#line 1 "Library/yu2.cc"
/*
 * @file template.cpp
 * @brief Template
 */

#include <bits/extc++.h>

#line 2 "Library/lib/alias"

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

#line 13 "Library/lib/alias"

namespace workspace {

constexpr char eol = '\n';

using namespace std;

using i32 = int_least32_t;
using i64 = int_least64_t;
using i128 = __int128_t;
using u32 = uint_least32_t;
using u64 = uint_least64_t;
using u128 = __uint128_t;

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>>;

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

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

#if __cplusplus <= 201703L

#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(std::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(std::remove(__cont.begin(), __cont.end(), __value),
               __cont.end());
  return __osz - __cont.size();
}

}  // namespace std

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

/*
 * @file option
 * @brief Optimize Options
 */

#ifdef ONLINE_JUDGE

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

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

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

#if __cplusplus >= 201703L

#include <cassert>
#include <cmath>
#include <vector>

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_type, class pred_type>
std::enable_if_t<
    std::is_convertible_v<std::invoke_result_t<pred_type, iter_type>, bool>,
    iter_type>
binary_search(iter_type ok, iter_type ng, pred_type pred) {
  assert(ok != ng);
  std::make_signed_t<decltype(ng - ok)> dist(ng - ok);
  while (1 < dist || dist < -1) {
    iter_type mid(ok + dist / 2);
    if (pred(mid))
      ok = mid, dist -= dist / 2;
    else
      ng = mid, dist /= 2;
  }
  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 iter_type, class pred_type>
std::enable_if_t<std::is_convertible_v<
                     std::invoke_result_t<pred_type, std::vector<iter_type>>,
                     std::vector<bool>>,
                 std::vector<iter_type>>
parallel_binary_search(std::vector<std::pair<iter_type, iter_type>> ends,
                       pred_type pred) {
  std::vector<iter_type> mids(ends.size());
  for (;;) {
    bool all_found = true;
    for (size_t i{}; i != ends.size(); ++i) {
      auto [ok, ng] = ends[i];
      iter_type mid(ok + (ng - ok) / 2);
      if (mids[i] != mid) {
        all_found = false;
        mids[i] = mid;
      }
    }
    if (all_found) break;
    auto res = pred(mids);
    for (size_t i{}; i != ends.size(); ++i) {
      (res[i] ? ends[i].first : ends[i].second) = mids[i];
    }
  }
  return mids;
}

/*
 * @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_type, class pred_type>
std::enable_if_t<
    std::is_convertible_v<std::invoke_result_t<pred_type, real_type>, bool>,
    real_type>
binary_search(real_type ok, real_type ng, const real_type eps, pred_type pred) {
  assert(ok != ng);
  for (auto loops = 0; loops != std::numeric_limits<real_type>::digits &&
                       (ok + eps < ng || ng + eps < ok);
       ++loops) {
    real_type mid{(ok + ng) / 2};
    (pred(mid) ? ok : ng) = mid;
  }
  return ok;
}

/*
 * @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 real_type, class pred_type>
std::enable_if_t<std::is_convertible_v<
                     std::invoke_result_t<pred_type, std::vector<real_type>>,
                     std::vector<bool>>,
                 std::vector<real_type>>
parallel_binary_search(std::vector<std::pair<real_type, real_type>> ends,
                       const real_type eps, pred_type pred) {
  std::vector<real_type> mids(ends.size());
  for (auto loops = 0; loops != std::numeric_limits<real_type>::digits;
       ++loops) {
    bool all_found = true;
    for (size_t i{}; i != ends.size(); ++i) {
      auto [ok, ng] = ends[i];
      if (ok + eps < ng || ng + eps < ok) {
        all_found = false;
        mids[i] = (ok + ng) / 2;
      }
    }
    if (all_found) break;
    auto res = pred(mids);
    for (size_t i{}; i != ends.size(); ++i) {
      (res[i] ? ends[i].first : ends[i].second) = mids[i];
    }
  }
  return mids;
}

}  // namespace workspace

#endif
#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/clock.hpp"

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

#line 9 "Library/src/utils/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 5 "Library/src/utils/coordinate_compression.hpp"

template <class T> class coordinate_compression {
  std::vector<T> uniquely;
  std::vector<size_t> compressed;

 public:
  coordinate_compression(const std::vector<T> &raw)
      : uniquely(raw), compressed(raw.size()) {
    std::sort(uniquely.begin(), uniquely.end());
    uniquely.erase(std::unique(uniquely.begin(), uniquely.end()),
                   uniquely.end());
    for (size_t i = 0; i != size(); ++i)
      compressed[i] =
          std::lower_bound(uniquely.begin(), uniquely.end(), raw[i]) -
          uniquely.begin();
  }

  size_t operator[](const size_t idx) const {
    assert(idx < size());
    return compressed[idx];
  }

  size_t size() const { return compressed.size(); }

  size_t count() const { return uniquely.size(); }

  T value(const size_t ord) const {
    assert(ord < count());
    return uniquely[ord];
  }

  size_t order(const T &value) const {
    return std::lower_bound(uniquely.begin(), uniquely.end(), value) -
           uniquely.begin();
  }

  auto begin() { return compressed.begin(); }
  auto end() { return compressed.end(); }
  auto rbegin() { return compressed.rbegin(); }
  auto rend() { return compressed.rend(); }
};
#line 2 "Library/src/utils/ejection.hpp"

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

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

namespace workspace {

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

}  // 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 6 "Library/src/utils/hash.hpp"

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

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

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

template <typename T, class = void> struct is_complete : std::false_type {};

template <typename T>
struct is_complete<T, decltype(void(sizeof(T)))> : std::true_type {};

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 = 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 {};
template <> struct is_integral_ext<__int128_t> : std::true_type {};
template <> struct is_integral_ext<__uint128_t> : std::true_type {};
#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))>::type> {
  using type = uint_least64_t;
};
template <typename T>
struct multiplicable_uint<T, typename std::enable_if<(4 < sizeof(T))>::type> {
  using type = __uint128_t;
};
#line 8 "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/casefmt.hpp"

/*
 * @file castfmt
 * @brief Case Output Format
 */

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

/*
 * @file iterate_case.hpp
 * @brief Iterate Testcases
 */

namespace workspace {

namespace internal {
// The 1-based index of the current testcase.
unsigned caseid;
}  // namespace internal

void main();
unsigned case_number();

/*
 * @fn iterate_main
 * @brief Iterate main function.
 */
void iterate_main() {
  for (unsigned total = case_number(), &counter = (internal::caseid = 1);
       counter <= total; ++counter) {
    try {
      main();
    } catch (std::nullptr_t) {
    }
  }
}

}  // namespace workspace
#line 9 "Library/src/utils/io/casefmt.hpp"

namespace workspace {

/*
 * @fn casefmt
 * @brief printf("Case #%u: ", internal::caseid)
 * @param os Reference to ostream
 * @return os
 */
std::ostream& casefmt(std::ostream& os) {
  return os << "Case #" << internal::caseid << ": ";
}

}  // namespace workspace
#line 3 "Library/src/utils/io/read.hpp"
namespace workspace {

namespace internal {
struct cast_read {
  template <class T> operator T() const {
    T value;
    workspace::cin >> value;
    return value;
  }
};
}  // namespace internal

/*
 * @fn read
 * @tparam Tp The type of input.
 * @brief Read from stdin.
 */
template <class Tp = void> auto read() {
  typename std::remove_const<Tp>::type value;
  cin >> value;
  return value;
}

/*
 * @fn read
 * @brief Read from stdin on type casting.
 */
template <> auto read<void>() { return internal::cast_read(); }

}  // 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 {

/*
 * @fn io_setup
 * @brief Setup I/O before main process.
 */
__attribute__((constructor)) void io_setup() {
  std::ios::sync_with_stdio(false);
  std::cin.tie(nullptr);
  std::cout << std::fixed << std::setprecision(15);

#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/io/stream.hpp"

/*
 * @file stream.hpp
 * @brief Stream
 */

#include <cxxabi.h>

#line 13 "Library/src/utils/io/stream.hpp"

namespace workspace {

/*
 * @class istream
 * @brief A wrapper class for std::istream.
 */
class istream : std::istream {
  template <class Tp, typename = std::nullptr_t> struct helper {
    helper(std::istream &is, Tp &x) {
      for (auto &&e : x) helper<decltype(e)>(is, e);
    }
  };

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

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

  template <class... Tps> struct helper<std::tuple<Tps...>> {
    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
        helper<typename std::tuple_element<N, Tp>::type>(is, std::get<N>(x)),
            iterate<Tp, N + 1>(is, x);
    }
  };

 public:
  template <typename Tp> istream &operator>>(Tp &x) {
    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;

// operator<< overloads

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 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(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(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(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 3 "Library/src/utils/random_number_generator.hpp"
template <typename num_type> class random_number_generator {
  typename std::conditional<std::is_integral<num_type>::value,
                            std::uniform_int_distribution<num_type>,
                            std::uniform_real_distribution<num_type>>::type
      unif;

  std::mt19937 engine;

 public:
  random_number_generator(num_type min = std::numeric_limits<num_type>::min(),
                          num_type max = std::numeric_limits<num_type>::max())
      : unif(min, max), engine(std::random_device{}()) {}

  num_type min() const { return unif.min(); }

  num_type max() const { return unif.max(); }

  // generate a random number in [min(), max()].
  num_type operator()() { return unif(engine); }
};
#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 4 "Library/src/utils/trinary_search.hpp"
// trinary search on discrete range.
template <class iter_type, class comp_type>
iter_type trinary(iter_type first, iter_type last, comp_type comp)
{
    assert(first < last);
    intmax_t dist(last - first);
    while(dist > 2)
    {
        iter_type left(first + dist / 3), right(first + dist * 2 / 3);
        if(comp(left, right)) last = right, dist = dist * 2 / 3;
        else first = left, dist -= dist / 3;
    }
    if(dist > 1 && comp(first + 1, first)) ++first;
    return first;
}
// trinary search on real numbers.
template <class comp_type>
long double trinary(long double first, long double last, const long double eps, comp_type comp)
{
    assert(first < last);
    while(last - first > eps)
    {
        long double left{(first * 2 + last) / 3}, right{(first + last * 2) / 3};
        if(comp(left, right)) last = right;
        else first = left;
    }
    return first;
}
#line 2 "Library/src/utils/wrapper.hpp"
template <class Container> class reversed {
  Container &ref, copy;

 public:
  constexpr reversed(Container &ref) : ref(ref) {}
  constexpr reversed(Container &&ref = Container()) : ref(copy), copy(ref) {}
  constexpr auto begin() const { return ref.rbegin(); }
  constexpr auto end() const { return ref.rend(); }
  constexpr operator Container() const { return ref; }
};
#line 12 "Library/yu2.cc"

int main() { workspace::iterate_main(); }

unsigned workspace::case_number() {
  // return -1; // unspecified
  // int t; std::cin >> t; std::cin.ignore(); return t; // given
  return 1;
}

#line 3 "Library/src/data_structure/convex_hull_trick/Li_Chao_tree.hpp"

template <class T = long long, class Comp = std::less<T>, T infty = std::numeric_limits<T>::max()>
class Li_Chao_tree
{
    struct line
    {
        T slop = 0, icpt = infty;
        line *lch = nullptr, *rch = nullptr;
        ~line() { delete lch; delete rch; }
        line *swap(line &rhs) { std::swap(slop, rhs.slop); std::swap(icpt, rhs.icpt); return this; }
        T eval(const T x) const { return slop * x + icpt; }
    }; // struct line

    T lower, upper, eps;
    Comp comp;
    line *root = nullptr;

    // // insert a line for the interval [l, r).
    line *insert(line *const p, const T l, const T r, line ln)
    {
        if(!p) return new line(ln);
        bool lcmp = comp(ln.eval(l), p->eval(l));
        bool rcmp = comp(ln.eval(r - eps), p->eval(r - eps));
        if(lcmp == rcmp) return lcmp ? p->swap(ln) : p;
        if(r - l <= eps) return p;
        T mid = (l + r) / 2;
        if(comp(ln.eval(mid), p->eval(mid)))
        {
            p->swap(ln);
            lcmp = !lcmp;
        }
        if(lcmp) p->lch = insert(p->lch, l, mid, ln);
        else p->rch = insert(p->rch, mid, r, ln);
        return p;
    }

    // // insert a segment for the interval [l, r).
    line *insert(line *const p, const T l, const T r, line ln, const T s, const T t)
    {
        if(t - eps < l || r - eps < s) return p;
        T mid = (l + r) / 2;
        if(l < s or t < r)
        {
            line *np = p ? p : new line;
            np->lch = insert(np->lch, l, mid, ln, s, t);
            np->rch = insert(np->rch, mid, r, ln, s, t);
            return np;
        }
        if(!p) return new line(ln);
        bool lcmp = comp(ln.eval(l), p->eval(l));
        bool rcmp = comp(ln.eval(r - eps), p->eval(r - eps));
        if(lcmp == rcmp) return lcmp ? p->swap(ln) : p;
        if(r - l <= eps) return p;
        if(comp(ln.eval(mid), p->eval(mid)))
        {
            p->swap(ln);
            lcmp = !lcmp;
        }
        if(lcmp) p->lch = insert(p->lch, l, mid, ln, s, t);
        else p->rch = insert(p->rch, mid, r, ln, s, t);
        return p;
    }

public:
    // domain set to be the interval [lower, upper).
    Li_Chao_tree(const T lower, const T upper, const T eps = 1, Comp comp = Comp())
        : lower(lower), upper(upper), eps(eps), comp(comp) {}

    ~Li_Chao_tree() { delete root; }

    bool empty() const { return !root; }

    // insert a line whose slope is p and inception is q.
    void insert(const T p, const T q) { root = insert(root, lower, upper, line{p, q}); }

    // insert a line(segment) whose slope is p, inception is q,
    // and domain is the interval [s, t).
    void insert(const T p, const T q, const T s, const T t) { if(s < t) root = insert(root, lower, upper, line{p, q}, s, t); }

    T get(const T x) const
    {
        line *p = root;
        T l = lower, r = upper;
        T res = infty;
        while(p)
        {
            T nval = p->eval(x);
            if(comp(nval, res)) res = nval;
            if(r - l <= eps) return res;
            T mid = (l + r) / 2;
            if(x < mid)
            {
                p = p->lch;
                r = mid;
            }
            else
            {
                p = p->rch;
                l = mid;
            }
        }
        return res;
    }
}; // class Li_Chao_tree
#line 4 "Library/src/data_structure/convex_hull_trick/monotone.hpp"

template <class T> class lower_convex_monotone {
  struct line {
    T slop, icpt;
    T eval(const T x) const { return slop * x + icpt; }
  };

  std::vector<line> lines;
  typename std::vector<line>::iterator lp, rp;

  void realloc() {
    if (rp != lines.end()) return;
    std::vector<line> copy((rp - lp) * 2 + 1);
    copy.swap(lines);
    rp = copy(lp, rp, lines.begin());
    lp = lines.begin();
  }

 public:
  lower_convex_monotone() : lines(2), lp(lines.begin()), rp(lp) {}

  bool empty() const { return lp == rp; }

  void clear() { rp = lp = lines.begin(); }

  void add(const T a, const T b) {
    while (rp - lp > 1) {
      auto [a1, b1] = *(rp - 1);
      auto [a2, b2] = *(rp - 2);
      if ((b - b1) * (a2 - a) > (b - b2) * (a1 - a)) break;
      --rp;
    }
    if (rp == lp) rp = lp = lines.begin();
    realloc();
    *rp++ = {a, b};
  }

  T get(const T x) {
    assert(!empty());
    while (rp - lp > 1 && lp->eval(x) > (lp + 1)->eval(x)) ++lp;
    return lp->eval(x);
  }
};  // class lower_convex_monotone
#line 4 "Library/src/data_structure/deque_aggregation.hpp"
// implementation with dynamic memory allocation.
template <class monoid>
class deque_aggregation
{
    template <bool left_operand_added>
    class stack_aggregation
    {
        friend deque_aggregation;
        struct data { monoid value, acc; };
        size_t capacity;
        data *stack, *end, *itr;
        bool top_referred;

        void recalc()
        {
            if(top_referred)
            {
                assert(itr != stack);
                top_referred = false;
                monoid top_val{top().value};
                pop();
                push(top_val);
            }
        }

    public:
        stack_aggregation() : capacity(1), stack(new data[1]), end(std::next(stack)), itr(stack), top_referred() {}
        ~stack_aggregation() { delete[] stack; }

        bool empty() const { return stack == itr; }
        size_t size() const { return itr - stack; }

        // copy of the element at the index.
        data operator[](size_t index) const
        {
            assert(index < size());
            recalc();
            return stack[index];
        }

        // reference to the last element
        data &top()
        {
            assert(itr != stack);
            top_referred = true;
            return *std::prev(itr);
        }

        void pop()
        {
            assert(itr != stack);
            --itr;
            top_referred = false;
        }

        void push(const monoid &mono)
        {
            recalc();
            if(itr == end)
            {
                data *tmp = new data[capacity << 1];
                std::swap(stack, tmp);
                end = (itr = std::copy(tmp, tmp + capacity, stack)) + capacity;
                capacity <<= 1;
                delete[] tmp;
            }
            if(left_operand_added) *itr = data{mono, mono + fold()};
            else *itr = data{mono, fold() + mono};
            ++itr;
        }

        monoid fold()
        {
            if(itr == stack) return monoid();
            recalc();
            return std::prev(itr)->acc;
        }
    }; // class stack_aggregation

    stack_aggregation<true> left;
    stack_aggregation<false> right;

    void balance_to_left()
    {
        if(!left.empty() || right.empty()) return;
        left.recalc(); right.recalc();
        size_t mid = (right.size() + 1) >> 1;
        auto *itr = right.stack + mid;
        do { left.push((--itr)->value); } while(itr != right.stack);
        monoid acc;
        for(auto *p = right.stack + mid; p != right.itr; ++p, ++itr)
        {
            *itr = {p->value, acc = acc + p->value};
        }
        right.itr = itr;
    }

    void balance_to_right()
    {
        if(!right.empty() || left.empty()) return;
        left.recalc(); right.recalc();
        size_t mid = (left.size() + 1) >> 1;
        auto *itr = left.stack + mid;
        do { right.push((--itr)->value); } while(itr != left.stack);
        monoid acc;
        for(auto *p = left.stack + mid; p != left.itr; ++p, ++itr)
        {
            *itr = {p->value, acc = p->value + acc};
        }
        left.itr = itr;
    }

public:
    bool empty() const { return left.empty() && right.empty(); }
    size_t size() const { return left.size() + right.size(); }

    // reference to the first element.
    monoid &front() { assert(!empty()); balance_to_left(); return left.top().value; }

    // reference to the last element.
    monoid &back() { assert(!empty()); balance_to_right(); return right.top().value; }

    // copy of the element at the index.
    monoid operator[](size_t index) const
    {
        assert(index < left.size() + right.size());
        return index < left.size() ? left[index].value : right[index - left.size()].value;
    }

    void push_front(const monoid &mono) { left.push(mono); }

    void push_back(const monoid &mono) { right.push(mono); }

    void pop_front()
    {
        assert(!empty());
        balance_to_left();
        left.pop();
    }

    void pop_back()
    {
        assert(!empty());
        balance_to_right();
        right.pop();
    }

    monoid fold() { return left.fold() + right.fold(); }
}; // class deque_aggregation
#line 2 "Library/src/data_structure/Mo.hpp"

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

#line 13 "Library/src/data_structure/Mo.hpp"

namespace workspace {

/*
 * @class Mo
 * @brief process queries about contiguous subarray
 * @tparam Push_back
 * @tparam Pop_back
 * @tparam Push_front Push_back as default
 * @tparam Pop_front Pop_back as default
 */
template <class Push_back, class Pop_back, class Push_front = Push_back,
          class Pop_front = Pop_back>
class Mo {
  Push_front push_front;
  Pop_front pop_front;
  Push_back push_back;
  Pop_back pop_back;
  std::vector<size_t> lft, rgt, ord;
  std::vector<size_t>::iterator itr;
  size_t lpos, rpos;

 public:
  /*
   * @param push_back
   * @param pop_back
   */
  Mo(Push_back push_back, Pop_back pop_back)
      : Mo(push_back, pop_back, push_back, pop_back) {}

  /*
   * @param push_front
   * @param pop_front
   * @param push_back
   * @param pop_back
   */
  Mo(Push_front push_front, Pop_front pop_front, Push_back push_back,
     Pop_back pop_back)
      : push_front(push_front),
        pop_front(pop_front),
        push_back(push_back),
        pop_back(pop_back),
        lpos(),
        rpos() {}

  /*
   * @return number of queries
   */
  size_t size() const { return lft.size(); }

  /*
   * @brief add query
   * @param l left end, inclusive
   * @param r right end, exclusive
   */
  void set(size_t l, size_t r) {
    assert(!(r < l));
    lft.emplace_back(l), rgt.emplace_back(r);
  }

  /*
   * @brief sort queries
   */
  void make() {
    assert(size());
    ord.resize(size());
    iota(ord.begin(), ord.end(), 0);
    const size_t width = sqrt(*max_element(rgt.begin(), rgt.end()));
    std::sort(ord.begin(), ord.end(), [&](size_t x, size_t y) {
      if (lft[x] / width != lft[y] / width) return lft[x] < lft[y];
      return rgt[x] < rgt[y];
    });
    itr = ord.begin();
  }

  /*
   * @brief process one query
   * @return index of query
   */
  size_t process() {
    if (itr == ord.end()) return ord.size();
    const size_t id = *itr++, l = lft[id], r = rgt[id];
    while (lpos > l) push_front(--lpos);
    while (rpos < r) push_back(rpos++);
    while (lpos < l) pop_front(lpos++);
    while (rpos > r) pop_back(--rpos);
    return id;
  }
};

}  // namespace workspace
#line 5 "Library/src/data_structure/segment_tree/basic.hpp"

#line 2 "Library/algebra/system/monoid.hpp"

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

#line 9 "Library/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 3 "Library/src/data_structure/segment_tree/waitlist.hpp"

namespace internal {
struct waitlist : std::queue<size_t> {
  waitlist(size_t n) : in(n) {}

  bool push(size_t index) {
    assert(index < in.size());
    if (in[index]) return false;
    emplace(index);
    return (in[index] = true);
  }

  size_t pop() {
    assert(!empty());
    auto index = front();
    std::queue<size_t>::pop();
    in[index] = false;
    return index;
  }

 private:
  std::vector<int_least8_t> in;
};
}
#line 9 "Library/src/data_structure/segment_tree/basic.hpp"

template <class Monoid, class Container = std::vector<Monoid>>
class segment_tree {
  static_assert(std::is_same<Monoid, mapped_type<Container>>::value);

  size_t size_orig, height, size_ext;
  Container data;
  internal::waitlist wait;

  void repair() {
    while (!wait.empty()) {
      const size_t index = wait.pop() >> 1;
      if (index && wait.push(index)) pull(index);
    }
  }

  void pull(const size_t node) {
    data[node] = data[node << 1] + data[node << 1 | 1];
  }

  template <class Pred>
  size_t left_partition_subtree(size_t index, const Pred pred,
                                Monoid mono) const {
    assert(index);
    while (index < size_ext) {
      const Monoid tmp = data[(index <<= 1) | 1] + mono;
      if (pred(tmp))
        mono = tmp;
      else
        ++index;
    }
    return ++index -= size_ext;
  }

  template <class Pred>
  size_t right_partition_subtree(size_t index, const Pred pred,
                                 Monoid mono) const {
    assert(index);
    while (index < size_ext) {
      const Monoid tmp = mono + data[index <<= 1];
      if (pred(tmp)) ++index, mono = tmp;
    }
    return (index -= size_ext) < size_orig ? index : size_orig;
  }

 public:
  using value_type = Monoid;

  segment_tree(const size_t n = 0)
      : size_orig{n},
        height(n > 1 ? 32 - __builtin_clz(n - 1) : 0),
        size_ext{1u << height},
        data(size_ext << 1),
        wait(size_ext << 1) {}

  segment_tree(const size_t n, const Monoid &init) : segment_tree(n) {
    std::fill(std::next(std::begin(data), size_ext), std::end(data), init);
    for (size_t i{size_ext}; --i;) pull(i);
  }

  template <class iter_type, class value_type = typename std::iterator_traits<
                                 iter_type>::value_type>
  segment_tree(iter_type first, iter_type last)
      : size_orig(std::distance(first, last)),
        height(size_orig > 1 ? 32 - __builtin_clz(size_orig - 1) : 0),
        size_ext{1u << height},
        data(size_ext << 1),
        wait(size_ext << 1) {
    static_assert(std::is_constructible<Monoid, value_type>::value,
                  "Monoid(iter_type::value_type) is not constructible.");
    for (auto iter{std::next(std::begin(data), size_ext)};
         iter != std::end(data) && first != last; ++iter, ++first)
      *iter = Monoid{*first};
    for (size_t i{size_ext}; --i;) pull(i);
  }

  template <class Cont, typename = typename Cont::value_type>
  segment_tree(const Cont &cont)
      : segment_tree(std::begin(cont), std::end(cont)) {}

  size_t size() const { return size_orig; }
  size_t capacity() const { return size_ext; }

  // reference to the element at the index.
  Monoid &operator[](size_t index) {
    assert(index < size_orig);
    wait.push(index |= size_ext);
    return data[index];
  }

  // const reference to the element at the index.
  const Monoid &operator[](size_t index) const {
    assert(index < size_orig);
    return data[index |= size_orig];
  }

  Monoid fold(size_t first, size_t last) {
    assert(last <= size_orig);
    repair();
    Monoid leftval{}, rightval{};
    first += size_ext, last += size_ext;
    while (first < last) {
      if (first & 1) leftval = leftval + data[first++];
      if (last & 1) rightval = data[--last] + rightval;
      first >>= 1, last >>= 1;
    }
    return leftval + rightval;
  }

  Monoid fold() { return fold(0, size_orig); }

  template <class Pred> size_t left_partition(size_t right, Pred pred) {
    assert(right <= size_orig);
    repair();
    right += size_ext;
    Monoid mono{};
    for (size_t left{size_ext}; left != right; left >>= 1, right >>= 1) {
      if ((left & 1) != (right & 1)) {
        const Monoid tmp = data[--right] + mono;
        if (!pred(tmp)) return left_partition_subtree(right, pred, mono);
        mono = tmp;
      }
    }
    return 0;
  }

  template <class Pred> size_t right_partition(size_t left, Pred pred) {
    assert(left <= size_orig);
    repair();
    left += size_ext;
    Monoid mono{};
    for (size_t right{size_ext << 1}; left != right; left >>= 1, right >>= 1) {
      if ((left & 1) != (right & 1)) {
        const Monoid tmp = mono + data[left];
        if (!pred(tmp)) return right_partition_subtree(left, pred, mono);
        mono = tmp;
        ++left;
      }
    }
    return size_orig;
  }
};  // class segment_tree
#line 5 "Library/src/data_structure/segment_tree/lazy.hpp"

#line 9 "Library/src/data_structure/segment_tree/lazy.hpp"

template <class Monoid, class Endomorphism,
          class Monoid_container = std::vector<Monoid>,
          class Endomorphism_container = std::vector<Endomorphism>>
class lazy_segment_tree {
  static_assert(std::is_same<Monoid, mapped_type<Monoid_container>>::value);

  static_assert(
      std::is_same<Endomorphism, mapped_type<Endomorphism_container>>::value);

  static_assert(std::is_same<Monoid, decltype(Monoid{} + Monoid{})>::value,
                "\'Monoid\' has no proper binary operator+.");

  static_assert(std::is_same<Endomorphism,
                             decltype(Endomorphism{} * Endomorphism{})>::value,
                "\'Endomorphism\' has no proper binary operator*.");

  static_assert(
      std::is_same<Monoid, decltype(Monoid{} * Endomorphism{})>::value,
      "\'Endomorphism\' is not applicable to \'Monoid\'.");

  size_t size_orig, height, size_ext;
  Monoid_container data;
  Endomorphism_container lazy;
  internal::waitlist wait;

  void repair() {
    while (!wait.empty()) {
      const size_t index = wait.pop() >> 1;
      if (index && wait.push(index)) pull(index);
    }
  }

  void apply(size_t node, const Endomorphism &endo) {
    data[node] = data[node] * endo;
    if (node < size_ext) lazy[node] = lazy[node] * endo;
  }

  void push(size_t node) {
    if (!(node < size_ext)) return;
    apply(node << 1, lazy[node]);
    apply(node << 1 | 1, lazy[node]);
    lazy[node] = Endomorphism{};
  }

  void pull(size_t node) { data[node] = data[node << 1] + data[node << 1 | 1]; }

  template <class Pred>
  size_t left_partition_subtree(size_t node, Pred pred, Monoid mono) {
    assert(node);
    while (node < size_ext) {
      push(node);
      const Monoid &tmp = data[(node <<= 1) | 1] + mono;
      if (pred(tmp))
        mono = tmp;
      else
        ++node;
    }
    return ++node -= size_ext;
  }

  template <class Pred>
  size_t right_partition_subtree(size_t node, Pred pred, Monoid mono) {
    assert(node);
    while (node < size_ext) {
      push(node);
      const Monoid &tmp = mono + data[node <<= 1];
      if (pred(tmp)) ++node, mono = tmp;
    }
    return (node -= size_ext) < size_orig ? node : size_orig;
  }

 public:
  using value_type = Monoid;

  lazy_segment_tree(size_t n = 0)
      : size_orig{n},
        height(n > 1 ? 32 - __builtin_clz(n - 1) : 0),
        size_ext{1u << height},
        data(size_ext << 1),
        lazy(size_ext),
        wait(size_ext << 1) {}

  lazy_segment_tree(size_t n, const Monoid &init) : lazy_segment_tree(n) {
    std::fill(std::next(std::begin(data), size_ext), std::end(data), init);
    for (size_t i{size_ext}; --i;) pull(i);
  }

  template <class iter_type, class value_type = typename std::iterator_traits<
                                 iter_type>::value_type>
  lazy_segment_tree(iter_type first, iter_type last)
      : size_orig(std::distance(first, last)),
        height(size_orig > 1 ? 32 - __builtin_clz(size_orig - 1) : 0),
        size_ext{1u << height},
        data(size_ext << 1),
        lazy(size_ext),
        wait(size_ext << 1) {
    static_assert(std::is_constructible<Monoid, value_type>::value,
                  "Monoid(iter_type::value_type) is not constructible.");
    for (auto iter{std::next(std::begin(data), size_ext)};
         iter != std::end(data) && first != last; ++iter, ++first)
      *iter = Monoid(*first);
    for (size_t i{size_ext}; --i;) pull(i);
  }

  template <class Container, typename = element_type<Container>>
  lazy_segment_tree(const Container &cont)
      : lazy_segment_tree(std::begin(cont), std::end(cont)) {}

  size_t size() const { return size_orig; }

  size_t capacity() const { return size_ext; }

  Monoid &operator[](size_t index) {
    assert(index < size_orig);
    index |= size_ext;
    wait.push(index);
    for (size_t i = height; i; --i) push(index >> i);
    return data[index];
  }

  void update(size_t index, const Endomorphism &endo) {
    update(index, index + 1, endo);
  }

  void update(size_t first, size_t last, const Endomorphism &endo) {
    assert(last <= size_orig);
    repair();
    if (first >= last) return;
    first += size_ext, last += size_ext - 1;
    for (size_t i = height; i; --i) push(first >> i), push(last >> i);
    for (size_t l = first, r = last + 1; last; l >>= 1, r >>= 1) {
      if (l < r) {
        if (l & 1) apply(l++, endo);
        if (r & 1) apply(--r, endo);
      }
      if (first >>= 1, last >>= 1) {
        pull(first), pull(last);
      }
    }
  }

  Monoid fold() { return fold(0, size_orig); }

  Monoid fold(size_t first, size_t last) {
    assert(last <= size_orig);
    repair();
    if (first >= last) return Monoid{};
    first += size_ext, last += size_ext - 1;
    Monoid left_val{}, right_val{};
    for (size_t l = first, r = last + 1; last; l >>= 1, r >>= 1) {
      if (l < r) {
        if (l & 1) left_val = left_val + data[l++];
        if (r & 1) right_val = data[--r] + right_val;
      }
      if (first >>= 1, last >>= 1) {
        left_val = left_val * lazy[first];
        right_val = right_val * lazy[last];
      }
    }
    return left_val + right_val;
  }

  template <class Pred> size_t left_partition(size_t right, Pred pred) {
    assert(right <= size_orig);
    repair();
    right += size_ext - 1;
    for (size_t i{height}; i; --i) push(right >> i);
    ++right;
    Monoid mono{};
    for (size_t left{size_ext}; left != right; left >>= 1, right >>= 1) {
      if ((left & 1) != (right & 1)) {
        const Monoid &tmp = data[--right] + mono;
        if (!pred(tmp)) return left_partition_subtree(right, pred, mono);
        mono = tmp;
      }
    }
    return 0;
  }

  template <class Pred> size_t right_partition(size_t left, Pred pred) {
    assert(left <= size_orig);
    repair();
    left += size_ext;
    for (size_t i{height}; i; --i) push(left >> i);
    Monoid mono{};
    for (size_t right{size_ext << 1}; left != right; left >>= 1, right >>= 1) {
      if ((left & 1) != (right & 1)) {
        const Monoid &tmp = mono + data[left];
        if (!pred(tmp)) return right_partition_subtree(left, pred, mono);
        mono = tmp;
        ++left;
      }
    }
    return size_orig;
  }
};  // class lazy_segment_tree
#line 2 "Library/src/data_structure/union_find/basic.hpp"

/*
 * @file basic.hpp
 * @brief Basic Union-Find
 */

#line 11 "Library/src/data_structure/union_find/basic.hpp"

namespace workspace {

template <typename Tp> struct union_find {
 protected:
  using signed_t = typename std::make_signed<Tp>::type;
  using unsigned_t = typename std::make_unsigned<Tp>::type;

  std::vector<signed_t> link;

 public:
  /*
   * @param n The number of nodes.
   */
  union_find(Tp n = 0) : link(n, 1) {}

  /*
   * @fn find
   * @param x A node.
   * @return The representative of the group.
   */
  virtual unsigned_t find(unsigned_t x) {
    assert(x < size());
    return link[x] > 0 ? x : -(link[x] = -(signed_t)find(-link[x]));
  }

  /*
   * @fn size
   * @return The number of nodes.
   */
  unsigned_t size() const { return link.size(); }

  /*
   * @fn size
   * @param x A node.
   * @return The number of nodes in the group.
   */
  virtual unsigned_t size(unsigned_t x) {
    assert(x < size());
    return link[find(x)];
  }

  /*
   * @fn same
   * @param x 1st node.
   * @param y 2nd node.
   * @return Whether or not the two nodes belong to the same group.
   */
  bool same(unsigned_t x, unsigned_t y) {
    assert(x < size());
    assert(y < size());
    return find(x) == find(y);
  }

  /*
   * @fn unite
   * @param x 1st node.
   * @param y 2nd node.
   * @return Whether or not the two groups were merged anew.
   */
  virtual bool unite(unsigned_t x, unsigned_t y) {
    assert(x < size()), x = find(x);
    assert(y < size()), y = find(y);
    if (x == y) return false;
    if (link[x] < link[y]) std::swap(x, y);
    link[x] += link[y];
    link[y] = -(signed_t)x;
    return true;
  }
};

}  // namespace workspace
#line 2 "Library/src/data_structure/union_find/bipartite.hpp"

/*
 * @file bipartite.hpp
 * @brief Bipartite Union-Find
 */

#line 9 "Library/src/data_structure/union_find/bipartite.hpp"

namespace workspace {

class bipartite_union_find : public union_find<uint_least64_t> {
  using base = union_find<uint_least64_t>;
  using base::union_find;

  constexpr static unsigned_t sgnb = unsigned_t(1) << 32, mask = sgnb - 1,
                              ubs = ~unsigned_t(0) ^ mask;

 public:
  unsigned_t find(unsigned_t x) override {
    assert(x < base::size());
    if (link[x] > 0) return x;
    const auto p = -link[x] & mask, r = find(p);
    if (r != p) link[x] = -(link[x] ^ p ^ link[p]);
    return r;
  }

  bool diff(unsigned_t x) {
    assert(x < base::size()), find(x);
    return link[x] > 0 ? 0 : link[x] >> 32 & 1;
  }

  // bool diff(unsigned_t x, unsigned_t y) {}

  unsigned_t size(size_t x, bool neq) {
    assert(x < base::size()), x = find(x);
    return neq ? link[x] >> 32 : link[x] & mask;
  }

  unsigned_t size(size_t x) override {
    assert(x < base::size()), x = find(x);
    return (link[x] >> 32) + (link[x] & mask);
  }

  /*
   * @fn unite
   * @param x 1st node.
   * @param y 2nd node.
   * @param flip
   * @return Whether or not the relation is consistent.
   */
  bool unite(size_t x, size_t y, bool flip = true) {
    assert(x < base::size()), flip ^= diff(x), x = find(x);
    assert(y < base::size()), flip ^= diff(y), y = find(y);
    if (x == y) return !flip;
    if (link[x] < link[y]) std::swap(x, y);
    link[x] += flip ? (unsigned_t)link[y] << 32 | link[y] >> 32 : link[y];
    link[y] = flip ? (mask + 1) ^ -x : -x;
    return true;
  }
};

}  // namespace workspace
#line 1 "Library/src/data_structure/union_find/partially_persistent_union_find.hpp"
// #line 2 "Partially_persistent_union_find.hpp"
// veryfied at https://atcoder.jp/contests/agc002/submissions/9514048
#ifndef Partially_persistent_union_find_hpp
#define Partially_persistent_union_find_hpp
#include <cstdint>
#include <cstddef>
#include <numeric>
#include <vector>

class partially_persistent_union_find
{
    using time_type = uint32_t;
    struct log_type { time_type time; size_t size; };
    const size_t n;
    std::vector<size_t> parent;
    std::vector<time_type> last;
    std::vector<std::vector<log_type>> size_log;
    time_type clock;

public:
    explicit partially_persistent_union_find(size_t _n) : n(_n), parent(n), last(n, UINT32_MAX), size_log(n, std::vector<log_type>(1, {0, 1})), clock()
    {
        std::iota(parent.begin(), parent.end(), 0);
    }

    size_t size(size_t x, time_type t = UINT32_MAX)
    {
        size_t root = find(x, t);
        auto __ok{size_log[root].begin()}, __ng{size_log[root].end()};
        auto dist = __ng - __ok;
        while(dist > 1)
        {
            auto mid{__ok + (dist >> 1)};
            if(mid->time > t) __ng = mid, dist >>= 1;
            else __ok = mid, ++dist >>= 1;
        }
        return __ok->size;
    }

    size_t find(size_t x, size_t t = UINT32_MAX) { return last[x] >= t ? x : find(parent[x], t); }

    bool same(size_t x, size_t y, time_type t = UINT32_MAX) { return find(x, t) == find(y, t); }

    time_type unite(size_t x, size_t y)
    {
        if((x = find(x)) != (y = find(y)))
        {
            size_t size_x = size_log[x].back().size;
            size_t size_y = size_log[y].back().size;
            if(size_x < size_y) std::swap(x, y), std::swap(size_x, size_y);
            size_log[x].push_back({clock + 1, size_x + size_y});
            parent[y] = x;
            last[y] = clock;
        }
        return ++clock;
    }
}; // class partially_persistent_union_find

#endif
#line 1 "Library/src/data_structure/union_find/potentialized_union_find.hpp"
// #line 2 "potentialized_union_find.hpp"
// verified at https://atcoder.jp/contests/abc087/submissions/9511701
#ifndef Potentialized_union_find_hpp
#define Potentialized_union_find_hpp
#include <cassert>
#include <cstddef>
#include <vector>

template <class Abelian>
class potentialized_union_find
{
    size_t n;
    std::vector<int> link;
    std::vector<Abelian> diff_weight;

public:
    explicit potentialized_union_find(size_t _n) : n(_n), link(n, -1), diff_weight(n) {}

    size_t find(const size_t x)
    {
        assert(x < n);
        if(link[x] < 0) return x;
        const size_t root = find(link[x]);
        diff_weight[x] += diff_weight[link[x]];
        return link[x] = root;
    }

    size_t size() const { return n; }
    size_t size(const size_t x) { return -link[find(x)]; }

    Abelian weight(size_t x) { return find(x), diff_weight[x]; }

    Abelian diff(size_t x, size_t y) { return weight(y) - weight(x); }

    bool same(const size_t x, const size_t y) { return find(x) == find(y); }

    bool unite(size_t x, size_t y, Abelian w)
    {
        w += weight(x) - weight(y);
        x = find(x), y = find(y);
        if(x == y) return false;
        if(link[x] > link[y]) std::swap(x, y), w = -w;
        link[x] += link[y], link[y] = x;
        diff_weight[y] = w;
        return true;
    }
}; // class potentialized_union_find

#endif
#line 2 "Library/src/data_structure/union_find/unbalanced.hpp"

/*
 * @file unbalanced.hpp
 * @brief Unbalanced Union-Find
 */

#line 9 "Library/src/data_structure/union_find/unbalanced.hpp"

namespace workspace {

class unbalanced_union_find : public union_find<uint_least32_t> {
  using base = union_find<uint_least32_t>;

 public:
  using base::union_find;

  bool unite(unsigned_t x, unsigned_t y) override {
    assert(x < size()), x = find(x);
    assert(y < size()), y = find(y);
    if (x == y) return false;
    link[x] += link[y];
    link[y] = -(signed_t)x;
    return true;
  }
};

}  // namespace workspace
#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 10 "Library/src/modular/modint.hpp"

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

namespace workspace {

namespace internal {

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

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

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

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

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

  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 ? mod + n : n) {}

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

  constexpr modint_base operator++(int) noexcept {
    modint_base t{*this};
    return operator+=(1), t;
  }

  constexpr modint_base operator--(int) noexcept {
    modint_base t{*this};
    return operator-=(1), t;
  }

  constexpr modint_base &operator++() noexcept { return operator+=(1); }

  constexpr modint_base &operator--() noexcept { return operator-=(1); }

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

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

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

  constexpr modint_base &operator*=(const modint_base &rhs) noexcept {
    return value = (typename multiplicable_uint<value_type>::type)value *
                   rhs.value % mod,
           *this;
  }

  constexpr modint_base &operator/=(const modint_base &rhs) noexcept {
    return operator*=(rhs.inverse());
  }

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

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

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

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

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

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

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

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

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

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

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

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

  constexpr modint_base inverse() const noexcept {
    assert(value);
    value_type a{mod}, b{value}, u{}, v{1}, t{};
    while (b)
      t = a / b, a ^= b ^= (a -= t * b) ^= b, u ^= v ^= (u -= t * v) ^= v;
    return {u};
  }

  template <class int_type>
  constexpr typename std::enable_if<is_integral_ext<int_type>::value,
                                    modint_base>::type
  power(int_type e) noexcept {
    return pow(*this, e);
  }

  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 {
    modint_base res{1};
    for (e < 0 ? b = b.inverse(), e = -e : 0; e; e >>= 1, b *= b)
      if (e & 1) res *= b;
    return res;
  }

  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;
  }

 protected:
  value_type value = 0;
};

template <auto Mod>
typename modint_base<Mod>::mod_type modint_base<Mod>::mod = Mod;

}  // namespace internal

/*
 * @typedef modint
 * @brief modular arithmetic.
 * @tparam Mod modulus
 */
template <auto Mod, typename std::enable_if<(Mod > 0)>::type * = nullptr>
using modint = internal::modint_base<Mod>;

/*
 * @typedef modint_runtime
 * @brief runtime modular arithmetic.
 * @tparam type_id uniquely assigned
 */
template <unsigned type_id = 0>
using modint_runtime = internal::modint_base<-(signed)type_id>;

// #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 23 "Library/yu2.cc"

namespace workspace {
using mint = modint<998244353>;

void main() {
  // start here!
  int n;
  cin >> n;
  vector<int> a(n);
  cin >> a;
  const auto mx = a.back();
  reverse(begin(a), end(a));
  mint ans;
  for (auto d = 1; d < mx; ++d) {  // maxdiff <= d
    if (d < a[0] - a[1]) continue;
    segment_tree<mint> z(n);
    bool fail = false;
    z[0] = 1;
    for (auto i = 2, j = 0; i < n; ++i) {
      while (a[i] + d < a[j]) ++j;
      if (i == j) {
        fail = true;
        break;
      }
      z[i - 1] = z.fold(j, i - 1);
    }
    if (!fail) ans += z.fold();
  }
  cout << mx * mint(2).power(n - 1) - ans * 2 << eol;
}
}  // namespace workspace