#line 1 "atcoder-workspace/177.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 1 "Library/lib/bit"
#if __cplusplus > 201703L

#include <bit>

#else

#ifndef _GLIBCXX_BIT
#define _GLIBCXX_BIT 1

#include <limits>
#include <type_traits>

namespace std {

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

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

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

  if (__x == 0) return _Nd;

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

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

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

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

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

  if (__x == 0) return _Nd;

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

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

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

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

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

  if (__x == 0) return 0;

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

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

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

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

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

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

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

}  // namespace std

#endif

#endif
#line 15 "Library/lib/alias"

namespace workspace {

constexpr char eol = '\n';

using namespace std;

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

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

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

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

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

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

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

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

#line 1 "Library/lib/bit"
#if __cplusplus > 201703L

#include <bit>

#else

#ifndef _GLIBCXX_BIT
#define _GLIBCXX_BIT 1

#include <limits>
#include <type_traits>

namespace std {

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

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

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

  if (__x == 0) return _Nd;

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

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

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

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

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

  if (__x == 0) return _Nd;

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

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

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

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

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

  if (__x == 0) return 0;

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

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

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

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

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

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

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

}  // namespace std

#endif

#endif
#line 9 "Library/lib/cxx20"

#if __cplusplus <= 201703L

#include <algorithm>
#include <vector>

namespace std {

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

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

}  // namespace std

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

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

#ifdef ONLINE_JUDGE

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

#endif
#line 2 "Library/src/opt/binary_search.hpp"

/*
 * @file binary_search.hpp
 * @brief Binary Search
 */

#line 12 "Library/src/opt/binary_search.hpp"

namespace workspace {

/*
 * @fn binary_search
 * @brief binary search on a discrete range.
 * @param ok pred(ok) is true
 * @param ng pred(ng) is false
 * @param pred the predicate
 * @return the closest point to (ng) where pred is true
 */
template <class Iter, class Pred>
typename std::enable_if<
    std::is_convertible<decltype(std::declval<Pred>()(std::declval<Iter>())),
                        bool>::value,
    Iter>::type
binary_search(Iter ok, Iter ng, Pred pred) {
  assert(ok != ng);
  typename std::make_signed<decltype(ng - ok)>::type dist(ng - ok);
  while (1 < dist || dist < -1) {
    const Iter mid(ok + dist / 2);
    if (pred(mid))
      ok = mid, dist -= dist / 2;
    else
      ng = mid, dist /= 2;
  }
  return ok;
}

/*
 * @fn binary_search
 * @brief binary search on the real number line.
 * @param ok pred(ok) is true
 * @param ng pred(ng) is false
 * @param eps the error tolerance
 * @param pred the predicate
 * @return the boundary point
 */
template <class Real, class Pred>
typename std::enable_if<
    std::is_convertible<decltype(std::declval<Pred>()(std::declval<Real>())),
                        bool>::value,
    Real>::type
binary_search(Real ok, Real ng, const Real eps, Pred pred) {
  assert(ok != ng);
  for (auto loops = 0; loops != std::numeric_limits<Real>::digits &&
                       (ok + eps < ng || ng + eps < ok);
       ++loops) {
    const Real mid{(ok + ng) / 2};
    (pred(mid) ? ok : ng) = mid;
  }
  return ok;
}

/*
 * @fn parallel_binary_search
 * @brief parallel binary search on discrete ranges.
 * @param ends a vector of pairs; pred(first) is true, pred(second) is false
 * @param pred the predicate
 * @return the closest points to (second) where pred is true
 */
template <class Array,
          class Iter = typename std::decay<
              decltype(std::get<0>(std::declval<Array>()[0]))>::type,
          class Pred>
typename std::enable_if<
    std::is_convertible<
        decltype(std::declval<Pred>()(std::declval<std::vector<Iter>>())[0]),
        bool>::value,
    std::vector<Iter>>::type
parallel_binary_search(Array ends, Pred pred) {
  std::vector<Iter> mids(std::size(ends));
  for (;;) {
    bool all_found = true;
    for (size_t i{}; i != std::size(ends); ++i) {
      const Iter &ok = std::get<0>(ends[i]);
      const Iter &ng = std::get<1>(ends[i]);
      const Iter mid(
          ok + typename std::make_signed<decltype(ng - ok)>::type(ng - ok) / 2);
      if (mids[i] != mid) {
        all_found = false;
        mids[i] = mid;
      }
    }
    if (all_found) break;
    const auto res = pred(mids);
    for (size_t i{}; i != std::size(ends); ++i) {
      (res[i] ? std::get<0>(ends[i]) : std::get<1>(ends[i])) = mids[i];
    }
  }
  return mids;
}

/*
 * @fn parallel_binary_search
 * @brief parallel binary search on the real number line.
 * @param ends a vector of pairs; pred(first) is true, pred(second) is false
 * @param eps the error tolerance
 * @param pred the predicate
 * @return the boundary points
 */
template <class Array,
          class Real = typename std::decay<
              decltype(std::get<0>(std::declval<Array>()[0]))>::type,
          class Pred>
typename std::enable_if<
    std::is_convertible<
        decltype(std::declval<Pred>()(std::declval<std::vector<Real>>())[0]),
        bool>::value,
    std::vector<Real>>::type
parallel_binary_search(Array ends, const Real eps, Pred pred) {
  std::vector<Real> mids(std::size(ends));
  for (auto loops = 0; loops != std::numeric_limits<Real>::digits; ++loops) {
    bool all_found = true;
    for (size_t i{}; i != std::size(ends); ++i) {
      const Real ok = std::get<0>(ends[i]);
      const Real ng = std::get<1>(ends[i]);
      if (ok + eps < ng || ng + eps < ok) {
        all_found = false;
        mids[i] = (ok + ng) / 2;
      }
    }
    if (all_found) break;
    const auto res = pred(mids);
    for (size_t i{}; i != std::size(ends); ++i) {
      (res[i] ? std::get<0>(ends[i]) : std::get<1>(ends[i])) = mids[i];
    }
  }
  return mids;
}

}  // namespace workspace
#line 2 "Library/src/opt/exponential_search.hpp"

/*
 * @file exponential_search.hpp
 * @brief Exponential Search
 */

#line 9 "Library/src/opt/exponential_search.hpp"

namespace workspace {

/*
 * @fn exponential_search
 * @brief Exponential search on a discrete range.
 * @param range Range of search, exclusive
 * @param pred Predicate
 * @return Minimum non-negative integer where pred is false.
 */
template <class Index, class Pred>
typename std::enable_if<
    std::is_convertible<decltype(std::declval<Pred>()(std::declval<Index>())),
                        bool>::value,
    Index>::type
exponential_search(Index range, Pred pred) {
  Index step(1);
  while (step < range && pred(step)) step <<= 1;
  if (range < step) step = range;
  return binary_search(Index(0), step, pred);
}

/*
 * @fn exponential_search
 * @brief Exponential search on the real number line.
 * @param range Range of search
 * @param eps Error tolerance
 * @param pred Predicate
 * @return Boundary point.
 */
template <class Real, class Pred>
typename std::enable_if<
    std::is_convertible<decltype(std::declval<Pred>()(std::declval<Real>())),
                        bool>::value,
    Real>::type
exponential_search(Real range, Real const &eps, Pred pred) {
  Real step(1);
  while (step < range && pred(step)) step += step;
  if (range < step) step = range;
  return binary_search(Real(0), step, eps, pred);
}

}  // namespace workspace
#line 2 "Library/src/opt/trinary_search.hpp"

/*
 * @file trinary_search.hpp
 * @brief Trinary Search
 */

#line 9 "Library/src/opt/trinary_search.hpp"
#include <type_traits>

namespace workspace {

/*
 * @brief Trinary search on discrete range.
 * @param first Left end, inclusive
 * @param last Right end, exclusive
 * @param comp Compare function
 * @return Local minimal point.
 */
template <class Iter, class Comp>
typename std::enable_if<
    std::is_convertible<decltype(std::declval<Comp>()(std::declval<Iter>(),
                                                      std::declval<Iter>())),
                        bool>::value,
    Iter>::type
trinary_search(Iter first, Iter last, Comp comp) {
  assert(first < last);
  typename std::make_signed<decltype(last - first)>::type dist(last - first);
  while (2 < dist) {
    Iter left(first + dist / 3), right(first + dist * 2 / 3);
    if (comp(left, right))
      last = right, dist = (dist + dist) / 3;
    else
      first = left, dist -= dist / 3;
  }
  if (1 < dist && comp(first + 1, first)) ++first;
  return first;
}

/*
 * @brief Trinary search on discrete range.
 * @param first Left end, inclusive
 * @param last Right end, exclusive
 * @param func Function
 * @return Local minimal point.
 */
template <class Iter, class Func>
typename std::enable_if<
    std::is_same<decltype(std::declval<Func>()(std::declval<Iter>()), nullptr),
                 std::nullptr_t>::value,
    Iter>::type
trinary_search(Iter const &first, Iter const &last, Func func) {
  return trinary_search(first, last, [&](Iter const &__i, Iter const &__j) {
    return func(__i) < func(__j);
  });
}

/*
 * @brief Trinary search on the real number line.
 * @param first Left end
 * @param last Right end
 * @param eps Error tolerance
 * @param comp Compare function
 * @return Local minimal point.
 */
template <class Real, class Comp>
typename std::enable_if<
    std::is_convertible<decltype(std::declval<Comp>()(std::declval<Real>(),
                                                      std::declval<Real>())),
                        bool>::value,
    Real>::type
trinary_search(Real first, Real last, Real const &eps, Comp comp) {
  assert(first < last);
  while (eps < last - first) {
    Real left{(first * 2 + last) / 3}, right{(first + last * 2) / 3};
    if (comp(left, right))
      last = right;
    else
      first = left;
  }
  return first;
}

/*
 * @brief Trinary search on the real number line.
 * @param first Left end
 * @param last Right end
 * @param eps Error tolerance
 * @param func Function
 * @return Local minimal point.
 */
template <class Real, class Func>
typename std::enable_if<
    std::is_same<decltype(std::declval<Func>()(std::declval<Real>()), nullptr),
                 std::nullptr_t>::value,
    Real>::type
trinary_search(Real const &first, Real const &last, Real const &eps,
               Func func) {
  return trinary_search(
      first, last, eps,
      [&](Real const &__i, Real const &__j) { return func(__i) < func(__j); });
}

}  // 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/src/utils/cat.hpp"

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

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

namespace workspace {

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

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

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

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

namespace workspace {

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

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

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

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

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

namespace workspace {

/*
 * @class fixed_point
 * @brief Recursive calling of lambda expression.
 */
template <class lambda_type> class fixed_point {
  lambda_type func;

 public:
  /*
   * @param func 1st arg callable with the rest of args, and the return type
   * specified.
   */
  fixed_point(lambda_type &&func) : func(std::move(func)) {}

  /*
   * @brief Recursively apply *this to 1st arg of func.
   * @param args Arguments of the recursive method.
   */
  template <class... Args> auto operator()(Args &&... args) const {
    return func(*this, std::forward<Args>(args)...);
  }
};

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

/**
 * @file grid.hpp
 * @brief Grid
 * @date 2021-01-09
 */

#line 10 "Library/src/utils/grid.hpp"

namespace workspace {

template <class Grid> Grid transpose(Grid const &grid) {
  Grid __t;
  for (auto &&__r : grid) {
    auto __i = std::begin(__t);
    for (auto &&__x : __r) {
      if (__i == std::end(__t))
        __i = __t.insert(__t.end(), typename std::decay<decltype(__r)>::type{});
      __i->insert(__i->end(), __x);
      ++__i;
    }
  }
  return __t;
}

// template <class _Tp, size_t _Row, size_t _Col>
// std::array<std::array<_Tp, _Row>, _Col> transpose(_Tp (&__g)[_Row][_Col]) {}

template <class Grid> Grid roll_ccw(Grid const &grid) {
  auto __t = transpose(grid);
  std::reverse(std::begin(__t), std::end(__t));
  return __t;
}

template <class Grid> Grid roll_cw(Grid const &grid) {
  auto __t = grid;
  std::reverse(std::begin(__t), std::end(__t));
  return transpose(__t);
}

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

#line 8 "Library/src/utils/hash.hpp"

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

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

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

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

namespace std {

#if __INT128_DEFINED__

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

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

#endif

}  // namespace std

namespace workspace {

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

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

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

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

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

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

#if __INT128_DEFINED__

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

#endif

#if __cplusplus >= 201402

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

#endif

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

#if __INT128_DEFINED__

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

#endif

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

}  // namespace workspace
#line 10 "Library/src/utils/hash.hpp"
namespace workspace {
template <class T, class = void> struct hash : std::hash<T> {};
#if __cplusplus >= 201703L
template <class Unique_bits_type>
struct hash<Unique_bits_type,
            enable_if_trait_type<Unique_bits_type,
                                 std::has_unique_object_representations>> {
  size_t operator()(uint64_t x) const {
    static const uint64_t m = std::random_device{}();
    x ^= x >> 23;
    x ^= m;
    x ^= x >> 47;
    return x - (x >> 32);
  }
};
#endif
template <class Key> size_t hash_combine(const size_t &seed, const Key &key) {
  return seed ^
         (hash<Key>()(key) + 0x9e3779b9 /* + (seed << 6) + (seed >> 2) */);
}
template <class T1, class T2> struct hash<std::pair<T1, T2>> {
  size_t operator()(const std::pair<T1, T2> &pair) const {
    return hash_combine(hash<T1>()(pair.first), pair.second);
  }
};
template <class... T> class hash<std::tuple<T...>> {
  template <class Tuple, size_t index = std::tuple_size<Tuple>::value - 1>
  struct tuple_hash {
    static uint64_t apply(const Tuple &t) {
      return hash_combine(tuple_hash<Tuple, index - 1>::apply(t),
                          std::get<index>(t));
    }
  };
  template <class Tuple> struct tuple_hash<Tuple, size_t(-1)> {
    static uint64_t apply(const Tuple &t) { return 0; }
  };

 public:
  uint64_t operator()(const std::tuple<T...> &t) const {
    return tuple_hash<std::tuple<T...>>::apply(t);
  }
};
template <class hash_table> struct hash_table_wrapper : hash_table {
  using key_type = typename hash_table::key_type;
  size_t count(const key_type &key) const {
    return hash_table::find(key) != hash_table::end();
  }
  template <class... Args> auto emplace(Args &&... args) {
    return hash_table::insert(typename hash_table::value_type(args...));
  }
};
template <class Key, class Mapped = __gnu_pbds::null_type>
using cc_hash_table =
    hash_table_wrapper<__gnu_pbds::cc_hash_table<Key, Mapped, hash<Key>>>;
template <class Key, class Mapped = __gnu_pbds::null_type>
using gp_hash_table =
    hash_table_wrapper<__gnu_pbds::gp_hash_table<Key, Mapped, hash<Key>>>;
template <class Key, class Mapped>
using unordered_map = std::unordered_map<Key, Mapped, hash<Key>>;
template <class Key> using unordered_set = std::unordered_set<Key, hash<Key>>;
}  // 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 15 "Library/src/utils/io/istream.hpp"

namespace workspace {

namespace internal {

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

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

#ifdef __SIZEOF_INT128__

template <> struct istream_helper<__int128_t, std::nullptr_t> {
  istream_helper(std::istream &is, __int128_t &x) {
    std::string s;
    is >> s;
    bool negative = s.front() == '-' ? s.erase(s.begin()), true : false;
    x = 0;
    for (char e : s) x = x * 10 + e - '0';
    if (negative) x = -x;
  }
};

template <> struct istream_helper<__uint128_t, std::nullptr_t> {
  istream_helper(std::istream &is, __uint128_t &x) {
    std::string s;
    is >> s;
    bool negative = s.front() == '-' ? s.erase(s.begin()), true : false;
    x = 0;
    for (char e : s) x = x * 10 + e - '0';
    if (negative) x = -x;
  }
};

#endif  // INT128

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

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

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

}  // namespace internal

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

namespace internal {
auto *const cin_ptr = (istream *)&std::cin;
}
auto &cin = *internal::cin_ptr;

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

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

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

namespace workspace {

template <class T, class U>
std::ostream &operator<<(std::ostream &os, const std::pair<T, U> &p) {
  return os << p.first << ' ' << p.second;
}
template <class tuple_t, size_t index> struct tuple_os {
  static std::ostream &apply(std::ostream &os, const tuple_t &t) {
    tuple_os<tuple_t, index - 1>::apply(os, t);
    return os << ' ' << std::get<index>(t);
  }
};
template <class tuple_t> struct tuple_os<tuple_t, 0> {
  static std::ostream &apply(std::ostream &os, const tuple_t &t) {
    return os << std::get<0>(t);
  }
};
template <class tuple_t> struct tuple_os<tuple_t, SIZE_MAX> {
  static std::ostream &apply(std::ostream &os, const tuple_t &t) { return os; }
};

template <class... T>
std::ostream &operator<<(std::ostream &os, const std::tuple<T...> &t) {
  return tuple_os<std::tuple<T...>,
                  std::tuple_size<std::tuple<T...>>::value - 1>::apply(os, t);
}

template <class Container,
          typename = decltype(std::begin(std::declval<Container>()))>
typename std::enable_if<
    !std::is_same<typename std::decay<Container>::type, std::string>::value &&
        !std::is_same<typename std::decay<Container>::type, char *>::value,
    std::ostream &>::type
operator<<(std::ostream &os, const Container &cont) {
  bool head = true;
  for (auto &&e : cont) head ? head = 0 : (os << ' ', 0), os << e;
  return os;
}

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

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

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

namespace workspace {

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

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

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

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

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

namespace workspace {

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

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

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

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

#if __cplusplus >= 201703L

#include <iterator>
#include <optional>

namespace workspace {

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

 public:
  using base_std::reverse_iterator;

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

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

}  // namespace workspace

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

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

#if __cplusplus >= 201703L

#include <tuple>
#include <vector>

namespace workspace {

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

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

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

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

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

}  // namespace workspace

#endif
#line 14 "Library/lib/utils"
// #include "src/utils/py-like/enumerate.hpp"
#line 2 "Library/src/utils/py-like/range.hpp"

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

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

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

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

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

namespace workspace {

namespace internal {

template <class Container> class reversed {
  Container cont;

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

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

}  // namespace internal

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

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

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

#if __cplusplus >= 201703L

namespace workspace {

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

 public:
  class iterator {
    Index current;

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

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

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

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

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

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

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

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

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

}  // namespace workspace

#endif
#line 17 "Library/lib/utils"
// #include "src/utils/py-like/zip.hpp"
#line 2 "Library/src/utils/rand/rng.hpp"

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

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

namespace workspace {

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

template <typename Arithmetic>
class random_number_generator : uniform_distribution<Arithmetic> {
  using base = uniform_distribution<Arithmetic>;

  std::mt19937 engine;

 public:
  template <class... Args>
  random_number_generator(Args&&... args)
      : base(args...), engine(std::random_device{}()) {}

  auto operator()() { 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>
void shuffle(RAIter const& __first, RAIter const& __last) {
  static std::mt19937 engine(std::random_device{}());
  std::shuffle(__first, __last, engine);
}

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

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

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

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

namespace workspace {

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

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

}  // namespace workspace
#line 11 "atcoder-workspace/177.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/modular/inverse.hpp"

/*
 * @file inverse.hpp
 * @brief Inverse Table
 */

#line 9 "Library/src/modular/inverse.hpp"

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

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

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

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

namespace workspace {

namespace internal {

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

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

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

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

  static mod_type mod;

  static value_type storage;

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

 protected:
  value_type value = 0;

 public:
  constexpr modint_base() noexcept = default;

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

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

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

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

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

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

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

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

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

  // }} unary operators

  // operator+= {{

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

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

  // }} operator+=

  // operator+ {{

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

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

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

  // }} operator+

  // operator-= {{

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

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

  // }} operator-=

  // operator- {{

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

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

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

  // }} operator-

  // operator*= {{

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

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

  // }} operator*=

  // operator* {{

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

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

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

  // }} operator*

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

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

 public:
  // operator/= {{

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

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

  // }} operator/=

  // operator/ {{

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

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

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

  // }} operator/

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

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

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

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

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

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

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

}  // namespace internal

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

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

// #define modint_newtype modint_runtime<__COUNTER__>

}  // namespace workspace
#line 11 "Library/src/modular/inverse.hpp"

namespace workspace {

// Modulus must be prime.
template <class Modint> struct inverse_table {
  static_assert(std::is_same<std::nullptr_t,
                             decltype((void *)Modint::mod, nullptr)>::value);

  using value_type = Modint;

  constexpr value_type operator()(int n) const {
    constexpr int_fast64_t mod = value_type::mod;
    assert(n %= mod);
    if (n < 0) n += mod;
    if (inv.empty()) inv = {1, mod != 1};
    for (int m(inv.size()); m <= n; ++m)
      inv.emplace_back(mod / m * -inv[mod % m]);
    return inv[n];
  }

 private:
  static std::vector<value_type> inv;
};

template <class Modint> std::vector<Modint> inverse_table<Modint>::inv;

}  // namespace workspace
#line 27 "atcoder-workspace/177.cc"

namespace workspace {

using mint = modint<998244353>;

void main() {
  // start here!

  int n;
  cin >> n;
  mint p = 1;
  mint ans;
  for (auto k : range(1, n + 1)) {
    p *= 1 - mint(n + 1 - k).inv();
    ans += p;
    ans += mint(k) * (n - k) / n;
  }
  for (auto k : range(n)) {
    ans *= k + 1;
  }
  cout << ans << "\n";
}

}  // namespace workspace