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

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

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

#line 13 "Library\\lib\\alias"

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

#if __cplusplus > 201703L

#include <bit>

#else

#ifndef _GLIBCXX_BIT
#define _GLIBCXX_BIT 1

#include <limits>
#include <type_traits>

namespace std {

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

  if (__x == 0) return _Nd;

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

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

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

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

  if (__x == 0) return _Nd;

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

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

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

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

  if (__x == 0) return 0;

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

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

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

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

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

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

}  // namespace std

#endif

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

#line 4 "Library\\lib\\limits"

namespace workspace {

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

#ifdef __SIZEOF_INT128__

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

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

#endif

}  // namespace workspace
#line 16 "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

namespace _alias_impl {

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

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

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

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;
};

}  // namespace _alias_impl

template <class _Tp = void, class _Compare = std::less<_Tp>>
decltype(auto) make_priority_queue(_Compare __x = _Compare()) noexcept {
  using type = std::conditional_t<
      std::is_void<_Tp>::value,
      std::decay_t<typename _alias_impl::parse_comp<_Compare>::type>, _Tp>;

  return std::priority_queue<type, std::vector<type>, _Compare>(__x);
}

template <class _Tp = void, class _Compare = std::less<_Tp>>
decltype(auto) make_set(_Compare __x = _Compare()) noexcept {
  using type = std::conditional_t<
      std::is_void<_Tp>::value,
      std::decay_t<typename _alias_impl::parse_comp<_Compare>::type>, _Tp>;

  return std::set<type, _Compare>(__x);
}

template <class _Key, class _Mapped, class _Compare = std::less<_Key>>
decltype(auto) make_map(_Compare __x = _Compare()) noexcept {
  return std::map<_Key, _Mapped, _Compare>(__x);
}

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);
}

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

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

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

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

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

namespace workspace {

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

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

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

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

namespace workspace {

using namespace std::chrono;

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

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

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

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

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

namespace workspace {

namespace internal {

struct ejection {
  bool exit = 0;
};

}  // namespace internal

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

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

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

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

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

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

namespace workspace {

void main();

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

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

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

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

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

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

namespace workspace {

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

  if constexpr (std::is_rvalue_reference<decltype(__c2)>::value)
    __c.insert(std::end(__c), std::move_iterator(std::begin(__c2)),
               std::move_iterator(std::end(__c2)));

  else
    __c.insert(std::end(__c), std::cbegin(__c2), std::cend(__c2));

  return __c;
}

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

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

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

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

namespace workspace {

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

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

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

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

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

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

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

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

#ifndef __INT128_DEFINED__

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

#endif

namespace std {

#if __INT128_DEFINED__

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

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

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

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

#endif

}  // namespace std

namespace workspace {

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

template <class... Args> struct variadic_back;

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

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

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

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

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

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

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

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

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

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

#if __INT128_DEFINED__

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

#endif

#if __cplusplus >= 201402

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

#endif

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

#if __INT128_DEFINED__

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

#endif

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

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

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

namespace workspace {

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

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

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

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

}  // namespace workspace
#line 6 "Library\\lib\\utils"
// #include "src/utils/fixed_point.hpp"
// #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"

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 {
  auto __h = std::size(__grid), __w = std::size(__grid[0]);

  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 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
 * @return
 */
template <class _Grid> decltype(auto) roll_ccw(_Grid &&__grid) noexcept {
  if 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
 * @return
 */
template <class _Grid> decltype(auto) roll_cw(_Grid &&__grid) noexcept {
  if constexpr (std::is_rvalue_reference<decltype(__grid)>::value) {
    std::reverse(std::begin(__grid), std::end(__grid));
    return transpose(std::move(__grid));
  }

  else {
    auto __t = transpose(__grid);
    for (auto &&__r : __t) std::reverse(std::begin(__r), std::end(__r));
    return __t;
  }
}

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

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

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

namespace workspace {

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

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

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

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

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

namespace workspace {

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

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

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

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

#if __cplusplus >= 201703L

#include <iterator>
#include <optional>

namespace workspace {

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

 public:
  using base_std::reverse_iterator;

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

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

}  // namespace workspace

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

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

#if __cplusplus >= 201703L

#include <tuple>
#include <vector>

namespace workspace {

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

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

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

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

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

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

}  // namespace workspace

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

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

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

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

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

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

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

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

namespace workspace {

namespace _reversed_impl {

template <class _Container> class reversed {
  _Container __cont;

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

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

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

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

}  // namespace _reversed_impl

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

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

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

#if __cplusplus >= 201703L

namespace workspace {

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

 public:
  class iterator {
    _Index current;

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

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

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

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

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

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

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

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

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

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

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

  constexpr range(_Index __first, _Index __last) noexcept
      : __first(__first), __last(__last) {}
  constexpr range(_Index __last) noexcept : __first(), __last(__last) {}

  constexpr iterator begin() const noexcept { return iterator{__first}; }
  constexpr iterator end() const noexcept { return iterator{__last}; }

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

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

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

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

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

}  // namespace workspace

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

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

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

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

#if __cplusplus >= 201703L

namespace workspace {

namespace internal {

template <class> struct zipped_iterator;

template <class...> struct zipped_iterator_tuple;

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

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

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

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

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

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

   protected:
    value_type current;

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  constexpr zipped_iterator(zipped_iterator const &__t) = default;

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

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

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

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

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

}  // namespace internal

}  // namespace workspace

namespace std {

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

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

}  // namespace std

namespace workspace {

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

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

}  // namespace workspace

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

#if __cplusplus >= 201703L

namespace workspace {

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 2 "Library\\src\\utils\\rand\\rng.hpp"

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

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

namespace workspace {

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

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

  _Engine __engine;

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

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

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

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

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

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

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

namespace workspace {

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

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

#line 4 "Library\\src\\utils\\rand\\tree.hpp"

#line 6 "Library\\src\\utils\\rand\\tree.hpp"

namespace workspace {

auto random_tree(std::size_t __n) {
  std::vector<std::pair<std::size_t, std::size_t>> __edges;
  random_number_generator rng(std::size_t(0), __n);
  for (std::size_t __i = 1; __i != __n; ++__i)
    __edges.emplace_back(__i + 1, rng() % __i + 1);
  return __edges;
}

}  // 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 11 "other-workspace\\b.cc"

signed main() {
  using namespace workspace;

  io_setup(15);

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

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

  return case_info.iterate();
}

#line 2 "Library\\src\\data_structure\\segment_tree\\basic.hpp"

/**
 * @file basic.hpp
 * @brief Segment Tree
 */

#line 10 "Library\\src\\data_structure\\segment_tree\\basic.hpp"

#if __cplusplus >= 201703L
#include <optional>
#endif

#line 2 "Library\\src\\algebra\\system\\monoid.hpp"

/*
 * @file monoid.hpp
 * @brief Monoid
 */

#line 9 "Library\\src\\algebra\\system\\monoid.hpp"

namespace workspace {

template <class T, class E = T> struct min_monoid {
  using value_type = T;
  static T min, max;
  T value;
  min_monoid() : value(max) {}
  min_monoid(const T &value) : value(value) {}
  operator T() const { return value; }
  min_monoid operator+(const min_monoid &rhs) const {
    return value < rhs.value ? *this : rhs;
  }
  min_monoid operator*(const E &rhs) const;
};

template <class T, class E>
T min_monoid<T, E>::min = std::numeric_limits<T>::min() / 2;
template <class T, class E>
T min_monoid<T, E>::max = std::numeric_limits<T>::max() / 2;

template <class T, class E = T> struct max_monoid : min_monoid<T, E> {
  using base = min_monoid<T, E>;
  using base::min_monoid;
  max_monoid() : base(base::min) {}
  max_monoid operator+(const max_monoid &rhs) const {
    return !(base::value < rhs.value) ? *this : rhs;
  }
  max_monoid operator*(const E &rhs) const;
};

}  // namespace workspace
#line 2 "Library\\src\\algebra\\system\\operation.hpp"

/**
 * @file operation.hpp
 * @brief Operation Traits
 */

#line 9 "Library\\src\\algebra\\system\\operation.hpp"

namespace workspace {

// Unary `+`
template <class _Tp>
using require_unary_plus = std::enable_if_t<
    std::is_convertible<decltype(+std::declval<const _Tp &>()), _Tp>::value>;

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

template <class _Tp>
struct has_unary_plus<_Tp, require_unary_plus<_Tp>> : std::true_type {};

// Unary `-`
template <class _Tp>
using require_unary_minus = std::enable_if_t<
    std::is_convertible<decltype(-std::declval<const _Tp &>()), _Tp>::value>;

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

template <class _Tp>
struct has_unary_minus<_Tp, require_unary_minus<_Tp>> : std::true_type {};

// Binary `+`
template <class _Tp1, class _Tp2 = _Tp1>
using require_binary_plus =
    std::enable_if_t<std::is_convertible<decltype(std::declval<const _Tp1 &>() +
                                                  std::declval<const _Tp2 &>()),
                                         _Tp1>::value>;

template <class _Tp1, class _Tp2 = _Tp1, class = void>
struct has_binary_plus : std::false_type {};

template <class _Tp1, class _Tp2>
struct has_binary_plus<_Tp1, _Tp2, require_binary_plus<_Tp1, _Tp2>>
    : std::true_type {};

// Binary `-`
template <class _Tp1, class _Tp2 = _Tp1>
using require_binary_minus = std::__void_t<decltype(
    std::declval<const _Tp1 &>() - std::declval<const _Tp2 &>())>;

template <class _Tp1, class _Tp2 = _Tp1, class = void>
struct has_binary_minus : std::false_type {};

template <class _Tp1, class _Tp2>
struct has_binary_minus<_Tp1, _Tp2, require_binary_minus<_Tp1, _Tp2>>
    : std::true_type {};

// Binary `*`
template <class _Tp1, class _Tp2 = _Tp1>
using require_binary_multiplies =
    std::enable_if_t<std::is_convertible<decltype(std::declval<const _Tp1 &>() *
                                                  std::declval<const _Tp2 &>()),
                                         _Tp1>::value>;

template <class _Tp1, class _Tp2 = _Tp1, class = void>
struct has_binary_multiplies : std::false_type {};

template <class _Tp1, class _Tp2>
struct has_binary_multiplies<_Tp1, _Tp2, require_binary_multiplies<_Tp1, _Tp2>>
    : std::true_type {};

// Binary `/`
template <class _Tp1, class _Tp2 = _Tp1>
using require_binary_divides =
    std::enable_if_t<std::is_convertible<decltype(std::declval<const _Tp1 &>() /
                                                  std::declval<const _Tp2 &>()),
                                         _Tp1>::value>;

template <class _Tp1, class _Tp2 = _Tp1, class = void>
struct has_binary_divides : std::false_type {};

template <class _Tp1, class _Tp2>
struct has_binary_divides<_Tp1, _Tp2, require_binary_divides<_Tp1, _Tp2>>
    : std::true_type {};

// Binary `%`
template <class _Tp1, class _Tp2 = _Tp1>
using require_binary_modulus =
    std::enable_if_t<std::is_convertible<decltype(std::declval<const _Tp1 &>() %
                                                  std::declval<const _Tp2 &>()),
                                         _Tp1>::value>;

template <class _Tp1, class _Tp2 = _Tp1, class = void>
struct has_binary_modulus : std::false_type {};

template <class _Tp1, class _Tp2>
struct has_binary_modulus<_Tp1, _Tp2, require_binary_modulus<_Tp1, _Tp2>>
    : std::true_type {};

}  // namespace workspace
#line 17 "Library\\src\\data_structure\\segment_tree\\basic.hpp"

namespace workspace {

/**
 * @tparam _Monoid `operator+`, `operator=`
 * @tparam Container_tmpl `operator[]`, `size_type`
 */
template <class _Monoid, class _Endomorphism = void,
          template <class...> class Container_tmpl = std::vector>
class segment_tree {
  static_assert(has_binary_plus<_Monoid>::value,
                "\'_Monoid\' has no proper binary \'operator+\'.");

  constexpr static bool __support_lazy = !std::is_void<_Endomorphism>::value;

#if __cplusplus < 201703L
  struct node_base {
    node_base() = default;
    node_base(_Monoid const &__x) : __v(__x) {}
    operator bool() const { return __f; }
    void operator=(_Monoid const &__x) {
      __v = __x;
      __f = true;
    }
    _Monoid &operator*() { return __v; }
    _Monoid const &operator*() const { return __v; }
    void reset() { __f = false; }

   private:
    _Monoid __v{};
    bool __f{true};
  };
#else
  struct node_base : std::optional<_Monoid> {
    using std::optional<_Monoid>::operator=;
    node_base() : std::optional<_Monoid>(_Monoid{}) {}
  };
#endif

  struct node_lazy : node_base {
    using node_base::operator=;
    std::optional<_Endomorphism> __z;
  };

  using node =
      typename std::conditional<__support_lazy, node_lazy, node_base>::type;

  using container_type = Container_tmpl<node>;

 public:
  using size_type = typename container_type::size_type;
  using difference_type = typename container_type::difference_type;

  class iterator {
    segment_tree *__p;
    size_type __i;

   public:
    using difference_type = segment_tree::difference_type;
    using value_type = _Monoid;
    using reference = _Monoid &;
    using pointer = iterator;
    using iterator_category = std::random_access_iterator_tag;

    /**
     * @brief Construct a new iterator object
     *
     */
    iterator() = default;

    /**
     * @brief Construct a new iterator object
     *
     * @param __p Pointer to a segment tree object
     * @param __i Index
     */
    iterator(segment_tree *__p, size_type __i) : __p(__p), __i(__i) {}

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

    bool operator<(iterator const &rhs) const { return __i < rhs.__i; }
    bool operator>(iterator const &rhs) const { return __i > rhs.__i; }
    bool operator<=(iterator const &rhs) const { return __i <= rhs.__i; }
    bool operator>=(iterator const &rhs) const { return __i >= rhs.__i; }

    iterator &operator++() { return ++__i, *this; }
    iterator &operator--() { return --__i, *this; }

    difference_type operator-(iterator const &rhs) const {
      return __i - rhs.__i;
    }

    /**
     * @brief
     *
     * @return reference
     */
    reference operator*() const { return __p->operator[](__i); }
  };

  using value_type = typename iterator::value_type;
  using reference = typename iterator::reference;

  iterator begin() { return {this, 0}; }
  iterator end() { return {this, size_orig}; }

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

 protected:
  size_type size_orig, height, size_ext;
  container_type data;

  node &pull(size_type __i) noexcept {
    if (!data[__i]) data[__i] = *pull(__i << 1) + *pull(__i << 1 | 1);
    return data[__i];
  }

  void push(size_type __i) {
    if (auto &__lz = data[__i].__z) {
      apply(data[__i << 1], *__lz);
      apply(data[__i << 1 | 1], *__lz);
      __lz.reset();
    }
  }

  void sync(size_type __i) {
    if (!data[__i])
      data[__i] = *pull(__i << 1) + *pull(__i << 1 | 1);
    else if (data[__i].__z) {
      apply(data[__i << 1], *data[__i].__z);
      apply(data[__i << 1 | 1], *data[__i].__z);
      data[__i].__z.reset();
    }
  }

  template <class _End = _Endomorphism>
  void apply(node &__nd, _End const &endo) {
    *__nd = *__nd * endo;
    __nd.__z = __nd.__z ? *__nd.__z * endo : endo;
  }

  // template <class _End = _Endomorphism>
  // void apply_top(size_t __i, _End const &endo) {
  //   auto &__nd = pull(__i);
  //   *__nd = *__nd * endo;
  //   __nd.__z = __nd.__z ? *__nd.__z * endo : endo;
  // }

  template <class Pred>
  constexpr decltype(std::declval<Pred>()(_Monoid{})) pass_args(
      Pred pred, _Monoid const &_1, [[maybe_unused]] size_type _2) {
    return pred(_1);
  }

  template <class Pred>
  constexpr decltype(std::declval<Pred>()(_Monoid{}, size_type{})) pass_args(
      Pred pred, _Monoid const &_1, size_type _2) {
    return pred(_1, _2);
  }

  template <class Pred>
  size_type left_partition_subtree(size_type __i, _Monoid mono, size_type step,
                                   Pred pred) {
    assert(__i);
    while (__i < size_ext) {
      if constexpr (__support_lazy) push(__i);
      const _Monoid tmp = *pull((__i <<= 1) | 1) + mono;
      if (pass_args(pred, tmp, ((__i | 1) << --step) ^ size_ext))
        mono = tmp;
      else
        ++__i;
    }
    return ++__i -= size_ext;
  }

  template <class Pred>
  size_type right_partition_subtree(size_type __i, _Monoid mono, size_type step,
                                    Pred pred) {
    assert(__i);
    while (__i < size_ext) {
      if constexpr (__support_lazy) push(__i);
      const _Monoid tmp = mono + *pull(__i <<= 1);
      if (pass_args(pred, tmp, ((__i | 1) << --step) ^ size_ext))
        ++__i, mono = tmp;
    }
    return (__i -= size_ext) < size_orig ? __i : size_orig;
  }

 public:
  /**
   * @brief Construct a new segment tree object.
   *
   * @param __n Number of elements.
   */
  segment_tree(size_type __n = 0)
      : size_orig{__n},
        height(__n > 1 ? 64 - __builtin_clzll(__n - 1) : 0),
        size_ext{size_type{1} << height} {
    if constexpr (std::is_constructible<container_type, size_t>::value)
      data = container_type(size_ext << 1);
    data[0].reset();
  }

  /**
   * @brief Construct a new segment tree object.
   *
   * @param __n Number of elements.
   * @param __x
   */
  segment_tree(size_type __n, const value_type &__x) : segment_tree(__n) {
    for (auto __i = begin(); __i != end(); ++__i) *__i = __x;
  }

  /**
   * @brief Construct a new segment tree object.
   *
   * @param __n Number of elements.
   * @param __x
   */
  template <class _Tp>
  segment_tree(size_type __n, _Tp &&__x) : segment_tree(__n) {
    for (auto __i = begin(); __i != end(); ++__i) *__i = __x;
  }

  /**
   * @brief Construct a new segment tree object.
   *
   * @param __first
   * @param __last
   */
  template <class _Iterator, typename = std::_RequireInputIter<_Iterator>>
  segment_tree(_Iterator __first, _Iterator __last)
      : segment_tree(std::distance(__first, __last)) {
    for (auto __i = begin(); __first != __last; ++__i, ++__first)
      *__i = *__first;
  }

  /**
   * @brief Conversion to container_type.
   */
  operator Container_tmpl<value_type>() const {
    Container_tmpl<value_type> __c(size());
    for (size_type __i = 0; __i != size(); ++__i)
      __c[__i] = *data[__i | size_ext];
    return __c;
  }

  /**
   * @return Number of elements.
   */
  size_type size() const { return size_orig; }

  /**
   * @return Whether %segment_tree is empty.
   */
  bool empty() const { return !size(); }

  /**
   * @brief Subscripting ( @c [] ) access.
   *
   * @param __i Index of the element
   * @return Reference to the element.
   */
  reference operator[](size_type __i) {
    assert(__i < size_orig);
    reference __ref = *data[__i |= size_ext];
    if constexpr (__support_lazy) {
      for (size_t __h{height}; __h; --__h) {
        push(__i >> __h);
        data[__i >> __h].reset();
      }
    } else {
      while (data[__i >>= 1]) data[__i].reset();
    }
    return __ref;
  }

  /**
   * @param first Left end, inclusive
   * @param last Right end, exclusive
   * @return Sum of elements in the interval.
   */
  value_type fold(size_type first, size_type last) {
    assert(last <= size_orig);
    if (!(first < last)) return {};
    first += size_ext, last += size_ext;
    value_type left{}, right{};
    for (size_t l = first, r = last--; l != r; l >>= 1, r >>= 1) {
      if (l & 1) left = left + *pull(l++);
      if (r & 1) right = *pull(--r) + right;
      if constexpr (__support_lazy) {
        if (data[first >>= 1].__z) left = left * *data[first].__z;
        if (data[last >>= 1].__z) right = right * *data[last].__z;
      }
    }
    if constexpr (__support_lazy) {
      while (first >>= 1, last >>= 1) {
        if (data[first].__z) left = left * *data[first].__z;
        if (data[last].__z) right = right * *data[last].__z;
      }
    }

    // if (first >= last) return _Monoid{};
    // first += size_ext, last += size_ext - 1;
    // _Monoid left{}, right{};
    // for (size_t l = first, r = last + 1; last; l >>= 1, r >>= 1) {
    //   if (l < r) {
    //     if (l & 1) left = left + data[l++];
    //     if (r & 1) right = data[--r] + right;
    //   }
    //   if (first >>= 1, last >>= 1) {
    //     left = left * lazy[first];
    //     right = right * lazy[last];
    //   }
    // }
    // return left + right;

    return left + right;
  }

  /**
   * @return The whole sum.
   */
  value_type fold() { return *pull(1); }

  template <class _End = _Endomorphism>
  void update(size_type first, size_type last, _End const &endo) {
    static_assert(__support_lazy);

    assert(last <= size_orig);
    if (!(first < last)) return;
    first += size_ext, last += size_ext;

    --last;
    for (auto i = height; i; --i) push(first >> i), push(last >> i);
    ++last;

    for (auto l = first, r = last; l < r; l >>= 1, r >>= 1) {
      if (l & 1) apply(pull(l++), endo);
      if (r & 1) apply(pull(--r), endo);
    }

    for (first >>= __builtin_ffs(first); data[first]; first >>= 1)
      data[first].reset();

    for (last >>= __builtin_ffs(last); data[last]; last >>= 1)
      data[last].reset();
  }

  /**
   * @brief Binary search for the partition point.
   * @param right Right fixed end of the interval, exclusive
   * @param pred Predicate in the form of either 'bool(_Monoid)' or
   * 'bool(_Monoid, size_type)'
   * @return Left end of the extremal interval satisfying the condition,
   * inclusive.
   */
  template <class Pred> size_type left_partition(size_type right, Pred pred) {
    assert(right <= size_orig);
    right += size_ext;

    if constexpr (__support_lazy)
      for (size_t i{height}; i; --i) push(right >> i);

    _Monoid mono{};
    for (size_type left{size_ext}, step{}; left != right;
         left >>= 1, right >>= 1, ++step) {
      if ((left & 1) != (right & 1)) {
        _Monoid tmp = *pull(--right) + mono;
        if (!pass_args(pred, tmp, (right << step) ^ size_ext))
          return left_partition_subtree(right, mono, step, pred);
        mono = tmp;
      }
    }

    return 0;
  }

  /**
   * @brief Binary search for the partition point.
   * @param left Left fixed end of the interval, inclusive
   * @param pred Predicate in the form of either 'bool(_Monoid)' or
   * 'bool(_Monoid, size_type)'
   * @return Right end of the extremal interval satisfying the condition,
   * exclusive.
   */
  template <class Pred> size_type right_partition(size_type left, Pred pred) {
    assert(left <= size_orig);
    left += size_ext;

    if constexpr (__support_lazy)
      for (size_t i{height}; i; --i) push(left >> i);

    _Monoid mono{};
    for (size_type right{size_ext << 1}, step{}; left != right;
         left >>= 1, right >>= 1, ++step) {
      if ((left & 1) != (right & 1)) {
        _Monoid tmp = mono + *pull(left);
        if (!pass_args(pred, tmp, ((left + 1) << step) ^ size_ext))
          return right_partition_subtree(left, mono, step, pred);
        mono = tmp;
        ++left;
      }
    }

    return size_orig;
  }
};

template <class _Iterator, typename = std::_RequireInputIter<_Iterator>>
segment_tree(_Iterator, _Iterator)
    -> segment_tree<typename std::iterator_traits<_Iterator>::value_type>;

template <class _Tp, typename = require_binary_plus<_Tp>>
segment_tree(typename segment_tree<_Tp>::size_type, _Tp &&)
    -> segment_tree<_Tp>;

}  // namespace workspace
#line 27 "other-workspace\\b.cc"

namespace workspace {

void main() {
  // start here!

  constexpr auto inf = INT64_MAX / 2;

  int n, K, x, y;
  cin >> n >> K >> x >> y;
  x %= K, y %= K;

  struct state {
    i64 val = -inf;
    state operator+(state const &x) const { return val < x.val ? x : *this; }
  };

  vector rows(K, segment_tree<state>(K)), cols(rows);
  rows[x][y] = cols[y][x] = {0};

  auto op = [&](int i, int j, auto &&f) -> auto {
    auto &r = rows[i], &c = cols[j];
    f(r[j].val);
    c[i] = r[j];
  };

  vector<i64> rm(K, -inf), cm(K, -inf);

  for (auto i : range(n)) {
    cin >> i;
    i %= K;

    for (auto j : range(K)) {
      op(j, (K * 2 - i - j) % K, [](auto &v) { ++v; });
    }

    for (auto &&[a, b] : zip(rm, rows)) {
      a = b.fold().val;
    }
    for (auto &&[a, b] : zip(cm, cols)) {
      a = b.fold().val;
    }

    for (auto j : range(K)) {
      op(j, i, [&](auto &x) { chgr(x, rm[j]); });
      op(i, j, [&](auto &x) { chgr(x, cm[j]); });
    }
  }

  i64 ans = -inf;
  for (auto &&x : rows) {
    chgr(ans, x.fold().val);
  }
  cout << ans << "\n";
}

}  // namespace workspace