   * @return Maximum flow.
   */
  _Cap run(size_type __src, size_type __dst) noexcept {
    return run(__src, __dst, std::numeric_limits<_Cap>::max());
  }

  /**
   * @brief Run Dinic's algorithm.
   * @param __src Source
   * @param __dst Destination
   * @param __limit Flow limit
   * @return Maximum flow.
   */
  _Cap run(size_type __src, size_type __dst, _Cap __limit) noexcept {
    assert(__src < _Base::size()), assert(__dst < _Base::size()),
        assert(__src != __dst);

    __level.resize(_Base::size(), nil);
    __iter.resize(_Base::size());

    if (!(static_cast<_Cap>(0) < __limit)) return 0;
    _Cap __flow = 0;

    for (std::vector<size_type> __q(_Base::size());;
         std::fill(__level.begin(), __level.end(), nil)) {
      __level[__q.front() = __src] = 0;

      for (auto __ql{__q.begin()}, __qr{std::next(__ql)};
           __level[__dst] == nil && __ql != __qr; ++__ql)
        for (const auto &__e : _Base::graph[*__ql])
          if (static_cast<_Cap>(0) < __e.capacity && __level[__e.head] == nil)
            __level[ *__qr++ = __e.head] = __level[*__ql] + 1;

      if (__level[__dst] == nil) break;

      for (size_type __x{}; __x != _Base::size(); ++__x)
        __iter[__x] = _Base::graph[__x].begin();

      __flow += dfs(__src, __dst, __limit);
    }

    return __flow;
  }

  // Minimum Cut.
  // Call it after `run`.
  auto min_cut() const noexcept {
    std::vector<typename _Base::edge> __cut;

    for (size_type __x{}; __x != _Base::size(); ++__x)
      if (~__level[__x])
        for (const auto &__e : _Base::operator[](__x))
          if (!~__level[__e.head]) __cut.emplace_back(__e);

    return __cut;
  }
};

}  // namespace workspace
#line 29 "other-workspace\\tmp.cpp"

namespace workspace {

void main() {
  // start here!

  input N;
  Dinic<i32> dinic;
  vector<pair<i32, i32>> od, ev;

  for (auto i : range(N)) {
    input a;
    if (popcount((i64)a) & 1) {
      od.emplace_back(dinic.add_node(), a);
    } else {
      ev.emplace_back(dinic.add_node(), a);
    }
  }

  for (auto [u, a] : od) {
    for (auto [v, b] : ev) {
      if (popcount(a ^ b) == 1) {
        dinic.add_edge(u, v, 1);
      }
    }
  }

  auto S = dinic.add_node();
  auto T = dinic.add_node();

  for (auto &&[u, a] : od) {
    dinic.add_edge(S, u, 1);
  }

  for (auto &&[v, b] : ev) {
    dinic.add_edge(v, T, 1);
  }

  print(dinic.size() - 2 - dinic.run(S, T));
}

}  // namespace workspace

/**
 * @note
 */