#line 2 "library/other/template.hpp"

#include <bits/stdc++.h>
#line 2 "library/template/macros.hpp"

#line 4 "library/template/macros.hpp"

#ifndef __COUNTER__
#define __COUNTER__ __LINE__
#endif

#define OVERLOAD5(a, b, c, d, e, ...) e
#define REP1_0(b, c) REP1_1(b, c)
#define REP1_1(b, c)                                                           \
    for (ll REP_COUNTER_##c = 0; REP_COUNTER_##c < (ll)(b); ++REP_COUNTER_##c)
#define REP1(b) REP1_0(b, __COUNTER__)
#define REP2(i, b) for (ll i = 0; i < (ll)(b); ++i)
#define REP3(i, a, b) for (ll i = (ll)(a); i < (ll)(b); ++i)
#define REP4(i, a, b, c) for (ll i = (ll)(a); i < (ll)(b); i += (ll)(c))
#define rep(...) OVERLOAD5(__VA_ARGS__, REP4, REP3, REP2, REP1)(__VA_ARGS__)
#define RREP2(i, a) for (ll i = (ll)(a)-1; i >= 0; --i)
#define RREP3(i, a, b) for (ll i = (ll)(a)-1; i >= (ll)(b); --i)
#define RREP4(i, a, b, c) for (ll i = (ll)(a)-1; i >= (ll)(b); i -= (ll)(c))
#define rrep(...) OVERLOAD5(__VA_ARGS__, RREP4, RREP3, RREP2)(__VA_ARGS__)
#define REPS2(i, b) for (ll i = 1; i <= (ll)(b); ++i)
#define REPS3(i, a, b) for (ll i = (ll)(a) + 1; i <= (ll)(b); ++i)
#define REPS4(i, a, b, c) for (ll i = (ll)(a) + 1; i <= (ll)(b); i += (ll)(c))
#define reps(...) OVERLOAD5(__VA_ARGS__, REPS4, REPS3, REPS2)(__VA_ARGS__)
#define RREPS2(i, a) for (ll i = (ll)(a); i > 0; --i)
#define RREPS3(i, a, b) for (ll i = (ll)(a); i > (ll)(b); --i)
#define RREPS4(i, a, b, c) for (ll i = (ll)(a); i > (ll)(b); i -= (ll)(c))
#define rreps(...) OVERLOAD5(__VA_ARGS__, RREPS4, RREPS3, RREPS2)(__VA_ARGS__)

#define each_for(...) for (auto&& __VA_ARGS__)
#define each_const(...) for (const auto& __VA_ARGS__)

#define all(v) std::begin(v), std::end(v)
#if __cplusplus >= 201402L
#define rall(v) std::rbegin(v), std::rend(v)
#else
#define rall(v) v.rbegin(), v.rend()
#endif

#if __cpp_constexpr >= 201304L
#define CONSTEXPR constexpr
#else
#define CONSTEXPR
#endif

#if __cpp_if_constexpr >= 201606L
#define IF_CONSTEXPR constexpr
#else
#define IF_CONSTEXPR
#endif

#define IO_BUFFER_SIZE 2048
#line 2 "library/template/alias.hpp"

#line 4 "library/template/alias.hpp"

using ll = long long;
using uint = unsigned int;
using ull = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;
using ld = long double;
using PLL = std::pair<ll, ll>;
template<class T>
using prique = std::priority_queue<T, std::vector<T>, std::greater<T>>;

template<class T> struct infinity {
    static constexpr T value = std::numeric_limits<T>::max() / 2;
    static constexpr T mvalue = std::numeric_limits<T>::lowest() / 2;
    static constexpr T max = std::numeric_limits<T>::max();
    static constexpr T min = std::numeric_limits<T>::lowest();
};

#if __cplusplus <= 201402L
template<class T> constexpr T infinity<T>::value;
template<class T> constexpr T infinity<T>::mvalue;
template<class T> constexpr T infinity<T>::max;
template<class T> constexpr T infinity<T>::min;
#endif

#if __cpp_variable_templates >= 201304L
template<class T> constexpr T INF = infinity<T>::value;
#endif

constexpr ll inf = infinity<ll>::value;
constexpr ld EPS = 1e-8;
constexpr ld PI = 3.1415926535897932384626;
#line 2 "library/template/type_traits.hpp"

#line 5 "library/template/type_traits.hpp"

template<class T, class... Args> struct function_traits_impl {
    using result_type = T;
    template<std::size_t idx>
    using argument_type =
        typename std::tuple_element<idx, std::tuple<Args...>>::type;
    using argument_tuple = std::tuple<Args...>;
    static constexpr std::size_t arg_size() { return sizeof...(Args); }
};

template<class> struct function_traits_helper;

template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...)> {
    using type = function_traits_impl<Res, Args...>;
};
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...)&> {
    using type = function_traits_impl<Res, Args...>;
};
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...) const> {
    using type = function_traits_impl<Res, Args...>;
};
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...) const&> {
    using type = function_traits_impl<Res, Args...>;
};

#if __cpp_noexcept_function_type >= 201510L
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...) noexcept> {
    using type = function_traits_impl<Res, Args...>;
};
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...)& noexcept> {
    using type = function_traits_impl<Res, Args...>;
};
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...) const noexcept> {
    using type = function_traits_impl<Res, Args...>;
};
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...) const& noexcept> {
    using type = function_traits_impl<Res, Args...>;
};
#endif

template<class F>
using function_traits = typename function_traits_helper<
    decltype(&std::remove_reference<F>::type::operator())>::type;

template<class F>
using function_result_type = typename function_traits<F>::result_type;
template<class F, std::size_t idx>
using function_argument_type =
    typename function_traits<F>::template argument_type<idx>;
template<class F>
using function_argument_tuple = typename function_traits<F>::argument_tuple;

template<class T>
using is_signed_int =
    std::integral_constant<bool, (std::is_integral<T>::value &&
                                  std::is_signed<T>::value) ||
                                     std::is_same<T, i128>::value>;
template<class T>
using is_unsigned_int =
    std::integral_constant<bool, (std::is_integral<T>::value &&
                                  std::is_unsigned<T>::value) ||
                                     std::is_same<T, u128>::value>;
template<class T>
using is_int = std::integral_constant<bool, is_signed_int<T>::value ||
                                                is_unsigned_int<T>::value>;
template<class T>
using make_signed_int = typename std::conditional<
    std::is_same<T, i128>::value || std::is_same<T, u128>::value,
    std::common_type<i128>, std::make_signed<T>>::type;
template<class T>
using make_unsigned_int = typename std::conditional<
    std::is_same<T, i128>::value || std::is_same<T, u128>::value,
    std::common_type<u128>, std::make_unsigned<T>>::type;


template<class T, class = void> struct is_range : std::false_type {};
template<class T>
struct is_range<
    T,
    decltype(all(std::declval<typename std::add_lvalue_reference<T>::type>()),
             (void)0)> : std::true_type {};

template<class T, bool = is_range<T>::value>
struct range_rank : std::integral_constant<std::size_t, 0> {};
template<class T>
struct range_rank<T, true>
    : std::integral_constant<std::size_t,
                             range_rank<typename T::value_type>::value + 1> {};

template<std::size_t size> struct int_least {
    static_assert(size <= 128, "size must be less than or equal to 128");

    using type = typename std::conditional<
        size <= 8, std::int_least8_t,
        typename std::conditional<
            size <= 16, std::int_least16_t,
            typename std::conditional<
                size <= 32, std::int_least32_t,
                typename std::conditional<size <= 64, std::int_least64_t,
                                          i128>::type>::type>::type>::type;
};

template<std::size_t size> using int_least_t = typename int_least<size>::type;

template<std::size_t size> struct uint_least {
    static_assert(size <= 128, "size must be less than or equal to 128");

    using type = typename std::conditional<
        size <= 8, std::uint_least8_t,
        typename std::conditional<
            size <= 16, std::uint_least16_t,
            typename std::conditional<
                size <= 32, std::uint_least32_t,
                typename std::conditional<size <= 64, std::uint_least64_t,
                                          u128>::type>::type>::type>::type;
};

template<std::size_t size> using uint_least_t = typename uint_least<size>::type;

template<class T>
using double_size_int = int_least<std::numeric_limits<T>::digits * 2 + 1>;
template<class T> using double_size_int_t = typename double_size_int<T>::type;
template<class T>
using double_size_uint = uint_least<std::numeric_limits<T>::digits * 2>;
template<class T> using double_size_uint_t = typename double_size_uint<T>::type;

template<class T>
using double_size =
    typename std::conditional<is_signed_int<T>::value, double_size_int<T>,
                              double_size_uint<T>>::type;
template<class T> using double_size_t = typename double_size<T>::type;
#line 2 "library/template/in.hpp"

#line 4 "library/template/in.hpp"
#include <unistd.h>
#line 8 "library/template/in.hpp"

template<std::size_t buf_size = IO_BUFFER_SIZE> class Reader {
private:
    int fd, idx, sz;
    bool state;
    std::array<char, buf_size> buffer;
    inline void read_buf() {
        sz = read(fd, buffer.begin(), buf_size);
        idx = 0;
        if (sz < 0) throw std::runtime_error("input failed");
    }

public:
    static constexpr int get_buf_size() { return buf_size; }
    Reader() noexcept : fd(0), idx(0), sz(0), state(true) {}
    Reader(int fd) noexcept : fd(fd), idx(0), sz(0), state(true) {}
    Reader(FILE* fp) noexcept : fd(fileno(fp)), idx(0), sz(0), state(true) {}

    class iterator {
    private:
        Reader* reader;

    public:
        using difference_type = void;
        using value_type = void;
        using pointer = void;
        using reference = void;
        using iterator_category = std::input_iterator_tag;

        iterator() : reader(nullptr) {}
        explicit iterator(Reader& reader) : reader(&reader) {}
        explicit iterator(Reader* reader) : reader(reader) {}

        iterator& operator++() {
            if (reader->idx == reader->sz) reader->read_buf();
            ++reader->idx;
            return *this;
        }
        iterator operator++(int) {
            iterator res = *this;
            ++(*this);
            return res;
        }
        char operator*() const {
            if (reader->idx == reader->sz) reader->read_buf();
            if (reader->idx < reader->sz) return reader->buffer[reader->idx];
            reader->state = false;
            return '\0';
        }
        bool rdstate() const { return reader->state; }
        void setstate(bool state) { reader->state = state; }
    };

    iterator begin() noexcept { return iterator(this); }
};

Reader<> reader(0);

template<class Iterator, std::size_t decimal_precision = 16> class Scanner {
public:
    using iterator_type = Iterator;

private:
    template<class, class = void> struct has_scan : std::false_type {};
    template<class T>
    struct has_scan<
        T, decltype(std::declval<T>().scan(std::declval<Scanner&>()), (void)0)>
        : std::true_type {};
    Iterator itr;

public:
    Scanner() = default;
    Scanner(const Iterator& itr) : itr(itr) {}

    char scan_char() {
        char c = *itr;
        ++itr;
        return c;
    }

    Scanner ignore(int n = 1) {
        rep (n) ++itr;
        return *this;
    }

    inline void discard_space() {
        while (('\t' <= *itr && *itr <= '\r') || *itr == ' ') ++itr;
    }
    void scan(char& a) {
        discard_space();
        a = *itr;
        ++itr;
    }
    void scan(bool& a) {
        discard_space();
        a = *itr != '0';
        ++itr;
    }
    void scan(std::string& a) {
        discard_space();
        a.clear();
        while ((*itr < '\t' || '\r' < *itr) && *itr != ' ' && *itr != '\0') {
            a += *itr;
            ++itr;
        }
    }
    template<std::size_t len> void scan(std::bitset<len>& a) {
        discard_space();
        rrep (i, len) {
            a[i] = *itr != '0';
            ++itr;
        }
    }
    template<class T,
             typename std::enable_if<is_signed_int<T>::value &&
                                     !has_scan<T>::value>::type* = nullptr>
    void scan(T& a) {
        discard_space();
        if (*itr == '-') {
            ++itr;
            a = 0;
            while ('0' <= *itr && *itr <= '9') {
                a = a * 10 - (*itr - '0');
                ++itr;
            }
        }
        else {
            a = 0;
            while ('0' <= *itr && *itr <= '9') {
                a = a * 10 + (*itr - '0');
                ++itr;
            }
        }
    }
    template<class T,
             typename std::enable_if<is_unsigned_int<T>::value &&
                                     !has_scan<T>::value>::type* = nullptr>
    void scan(T& a) {
        discard_space();
        a = 0;
        while ('0' <= *itr && *itr <= '9') {
            a = a * 10 + *itr - '0';
            ++itr;
        }
    }
    template<class T,
             typename std::enable_if<std::is_floating_point<T>::value &&
                                     !has_scan<T>::value>::type* = nullptr>
    void scan(T& a) {
        discard_space();
        bool sgn = false;
        if (*itr == '-') {
            sgn = true;
            ++itr;
        }
        a = 0;
        while ('0' <= *itr && *itr <= '9') {
            a = a * 10 + *itr - '0';
            ++itr;
        }
        if (*itr == '.') {
            ++itr;
            T n = 0, d = 1;
            for (int i = 0;
                 '0' <= *itr && *itr <= '9' && i < (int)decimal_precision;
                 ++i) {
                n = n * 10 + *itr - '0';
                d *= 10;
                ++itr;
            }
            while ('0' <= *itr && *itr <= '9') ++itr;
            a += n / d;
        }
        if (sgn) a = -a;
    }

private:
    template<std::size_t i, class... Args> void scan(std::tuple<Args...>& a) {
        if IF_CONSTEXPR (i < sizeof...(Args)) {
            scan(std::get<i>(a));
            scan<i + 1, Args...>(a);
        }
    }

public:
    template<class... Args> void scan(std::tuple<Args...>& a) {
        scan<0, Args...>(a);
    }
    template<class T, class U> void scan(std::pair<T, U>& a) {
        scan(a.first);
        scan(a.second);
    }
    template<class T,
             typename std::enable_if<is_range<T>::value &&
                                     !has_scan<T>::value>::type* = nullptr>
    void scan(T& a) {
        each_for (i : a) scan(i);
    }
    template<class T,
             typename std::enable_if<has_scan<T>::value>::type* = nullptr>
    void scan(T& a) {
        a.scan(*this);
    }

    void operator()() {}
    template<class Head, class... Args>
    void operator()(Head& head, Args&... args) {
        scan(head);
        operator()(args...);
    }

    template<class T> Scanner& operator>>(T& a) {
        scan(a);
        return *this;
    }

    explicit operator bool() const { return itr.rdstate(); }

    friend Scanner& getline(Scanner& scan, std::string& a) {
        a.erase();
        char c;
        if ((c = scan.scan_char()) == '\n' || c == '\0') return scan;
        a += c;
        while ((c = scan.scan_char()) != '\n' && c != '\0') a += c;
        scan.itr.setstate(true);
        return scan;
    }
};

Scanner<Reader<>::iterator> scan(reader.begin());
#line 2 "library/template/out.hpp"

#line 8 "library/template/out.hpp"

template<std::size_t buf_size = IO_BUFFER_SIZE> class Writer {
private:
    int fd, idx;
    std::array<char, buf_size> buffer;
    inline void write_buf() {
        int num = write(fd, buffer.begin(), idx);
        idx = 0;
        if (num < 0) throw std::runtime_error("output failed");
    }

public:
    Writer() noexcept : fd(1), idx(0) {}
    Writer(int fd) noexcept : fd(fd), idx(0) {}
    Writer(FILE* fp) noexcept : fd(fileno(fp)), idx(0) {}

    ~Writer() { write_buf(); }

    class iterator {
    private:
        Writer* writer;

    public:
        using difference_type = void;
        using value_type = void;
        using pointer = void;
        using reference = void;
        using iterator_category = std::output_iterator_tag;

        iterator() noexcept : writer(nullptr) {}
        explicit iterator(Writer& writer) noexcept : writer(&writer) {}
        explicit iterator(Writer* writer) noexcept : writer(writer) {}

        iterator& operator++() {
            ++writer->idx;
            if (writer->idx == buf_size) writer->write_buf();
            return *this;
        }
        iterator operator++(int) {
            iterator res = *this;
            ++(*this);
            return res;
        }
        char& operator*() const { return writer->buffer[writer->idx]; }
        void flush() const { writer->write_buf(); }
    };

    iterator begin() noexcept { return iterator(this); }
};

Writer<> writer(1), ewriter(2);

template<class Iterator, bool debug = false> class Printer {
public:
    using iterator_type = Iterator;

private:
    template<class, bool = debug, class = void>
    struct has_print : std::false_type {};
    template<class T>
    struct has_print<T, false,
                     decltype(std::declval<T>().print(std::declval<Printer&>()),
                              (void)0)> : std::true_type {};
    template<class T>
    struct has_print<T, true,
                     decltype(std::declval<T>().debug(std::declval<Printer&>()),
                              (void)0)> : std::true_type {};
    Iterator itr;
    std::size_t decimal_precision;

public:
    void print_char(char c) {
        *itr = c;
        ++itr;
    }

    void flush() { itr.flush(); }

    Printer() noexcept = default;
    explicit Printer(const Iterator& itr) noexcept
        : itr(itr), decimal_precision(16) {}

    void set_decimal_precision(std::size_t decimal_precision) {
        this->decimal_precision = decimal_precision;
    }

    void print(char c) {
        if IF_CONSTEXPR (debug) print_char('\'');
        print_char(c);
        if IF_CONSTEXPR (debug) print_char('\'');
    }
    void print(bool b) { print_char((char)(b + '0')); }
    void print(const char* a) {
        if IF_CONSTEXPR (debug) print_char('"');
        for (; *a != '\0'; ++a) print_char(*a);
        if IF_CONSTEXPR (debug) print_char('"');
    }
    template<std::size_t len> void print(const char (&a)[len]) {
        if IF_CONSTEXPR (debug) print_char('"');
        for (auto i : a) print_char(i);
        if IF_CONSTEXPR (debug) print_char('"');
    }
    void print(const std::string& a) {
        if IF_CONSTEXPR (debug) print_char('"');
        for (auto i : a) print_char(i);
        if IF_CONSTEXPR (debug) print_char('"');
    }
    template<std::size_t len> void print(const std::bitset<len>& a) {
        rrep (i, len) print_char((char)(a[i] + '0'));
    }
    template<class T,
             typename std::enable_if<is_int<T>::value &&
                                     !has_print<T>::value>::type* = nullptr>
    void print(T a) {
        if (!a) {
            print_char('0');
            return;
        }
        if IF_CONSTEXPR (is_signed_int<T>::value) {
            if (a < 0) {
                print_char('-');
                using U = typename make_unsigned_int<T>::type;
                print(static_cast<U>(-static_cast<U>(a)));
                return;
            }
        }
        std::string s;
        while (a) {
            s += (char)(a % 10 + '0');
            a /= 10;
        }
        for (auto i = s.rbegin(); i != s.rend(); ++i) print_char(*i);
    }
    template<class T,
             typename std::enable_if<std::is_floating_point<T>::value &&
                                     !has_print<T>::value>::type* = nullptr>
    void print(T a) {
        if (a == std::numeric_limits<T>::infinity()) {
            print("inf");
            return;
        }
        if (a == -std::numeric_limits<T>::infinity()) {
            print("-inf");
            return;
        }
        if (std::isnan(a)) {
            print("nan");
            return;
        }
        if (a < 0) {
            print_char('-');
            a = -a;
        }
        T b = a;
        if (b < 1) {
            print_char('0');
        }
        else {
            std::string s;
            while (b >= 1) {
                s += (char)('0' + (int)std::fmod(b, 10.0));
                b /= 10;
            }
            for (auto i = s.rbegin(); i != s.rend(); ++i) print_char(*i);
        }
        print_char('.');
        rep (decimal_precision) {
            a *= 10;
            print_char((char)('0' + (int)std::fmod(a, 10.0)));
        }
    }

private:
    template<std::size_t i, class... Args>
    void print(const std::tuple<Args...>& a) {
        if IF_CONSTEXPR (i < sizeof...(Args)) {
            if IF_CONSTEXPR (debug) print_char(',');
            print_char(' ');
            print(std::get<i>(a));
            print<i + 1, Args...>(a);
        }
    }

public:
    template<class... Args> void print(const std::tuple<Args...>& a) {
        if IF_CONSTEXPR (debug) print_char('(');
        if IF_CONSTEXPR (sizeof...(Args) != 0) print(std::get<0>(a));
        print<1, Args...>(a);
        if IF_CONSTEXPR (debug) print_char(')');
    }
    template<class T, class U> void print(const std::pair<T, U>& a) {
        if IF_CONSTEXPR (debug) print_char('(');
        print(a.first);
        if IF_CONSTEXPR (debug) print_char(',');
        print_char(' ');
        print(a.second);
        if IF_CONSTEXPR (debug) print_char(')');
    }
    template<class T,
             typename std::enable_if<is_range<T>::value &&
                                     !has_print<T>::value>::type* = nullptr>
    void print(const T& a) {
        if IF_CONSTEXPR (debug) print_char('{');
        for (auto i = std::begin(a); i != std::end(a); ++i) {
            if (i != std::begin(a)) {
                if IF_CONSTEXPR (debug) print_char(',');
                print_char(' ');
            }
            print(*i);
        }
        if IF_CONSTEXPR (debug) print_char('}');
    }
    template<class T, typename std::enable_if<has_print<T>::value &&
                                              !debug>::type* = nullptr>
    void print(const T& a) {
        a.print(*this);
    }
    template<class T, typename std::enable_if<has_print<T>::value &&
                                              debug>::type* = nullptr>
    void print(const T& a) {
        a.debug(*this);
    }

    void operator()() {}
    template<class Head, class... Args>
    void operator()(const Head& head, const Args&... args) {
        print(head);
        operator()(args...);
    }

    template<class T> Printer& operator<<(const T& a) {
        print(a);
        return *this;
    }

    Printer& operator<<(Printer& (*pf)(Printer&)) { return pf(*this); }
};

template<class Iterator, bool debug>
Printer<Iterator, debug>& endl(Printer<Iterator, debug>& pr) {
    pr.print_char('\n');
    pr.flush();
    return pr;
}
template<class Iterator, bool debug>
Printer<Iterator, debug>& flush(Printer<Iterator, debug>& pr) {
    pr.flush();
    return pr;
}

struct SetPrec {
    int n;
    template<class Pr> void print(Pr& pr) const { pr.set_decimal_precision(n); }
    template<class Pr> void debug(Pr& pr) const { pr.set_decimal_precision(n); }
};
SetPrec setprec(int n) { return SetPrec{n}; };

Printer<Writer<>::iterator> print(writer.begin()), eprint(ewriter.begin());

void prints() { print.print_char('\n'); }

template<class T> auto prints(const T& v) -> decltype(print << v, (void)0) {
    print << v;
    print.print_char('\n');
}

template<class Head, class... Tail>
auto prints(const Head& head, const Tail&... tail)
    -> decltype(print << head, (void)0) {
    print << head;
    print.print_char(' ');
    prints(tail...);
}

#ifdef SHIO_LOCAL
Printer<Writer<>::iterator, true> debug(writer.begin()),
    edebug(ewriter.begin());
#else
char debug_iterator_character;
class DebugIterator {
public:
    DebugIterator() noexcept = default;
    DebugIterator& operator++() { return *this; }
    DebugIterator& operator++(int) { return *this; }
    char& operator*() const { return debug_iterator_character; }
    void flush() const {}
};
Printer<DebugIterator> debug, edebug;
#endif

void debugs() { debug.print_char('\n'); }

template<class T> auto debugs(const T& v) -> decltype(debug << v, (void)0) {
    debug << v;
    debug.print_char('\n');
}

template<class Head, class... Tail>
auto debugs(const Head& head, const Tail&... tail)
    -> decltype(debug << head, (void)0) {
    debug << head;
    debug.print_char(' ');
    debugs(tail...);
}
#line 2 "library/template/bitop.hpp"

#line 6 "library/template/bitop.hpp"

namespace bitop {

#define KTH_BIT(b, k) (((b) >> (k)) & 1)
#define POW2(k) (1ull << (k))

inline ull next_combination(int n, ull x) {
    if (n == 0) return 1;
    ull a = x & -x;
    ull b = x + a;
    return (x & ~b) / a >> 1 | b;
}

#define rep_comb(i, n, k)                                                      \
    for (ull i = (1ull << (k)) - 1; i < (1ull << (n));                         \
         i = bitop::next_combination((n), i))

inline CONSTEXPR int msb(ull x) {
    int res = x ? 0 : -1;
    if (x & 0xFFFFFFFF00000000) x &= 0xFFFFFFFF00000000, res += 32;
    if (x & 0xFFFF0000FFFF0000) x &= 0xFFFF0000FFFF0000, res += 16;
    if (x & 0xFF00FF00FF00FF00) x &= 0xFF00FF00FF00FF00, res += 8;
    if (x & 0xF0F0F0F0F0F0F0F0) x &= 0xF0F0F0F0F0F0F0F0, res += 4;
    if (x & 0xCCCCCCCCCCCCCCCC) x &= 0xCCCCCCCCCCCCCCCC, res += 2;
    return res + ((x & 0xAAAAAAAAAAAAAAAA) ? 1 : 0);
}

inline CONSTEXPR int ceil_log2(ull x) { return x ? msb(x - 1) + 1 : 0; }

inline CONSTEXPR ull reverse(ull x) {
    x = ((x & 0xAAAAAAAAAAAAAAAA) >> 1) | ((x & 0x5555555555555555) << 1);
    x = ((x & 0xCCCCCCCCCCCCCCCC) >> 2) | ((x & 0x3333333333333333) << 2);
    x = ((x & 0xF0F0F0F0F0F0F0F0) >> 4) | ((x & 0x0F0F0F0F0F0F0F0F) << 4);
    x = ((x & 0xFF00FF00FF00FF00) >> 8) | ((x & 0x00FF00FF00FF00FF) << 8);
    x = ((x & 0xFFFF0000FFFF0000) >> 16) | ((x & 0x0000FFFF0000FFFF) << 16);
    return (x >> 32) | (x << 32);
}

inline CONSTEXPR ull reverse(ull x, int n) { return reverse(x) >> (64 - n); }

} // namespace bitop

inline CONSTEXPR int popcnt(ull x) noexcept {
#if __cplusplus >= 202002L
    return std::popcount(x);
#endif
    x = (x & 0x5555555555555555) + ((x >> 1) & 0x5555555555555555);
    x = (x & 0x3333333333333333) + ((x >> 2) & 0x3333333333333333);
    x = (x & 0x0f0f0f0f0f0f0f0f) + ((x >> 4) & 0x0f0f0f0f0f0f0f0f);
    x = (x & 0x00ff00ff00ff00ff) + ((x >> 8) & 0x00ff00ff00ff00ff);
    x = (x & 0x0000ffff0000ffff) + ((x >> 16) & 0x0000ffff0000ffff);
    return (x & 0x00000000ffffffff) + ((x >> 32) & 0x00000000ffffffff);
}
#line 2 "library/template/func.hpp"

#line 6 "library/template/func.hpp"

template<class T, class U, class Comp = std::less<>>
inline constexpr bool chmin(T& a, const U& b,
                            Comp cmp = Comp()) noexcept(noexcept(cmp(b, a))) {
    return cmp(b, a) ? a = b, true : false;
}
template<class T, class U, class Comp = std::less<>>
inline constexpr bool chmax(T& a, const U& b,
                            Comp cmp = Comp()) noexcept(noexcept(cmp(a, b))) {
    return cmp(a, b) ? a = b, true : false;
}

inline CONSTEXPR ll gcd(ll a, ll b) {
    if (a < 0) a = -a;
    if (b < 0) b = -b;
    while (b) {
        const ll c = a;
        a = b;
        b = c % b;
    }
    return a;
}
inline CONSTEXPR ll lcm(ll a, ll b) { return a / gcd(a, b) * b; }

inline CONSTEXPR bool is_prime(ll N) {
    if (N <= 1) return false;
    for (ll i = 2; i * i <= N; ++i) {
        if (N % i == 0) return false;
    }
    return true;
}
inline std::vector<ll> prime_factor(ll N) {
    std::vector<ll> res;
    for (ll i = 2; i * i <= N; ++i) {
        while (N % i == 0) {
            res.push_back(i);
            N /= i;
        }
    }
    if (N != 1) res.push_back(N);
    return res;
}

inline CONSTEXPR ll my_pow(ll a, ll b) {
    ll res = 1;
    while (b) {
        if (b & 1) res *= a;
        b >>= 1;
        a *= a;
    }
    return res;
}
inline CONSTEXPR ll mod_pow(ll a, ll b, ll mod) {
    assert(mod > 0);
    if (mod == 1) return 0;
    a %= mod;
    ll res = 1;
    while (b) {
        if (b & 1) (res *= a) %= mod;
        b >>= 1;
        (a *= a) %= mod;
    }
    return res;
}

inline PLL extGCD(ll a, ll b) {
    const ll n = a, m = b;
    ll x = 1, y = 0, u = 0, v = 1;
    ll t;
    while (b) {
        t = a / b;
        std::swap(a -= t * b, b);
        std::swap(x -= t * u, u);
        std::swap(y -= t * v, v);
    }
    if (x < 0) {
        x += m;
        y -= n;
    }
    return {x, y};
}
inline ll mod_inv(ll a, ll mod) {
    ll b = mod;
    ll x = 1, u = 0;
    ll t;
    while (b) {
        t = a / b;
        std::swap(a -= t * b, b);
        std::swap(x -= t * u, u);
    }
    if (x < 0) x += mod;
    assert(a == 1);
    return x;
}
#line 2 "library/template/util.hpp"

#line 6 "library/template/util.hpp"

template<class F> class RecLambda {
private:
    F f;

public:
    explicit constexpr RecLambda(F&& f_) : f(std::forward<F>(f_)) {}
    template<class... Args>
    constexpr auto operator()(Args&&... args)
        -> decltype(f(*this, std::forward<Args>(args)...)) {
        return f(*this, std::forward<Args>(args)...);
    }
};

template<class F> inline constexpr RecLambda<F> rec_lambda(F&& f) {
    return RecLambda<F>(std::forward<F>(f));
}


template<class Head, class... Tail> struct multi_dim_vector {
    using type = std::vector<typename multi_dim_vector<Tail...>::type>;
};
template<class T> struct multi_dim_vector<T> { using type = T; };

template<class T, class Arg>
constexpr std::vector<T> make_vec(int n, Arg&& arg) {
    return std::vector<T>(n, std::forward<Arg>(arg));
}
template<class T, class... Args>
constexpr typename multi_dim_vector<Args..., T>::type make_vec(int n,
                                                               Args&&... args) {
    return typename multi_dim_vector<Args..., T>::type(
        n, make_vec<T>(std::forward<Args>(args)...));
}


template<class T, class Comp = std::less<T>> class compressor {
private:
    std::vector<T> dat;
    Comp cmp;
    bool sorted = false;

public:
    compressor() : compressor(Comp()) {}
    compressor(const Comp& cmp) : cmp(cmp) {}
    compressor(const std::vector<T>& vec, bool f = false,
               const Comp& cmp = Comp())
        : dat(vec), cmp(cmp) {
        if (f) build();
    }
    compressor(std::vector<T>&& vec, bool f = false, const Comp& cmp = Comp())
        : dat(std::move(vec)), cmp(cmp) {
        if (f) build();
    }
    compressor(std::initializer_list<T> il, bool f = false,
               const Comp& cmp = Comp())
        : dat(all(il)), cmp(cmp) {
        if (f) build();
    }
    void reserve(int n) {
        assert(!sorted);
        dat.reserve(n);
    }
    void push_back(const T& v) {
        assert(!sorted);
        dat.push_back(v);
    }
    void push_back(T&& v) {
        assert(!sorted);
        dat.push_back(std::move(v));
    }
    template<class... Args> void emplace_back(Args&&... args) {
        assert(!sorted);
        dat.emplace_back(std::forward<Args>(args)...);
    }
    void push(const std::vector<T>& vec) {
        assert(!sorted);
        const int n = dat.size();
        dat.resize(n + vec.size());
        rep (i, vec.size()) dat[n + i] = vec[i];
    }
    int build() {
        assert(!sorted);
        sorted = true;
        std::sort(all(dat), cmp);
        dat.erase(std::unique(all(dat),
                              [&](const T& a, const T& b) -> bool {
                                  return !cmp(a, b) && !cmp(b, a);
                              }),
                  dat.end());
        return dat.size();
    }
    const T& operator[](int k) const& {
        assert(sorted);
        assert(0 <= k && k < (int)dat.size());
        return dat[k];
    }
    int get(const T& val) const {
        assert(sorted);
        auto itr = std::lower_bound(all(dat), val, cmp);
        assert(itr != dat.end() && !cmp(val, *itr));
        return itr - dat.begin();
    }
    int lower_bound(const T& val) const {
        assert(sorted);
        auto itr = std::lower_bound(all(dat), val, cmp);
        return itr - dat.begin();
    }
    int upper_bound(const T& val) const {
        assert(sorted);
        auto itr = std::upper_bound(all(dat), val, cmp);
        return itr - dat.begin();
    }
    bool contains(const T& val) const {
        assert(sorted);
        return std::binary_search(all(dat), val, cmp);
    }
    std::vector<int> pressed(const std::vector<T>& vec) const {
        assert(sorted);
        std::vector<int> res(vec.size());
        rep (i, vec.size()) res[i] = get(vec[i]);
        return res;
    }
    void press(std::vector<T>& vec) const {
        assert(sorted);
        each_for (i : vec) i = get(i);
    }
    int size() const {
        assert(sorted);
        return dat.size();
    }
};
#line 2 "main.cpp"
#define GEOMETRY_REAL_TYPE ll
#line 2 "library/geometry/All.hpp"

#line 2 "library/geometry/template.hpp"

#line 4 "library/geometry/template.hpp"

#ifdef GEOMETRY_EPS
constexpr ld geom_eps = GEOMETRY_EPS;
#else
constexpr ld geom_eps = EPS;
#endif

#ifdef GEOMETRY_REAL_TYPE
using Real = GEOMETRY_REAL_TYPE;
// a <=> b  :  cmp(a, b) <=> 0
inline int cmp(Real a, Real b) {
    if (a > b) return 1;
    if (a < b) return -1;
    return 0;
}
#else
using Real = ld;
// a <=> b  :  cmp(a, b) <=> 0
inline int cmp(ld a, ld b) {
    if (a > b + geom_eps) return 1;
    if (a < b - geom_eps) return -1;
    return 0;
}
#endif

#ifdef GEOMETRY_ANGLE_TYPE
using angle_t = GEOMETRY_ANGLE_TYPE;
#else
using angle_t = ld;
#endif
#line 2 "library/geometry/Point.hpp"

#line 4 "library/geometry/Point.hpp"

class Point {
public:
    Real x, y;
    Point() : x(0), y(0) {}
    Point(Real x, Real y) : x(x), y(y) {}
    Point& operator+=(const Point& p) {
        x += p.x;
        y += p.y;
        return *this;
    }
    Point& operator-=(const Point& p) {
        x -= p.x;
        y -= p.y;
        return *this;
    }
    Point& operator*=(Real a) {
        x *= a;
        y *= a;
        return *this;
    }
    Point& operator/=(Real a) {
        x /= a;
        y /= a;
        return *this;
    }
    Point operator+() const { return *this; }
    Point operator-() const { return Point(-x, -y); }
    friend Point operator+(const Point& p1, const Point& p2) {
        return Point(p1) += p2;
    }
    friend Point operator-(const Point& p1, const Point& p2) {
        return Point(p1) -= p2;
    }
    friend Point operator*(const Point& p, Real a) { return Point(p) *= a; }
    friend Point operator*(Real a, const Point& p) { return Point(p) *= a; }
    friend Point operator/(const Point& p, Real a) { return Point(p) /= a; }
    friend bool operator==(const Point& p1, const Point& p2) {
        return cmp(p1.x, p2.x) == 0 && cmp(p1.y, p2.y) == 0;
    }
    friend bool operator!=(const Point& p1, const Point& p2) {
        return !(p1 == p2);
    }
    friend bool operator<(const Point& p1, const Point& p2) {
        return cmp(p1.x, p2.x) < 0 ||
               (cmp(p1.x, p2.x) == 0 && cmp(p1.y, p2.y) < 0);
    }
    friend bool operator>(const Point& p1, const Point& p2) { return p2 < p1; }
    friend bool operator<=(const Point& p1, const Point& p2) {
        return !(p2 < p1);
    }
    friend bool operator>=(const Point& p1, const Point& p2) {
        return !(p1 < p2);
    }
    friend bool comp_arg(const Point& p1, const Point& p2) {
        // -pi < theta <= pi
        int a1 = p1.y < 0 ? 0 : p1.y > 0 ? 2 : p1.x >= 0 ? 1 : 3;
        int a2 = p2.y < 0 ? 0 : p2.y > 0 ? 2 : p2.x >= 0 ? 1 : 3;
        if (a1 != a2) return a1 < a2;
        return cross(p1, p2) > 0;
    }
    Real norm() const { return x * x + y * y; }
    friend Real norm(const Point& p) { return p.norm(); }
    Real abs() const { return sqrt(norm()); }
    friend Real abs(const Point& p) { return p.abs(); }
    inline angle_t arg() const { return atan2((ld)y, (ld)x); }
    friend angle_t arg(const Point& p) { return p.arg(); }
    Point& rotate(angle_t theta) {
        Real c = cos(theta), s = sin(theta);
        Real nx = x * c - y * s, ny = x * s + y * c;
        x = nx;
        y = ny;
        return *this;
    }
    friend Point rotate(const Point& p, angle_t theta) {
        return Point(p).rotate(theta);
    }
    Point& rotate90() {
        Real nx = -y, ny = x;
        x = nx;
        y = ny;
        return *this;
    }
    friend Point rotate90(const Point& p) { return Point(p).rotate90(); }
    // inner product(内積), p1 * p2 = |p1| * |p2| * cos(theta)
    friend Real dot(const Point& p1, const Point& p2) {
        return p1.x * p2.x + p1.y * p2.y;
    }
    // outer product(外積), p1 ^ p2 = |p1| * |p2| * sin(theta)
    friend Real cross(const Point& p1, const Point& p2) {
        return p1.x * p2.y - p1.y * p2.x;
    }
    template<class Sc> void scan(Sc& scan) { scan >> x >> y; }
    template<class Pr> void print(Pr& print) const { print << x << ' ' << y; }
    template<class Pr> void debug(Pr& print) const {
        print.print_char('(');
        print << x;
        print.print_char(',');
        print << y;
        print.print_char(')');
    }
};

Real distance(const Point& p1, const Point& p2) { return abs(p1 - p2); }

enum class CCW {
    COUNTER_CLOCKWISE = 1,
    CLOCKWISE = -1,
    ONLINE_BACK = 2,
    ONLINE_FRONT = -2,
    ON_SEGMENT = 0,
};

CCW ccw(const Point& p0, const Point& p1, const Point& p2) {
    Point a = p1 - p0, b = p2 - p0;
    if (cmp(cross(a, b), 0) > 0) return CCW::COUNTER_CLOCKWISE;
    if (cmp(cross(a, b), 0) < 0) return CCW::CLOCKWISE;
    if (cmp(dot(a, b), 0) < 0) return CCW::ONLINE_BACK;
    if (a.norm() < b.norm()) return CCW::ONLINE_FRONT;
    return CCW::ON_SEGMENT;
}
#line 2 "library/geometry/Line.hpp"

#line 5 "library/geometry/Line.hpp"

class Line {
public:
    Real a, b, c; // ax + by + c = 0
    Line() : a(0), b(1), c(0) {}
    Line(Real a, Real b, Real c) : a(a), b(b), c(c) {}
    Line(const Point& p1, const Point& p2) {
        a = p2.y - p1.y;
        b = p1.x - p2.x;
        c = p2.x * p1.y - p1.x * p2.y;
    }
    friend bool operator==(const Line& l1, const Line& l2) {
        return cmp(l1.a * l2.b, l2.a * l1.b) == 0 &&
               cmp(l1.b * l2.c, l2.b * l1.c) == 0;
    }
    friend bool operator!=(const Line& l1, const Line& l2) {
        return !(l1 == l2);
    }
    friend bool operator<(const Line& l1, const Line& l2) {
        return cmp(l1.a * l2.b, l2.a * l1.b) < 0 ||
               (cmp(l1.a * l2.b, l2.a * l1.b) == 0 &&
                cmp(l1.b * l2.c, l2.b * l1.c) < 0);
    }
    friend bool operator>(const Line& l1, const Line& l2) { return l2 < l1; }
    friend bool operator<=(const Line& l1, const Line& l2) {
        return !(l2 < l1);
    }
    friend bool operator>=(const Line& l1, const Line& l2) {
        return !(l1 < l2);
    }
    bool is_on(const Point& p) const {
        return cmp(a * p.x + b * p.y + c, 0) == 0;
    }
    template<class Pr> void debug(Pr& print) const {
        print << a;
        print.print_char('x');
        print.print_char('+');
        print << b;
        print.print_char('y');
        print.print_char('+');
        print << c;
        print.print_char('=');
        print.print_char('0');
    }
};

Real distance(const Point& p, const Line& l) {
    return std::abs(l.a * p.x + l.b * p.y + l.c) /
           std::sqrt(l.a * l.a + l.b * l.b);
}
Real distance(const Line& l, const Point& p) { return distance(p, l); }

// 垂直二等分線
Line perpendicular_bisector(const Point& p1, const Point& p2) {
    return Line((p1 + p2) / 2, (p1 + p2) / 2 + (p2 - p1).rotate90());
}

// 平行判定
bool is_parallel(const Line& l1, const Line& l2) {
    return cmp(l1.a * l2.b, l2.a * l1.b) == 0;
}
// 直交判定
bool is_orthogonal(const Line& l1, const Line& l2) {
    return cmp(l1.a * l2.a + l1.b * l2.b, 0) == 0;
}
// 平行線
Line parallel(const Line& l, const Point& p) {
    return Line(l.a, l.b, -l.a * p.x - l.b * p.y);
}
// 垂直線
Line perpendicular(const Line& l, const Point& p) {
    return Line(l.b, -l.a, -l.b * p.x + l.a * p.y);
}

// 交叉判定
bool is_intersect(const Line& l1, const Line& l2) {
    return l1 == l2 || !is_parallel(l1, l2);
}
// 交点
Point intersection(const Line& l1, const Line& l2) {
    assert(!is_parallel(l1, l2));
    Real d = l1.a * l2.b - l2.a * l1.b;
    return Point((l1.b * l2.c - l2.b * l1.c) / d,
                 (l1.c * l2.a - l2.c * l1.a) / d);
}
// 射影
Point projection(const Line& l, const Point& p) {
    return intersection(l, perpendicular(l, p));
}
// 反射
Point reflection(const Line& l, const Point& p) {
    return projection(l, p) * 2 - p;
}
#line 2 "library/geometry/Segment.hpp"

#line 6 "library/geometry/Segment.hpp"

class Segment {
public:
    Point p1, p2;
    Segment() = default;
    Segment(const Point& p1, const Point& p2) : p1(p1), p2(p2) {}
    friend bool operator==(const Segment& s1, const Segment& s2) {
        return s1.p1 == s2.p1 && s1.p2 == s2.p2;
    }
    friend bool operator!=(const Segment& s1, const Segment& s2) {
        return !(s1 == s2);
    }
    friend bool operator<(const Segment& s1, const Segment& s2) {
        return s1.p1 < s2.p1 || (s1.p1 == s2.p1 && s1.p2 < s2.p2);
    }
    friend bool operator>(const Segment& s1, const Segment& s2) {
        return s2 < s1;
    }
    friend bool operator<=(const Segment& s1, const Segment& s2) {
        return !(s2 < s1);
    }
    friend bool operator>=(const Segment& s1, const Segment& s2) {
        return !(s1 < s2);
    }
    bool is_on(const Point& p) const {
        return p == p1 || p == p2 || ccw(p1, p2, p) == CCW::ON_SEGMENT;
    }
    explicit operator Line() const { return Line(p1, p2); }
    template<class Pr> void debug(Pr& print) const {
        print << p1;
        print.print_char('-');
        print.print_char('>');
        print << p2;
    }
    template<class Sc> void scan(Sc& scan) { scan >> p1 >> p2; }
};

bool is_parallel(const Segment& s1, const Segment& s2) {
    return is_parallel(Line(s1), Line(s2));
}
bool is_orthogonal(const Segment& s1, const Segment& s2) {
    return is_orthogonal(Line(s1), Line(s2));
}
Line perpendicular_bisector(const Segment& s) {
    return perpendicular_bisector(s.p1, s.p2);
}

bool is_intersect(const Segment& s1, const Segment& s2) {
    if (is_parallel(s1, s2)) {
        return s1.is_on(s2.p1) || s1.is_on(s2.p2) || s2.is_on(s1.p1) ||
               s2.is_on(s1.p2);
    }
    Point p = intersection(Line(s1), Line(s2));
    return s1.is_on(p) && s2.is_on(p);
}
bool is_intersect(const Segment& s1, const Line& l) {
    if (!is_intersect(Line(s1), l)) return false;
    Point p = intersection(Line(s1), l);
    return s1.is_on(p);
}
bool is_intersect(const Line& l, const Segment& s1) {
    return is_intersect(s1, l);
}

Real distance(const Point& p, const Segment& s) {
    if (s.p1 == s.p2) return distance(p, s.p1);
    if (dot(s.p2 - s.p1, p - s.p1) < 0) return distance(p, s.p1);
    if (dot(s.p1 - s.p2, p - s.p2) < 0) return distance(p, s.p2);
    return distance(p, Line(s));
}
Real distance(const Segment& s, const Point& p) { return distance(p, s); }
Real distance(const Segment& s1, const Segment& s2) {
    if (is_intersect(s1, s2)) return 0;
    return std::min({distance(s1.p1, s2), distance(s1.p2, s2),
                     distance(s2.p1, s1), distance(s2.p2, s1)});
}
Real distance(const Segment& s, const Line& l) {
    if (is_intersect(s, l)) return 0;
    return std::min(distance(s.p1, l), distance(s.p2, l));
}
Real distance(const Line& l, const Segment& s) { return distance(s, l); }
#line 2 "library/geometry/Polygon.hpp"

#line 6 "library/geometry/Polygon.hpp"

class Polygon : public std::vector<Point> {
public:
    using std::vector<Point>::vector;
    explicit Polygon(const std::vector<Point>& v) : std::vector<Point>(v) {}
    explicit Polygon(std::vector<Point>&& v)
        : std::vector<Point>(std::move(v)) {}
};

Real area(const Polygon& p) {
    const int n = p.size();
    Real res = 0;
    rep (i, n) {
        res += cross(p[i], p[(i + 1) % n]);
    }
    return res / 2;
}

bool is_convex(const Polygon& p, bool allow_straight = false) {
    const int n = p.size();
    rep (i, n) {
        CCW c = ccw(p[(i + 1) % n], p[i], p[(i + 2) % n]);
        if (c == CCW::COUNTER_CLOCKWISE ||
            (!allow_straight && c == CCW::ONLINE_BACK)) {
            return false;
        }
    }
    return true;
}

bool contains(const Polygon& p, const Point& q, bool true_when_on_edge = true) {
    const int n = p.size();
    rep (i, n) {
        if (p[i] == q) return true_when_on_edge;
        Point a = p[i] - q;
        Point b = p[(i + 1) % n] - q;
        if (cmp(cross(a, b), 0) == 0 && cmp(dot(a, b), 0) <= 0) {
            return true_when_on_edge;
        }
    }
    bool res = false;
    rep (i, n) {
        Point a = p[i] - q;
        Point b = p[(i + 1) % n] - q;
        if (cmp(a.y, b.y) > 0) std::swap(a, b);
        if (cmp(a.y, 0) <= 0 && cmp(b.y, 0) > 0 && cmp(cross(a, b), 0) < 0) {
            res = !res;
        }
    }
    return res;
}

Polygon convex_hull(std::vector<Point> A, bool allow_straight = false) {
    const int n = A.size();
    if (n <= 2) return Polygon{A};
    std::sort(A.begin(), A.end(), [](const Point& a, const Point& b) {
        return cmp(a.x, b.x) != 0 ? cmp(a.x, b.x) < 0 : cmp(a.y, b.y) < 0;
    });
    Polygon res;
    rep (i, n) {
        while ((int)res.size() >= 2) {
            CCW c = ccw(res[res.size() - 2], res.back(), A[i]);
            if (c == CCW::CLOCKWISE ||
                (!allow_straight && c == CCW::ONLINE_FRONT)) {
                res.pop_back();
            }
            else break;
        }
        res.push_back(A[i]);
    }
    int t = res.size();
    rrep (i, n - 1) {
        while ((int)res.size() >= t + 1) {
            CCW c = ccw(res[res.size() - 2], res.back(), A[i]);
            if (c == CCW::CLOCKWISE ||
                (!allow_straight && c == CCW::ONLINE_FRONT)) {
                res.pop_back();
            }
            else break;
        }
        res.push_back(A[i]);
    }
    res.pop_back();
    return res;
}

std::pair<Point, Point> diameter(const Polygon& p) {
    const int n = p.size();
    int i = 0, j = 0;
    rep (k, n) {
        if (cmp(p[k].x, p[i].x) > 0) i = k;
        if (cmp(p[k].x, p[j].x) < 0) j = k;
    }
    Real res = abs(p[i] - p[j]);
    int ri = i, rj = j;
    int si = i, sj = j;
    do {
        if (cross(p[(i + 1) % n] - p[i], p[(j + 1) % n] - p[j]) < 0) {
            i = (i + 1) % n;
        }
        else {
            j = (j + 1) % n;
        }
        if (chmax(res, abs(p[i] - p[j]),
                  [](const Real& a, const Real& b) { return cmp(a, b) < 0; })) {
            ri = i;
            rj = j;
        }
    } while (i != si || j != sj);
    return {p[ri], p[rj]};
}

std::pair<Point, Point> farthest_pair(const std::vector<Point>& p) {
    auto poly = convex_hull(p);
    return diameter(poly);
}

std::pair<Point, Point> closest_pair(std::vector<Point> p) {
    assert(p.size() >= 2);
    const int n = p.size();
    std::sort(all(p));
    Real res = infinity<Real>::value;
    Point a, b;
    rec_lambda([&](auto&& self, int l, int r) -> void {
        const int m = (l + r) / 2;
        if (r - l <= 1) return;
        const Real x = p[m].x;
        self(l, m);
        self(m, r);
        std::inplace_merge(
            p.begin() + l, p.begin() + m, p.begin() + r,
            [](const Point& a, const Point& b) { return cmp(a.y, b.y) < 0; });
        std::vector<int> B;
        rep (i, l, r) {
            if (cmp(std::abs(p[i].x - x), res) >= 0) continue;
            rrep (j, B.size()) {
                if (cmp(p[i].y - p[B[j]].y, res) >= 0) break;
                if (chmin(res, distance(p[i], p[B[j]]),
                          [](const Real& a, const Real& b) {
                              return cmp(a, b) < 0;
                          })) {
                    a = p[i];
                    b = p[B[j]];
                }
            }
            B.push_back(i);
        }
    })(0, n);
    return {a, b};
}

// cut with line p0-p1 and return left side
Polygon polygon_cut(const Polygon& p, const Point& p0, const Point& p1) {
    const int n = p.size();
    Polygon res;
    rep (i, n) {
        Point a = p[i], b = p[(i + 1) % n];
        Real ca = cross(p0 - a, p1 - a);
        Real cb = cross(p0 - b, p1 - b);
        if (cmp(ca, 0) >= 0) res.push_back(a);
        if (cmp(ca, 0) * cmp(cb, 0) < 0) {
            res.push_back(intersection(Line(a, b), Line(p0, p1)));
        }
    }
    return res;
}
#line 2 "library/geometry/Triangle.hpp"

#line 6 "library/geometry/Triangle.hpp"

class Triangle {
public:
    Point p1, p2, p3;
    Triangle() = default;
    Triangle(const Point& p1, const Point& p2, const Point& p3)
        : p1(p1), p2(p2), p3(p3) {}

    Real area() const { return std::abs(cross(p2 - p1, p3 - p1)) / 2; }
    Point centroid() const { return (p1 + p2 + p3) / 3; }
    Point circumcenter() const {
        Line l1 = perpendicular_bisector(p1, p2);
        Line l2 = perpendicular_bisector(p2, p3);
        return intersection(l1, l2);
    }
    Real circumradius() const { return distance(p1, circumcenter()); }
    Point incenter() const {
        Real a = distance(p2, p3);
        Real b = distance(p3, p1);
        Real c = distance(p1, p2);
        return (a * p1 + b * p2 + c * p3) / (a + b + c);
    }
    Real inradius() const {
        return 2 * area() /
               (distance(p1, p2) + distance(p2, p3) + distance(p3, p1));
    }
    Point orthocenter() const {
        return intersection(perpendicular(Line(p1, p2), p3),
                            perpendicular(Line(p2, p3), p1));
    }
    std::array<Point, 3> excenter() const {
        Real a = distance(p2, p3);
        Real b = distance(p3, p1);
        Real c = distance(p1, p2);
        return {(-a * p1 + b * p2 + c * p3) / (-a + b + c),
                (a * p1 - b * p2 + c * p3) / (a - b + c),
                (a * p1 + b * p2 - c * p3) / (a + b - c)};
    }
    std::array<Real, 3> exradius() const {
        auto a = excenter();
        Line l(p1, p2);
        return {distance(a[0], l), distance(a[1], l), distance(a[2], l)};
    }
    Point nine_point_center() const {
        return (orthocenter() + circumcenter()) / 2;
    }
    Real nine_point_radius() const { return circumradius() / 2; }

    template<class Sc> void scan(Sc& scan) { scan >> p1 >> p2 >> p3; }
    template<class Pr> void debug(Pr& print) const {
        print.print_char('{');
        print << p1;
        print.print_char(' ');
        print << p2;
        print.print_char(' ');
        print << p3;
        print.print_char('}');
    }
};
#line 2 "library/geometry/Circle.hpp"

#line 6 "library/geometry/Circle.hpp"

class Circle {
public:
    Point c;
    Real r;
    Circle() : c(Point()), r(0) {}
    Circle(Point c, Real r) : c(c), r(r) {}
    friend bool operator==(const Circle& c1, const Circle& c2) {
        return c1.c == c2.c && cmp(c1.r, c2.r) == 0;
    }
    friend bool operator!=(const Circle& c1, const Circle& c2) {
        return !(c1 == c2);
    }
    friend bool operator<(const Circle& c1, const Circle& c2) {
        return c1.c < c2.c || (c1.c == c2.c && cmp(c1.r, c2.r) < 0);
    }
    friend bool operator>(const Circle& c1, const Circle& c2) {
        return c2 < c1;
    }
    friend bool operator<=(const Circle& c1, const Circle& c2) {
        return !(c2 < c1);
    }
    friend bool operator>=(const Circle& c1, const Circle& c2) {
        return !(c1 < c2);
    }
    template<class Sc> void scan(Sc& scan) { scan >> c >> r; }
    template<class Pr> void print(Pr& print) { print << c << ' ' << r; }
    template<class Pr> void debug(Pr& print) {
        print.print_char('{');
        print << c;
        print.print_char(':');
        print << r;
        print.print_char('}');
    }
};

enum class circle_relation {
    IN = 0,           // 内包
    INSCRIBE = 1,     // 内接
    INTERSECT = 2,    // 交わる
    CIRCUMSCRIBE = 3, // 外接
    SEPARATE = 4,     // 離れている
    SAME = 5,         // 等しい
};

circle_relation relation(const Circle& c1, const Circle& c2) {
    if (c1 == c2) return circle_relation::SAME;
    const Real d = norm(c1.c - c2.c);
    const Real r1 = c1.r + c2.r, r2 = c1.r - c2.r;
    if (cmp(d, r1 * r1) > 0) return circle_relation::SEPARATE;
    if (cmp(d, r1 * r1) == 0) return circle_relation::CIRCUMSCRIBE;
    if (cmp(d, r2 * r2) > 0) return circle_relation::INTERSECT;
    if (cmp(d, r2 * r2) == 0) return circle_relation::INSCRIBE;
    return circle_relation::IN;
}

std::vector<Point> intersections(const Circle& c, const Line& l) {
    const Point h = projection(l, c.c);
    const Real d = norm(h - c.c);
    if (cmp(d, c.r * c.r) > 0) return {};
    if (cmp(d, c.r * c.r) == 0) return {h};
    const Point v =
        Point(l.b, -l.a) *
        std::sqrt(std::max<Real>((c.r * c.r - d) / (l.a * l.a + l.b * l.b), 0));
    return {h - v, h + v};
}

Line radical_axis(const Circle& c1, const Circle& c2) {
    const Real a = c1.c.x, b = c1.c.y, r = c1.r;
    const Real c = c2.c.x, d = c2.c.y, s = c2.r;
    const Real p = -2 * a + 2 * c, q = -2 * b + 2 * d;
    const Real r2 = a * a + b * b - c * c - d * d - r * r + s * s;
    return Line(p, q, r2);
}

std::vector<Point> intersections(const Circle& c1, const Circle& c2) {
    const Line l = radical_axis(c1, c2);
    return intersections(c1, l);
}

Line tangent_at_point(const Circle& c, const Point& p) {
    assert(cmp(norm(c.c - p), c.r * c.r) == 0);
    const Real a = c.c.x, b = c.c.y;
    const Real px = p.x, py = p.y;
    return Line(px - a, py - b, (a - px) * a + (b - py) * b - c.r * c.r);
}

std::vector<Point> tangent_points(const Circle& c, const Point& p) {
    const Real d = norm(c.c - p);
    const Real r2 = c.r * c.r;
    if (cmp(d, r2) < 0) return {};
    if (cmp(d, r2) == 0) return {p};
    const Circle c2(p, std::sqrt(std::max<Real>(d - r2, 0)));
    return intersections(c, c2);
}

std::vector<Point> common_tangents(const Circle& c1, const Circle& c2) {
    assert(c1 != c2);
    const Real d = norm(c1.c - c2.c);
    const Real r1 = c1.r, r2 = c2.r;
    std::vector<Point> res;
    if (cmp(d, (r1 - r2) * (r1 - r2)) == 0) {
        const Point v = (c2.c - c1.c) * (r1 / std::sqrt(d));
        res.push_back(c1.c + (cmp(r1, r2) < 0 ? -v : v));
    }
    else if (cmp(d, (r1 - r2) * (r1 - r2)) > 0) {
        if (cmp(r1, r2) == 0) {
            const Point v = (c2.c - c1.c).rotate90() * (r1 / std::sqrt(d));
            res.push_back(c1.c + v);
            res.push_back(c1.c - v);
        }
        else {
            const Point v = (c1.c * r2 - c2.c * r1) / (-r1 + r2);
            auto ps = tangent_points(c1, v);
            std::copy(all(ps), std::back_inserter(res));
        }
        if (cmp(d, (r1 + r2) * (r1 + r2)) == 0) {
            const Point v = (c2.c - c1.c) * (r1 / std::sqrt(d));
            res.push_back(c1.c + v);
        }
        else if (cmp(d, (r1 + r2) * (r1 + r2)) > 0) {
            const Point v = (c1.c * r2 + c2.c * r1) / (r1 + r2);
            auto ps = tangent_points(c1, v);
            std::copy(all(ps), std::back_inserter(res));
        }
    }
    return res;
}
#line 4 "main.cpp"

using namespace std;

int main() {
    int q; scan >> q;
    rep (q) {
        Point a, b, c, d; scan >> a >> b >> c >> d;
        auto cc = ccw({0, 0}, a - b, c - d);
        if ((cc == CCW::ON_SEGMENT || cc == CCW::ONLINE_FRONT) && ((a == c && b == d) || (a - b).norm() > (c - d).norm())) prints("Yes");
        else prints("No");
    }
}