// BEGIN: ../sakumon/gosayurusanai/main.cpp
#line 1 "..::sakumon::gosayurusanai::main.cpp"
// BEGIN: pch.hpp
#line 3 "pch.hpp"

#if defined(__GNUC__) && !defined(__clang__)
#pragma GCC optimize("O3")
#pragma GCC optimize("unroll-loops")
#endif

#define dump(...)
#define CPP_DUMP_SET_OPTION(...)
#define CPP_DUMP_DEFINE_EXPORT_OBJECT(...)
#define CPP_DUMP_DEFINE_EXPORT_ENUM(...)
#define CPP_DUMP_DEFINE_DANGEROUS_EXPORT_OBJECT(...)

// BEGIN: template.hpp
#line 3 "template.hpp"

#include <algorithm>
#include <any>
#include <array>
#include <atomic>
#include <barrier>
#include <bit>
#include <bitset>
#include <cassert>
#include <cctype>
#include <cerrno>
#include <cfenv>
#include <cfloat>
#include <charconv>
#include <chrono>
#include <cinttypes>
#include <climits>
#include <clocale>
#include <cmath>
#include <codecvt>
#include <compare>
#include <complex>
#include <concepts>
#include <condition_variable>
#include <coroutine>
#include <cstdint>
#include <cstdio>
#include <cstdlib>
#include <csetjmp>
#include <csignal>
#include <cstdarg>
#include <cstddef>
#include <cstring>
#include <ctime>
#include <cuchar>
#include <cwchar>
#include <cwctype>
#include <deque>
#include <exception>
#include <execution>
#include <filesystem>
#include <format>
#include <forward_list>
#include <fstream>
#include <functional>
#include <future>
#include <iomanip>
#include <initializer_list>
#include <iostream>
#include <ios>
#include <iosfwd>
#include <istream>
#include <iterator>
#include <latch>
#include <limits>
#include <list>
#include <locale>
#include <map>
#include <memory>
#include <memory_resource>
#include <mutex>
#include <new>
#include <numbers>
#include <numeric>
#include <optional>
#include <ostream>
#include <queue>
#include <random>
#include <ranges>
#include <ratio>
#include <regex>
#include <scoped_allocator>
#include <semaphore>
#include <set>
#include <shared_mutex>
#include <source_location>
#include <span>
#include <sstream>
#include <stack>
#include <stdexcept>
#include <stop_token>
#include <streambuf>
#include <string>
#include <string_view>
#include <syncstream>
#include <system_error>
#include <thread>
#include <tuple>
#include <type_traits>
#include <typeindex>
#include <typeinfo>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <valarray>
#include <variant>
#include <vector>
#include <version>

// BEGIN: utilities/fast_io.hpp
#line 3 "utilities::fast_io.hpp"

#include <algorithm>
#include <array>
#include <cerrno>
#include <charconv>
#include <cstddef>
#include <cstdio>
#include <cstdlib>
#include <cstdint>
#include <cstring>
#include <iterator>
#include <string>
#include <sys/stat.h>
#include <type_traits>
#include <utility>
#include <unistd.h>

namespace m1une {
namespace utilities {
namespace internal {

// Detect std::begin(x), std::end(x).
template <class T, class = void>
struct is_range : std::false_type {};

template <class T>
struct is_range<T, std::void_t<
    decltype(std::begin(std::declval<T&>())),
    decltype(std::end(std::declval<T&>()))
>> : std::true_type {};

template <class T>
inline constexpr bool is_range_v = is_range<T>::value;

template <class T>
using range_reference_t = decltype(*std::begin(std::declval<T&>()));

template <class T>
using range_value_t = std::remove_cv_t<std::remove_reference_t<range_reference_t<T>>>;

template <class T, class = void>
struct range_stored_value {
    using type = range_value_t<T>;
};

template <class T>
struct range_stored_value<T, std::void_t<typename std::remove_cv_t<std::remove_reference_t<T>>::value_type>> {
    using type = typename std::remove_cv_t<std::remove_reference_t<T>>::value_type;
};

template <class T>
using range_stored_value_t = typename range_stored_value<T>::type;

// Treat strings and C strings as scalar output objects, not as ranges.
template <class T>
struct is_char_array : std::false_type {};

template <class T, std::size_t N>
struct is_char_array<T[N]>
    : std::bool_constant<std::is_same_v<std::remove_cv_t<T>, char>> {};

template <class T>
struct is_string_like
    : std::bool_constant<
          std::is_same_v<std::decay_t<T>, std::string>
          || std::is_same_v<std::decay_t<T>, const char*>
          || std::is_same_v<std::decay_t<T>, char*>
          || is_char_array<std::remove_reference_t<T>>::value
      > {};

template <class T>
inline constexpr bool is_string_like_v = is_string_like<T>::value;

// ModInt-like type: x.val() is printable, and x can be assigned from long long.
template <class T, class = void>
struct has_val_method : std::false_type {};

template <class T>
struct has_val_method<T, std::void_t<decltype(std::declval<const T&>().val())>>
    : std::true_type {};

template <class T>
inline constexpr bool has_val_method_v = has_val_method<T>::value;

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

template <class T>
struct has_static_mod_raw<
    T, std::void_t<decltype(T::mod()), decltype(T::raw(std::declval<uint32_t>()))>>
    : std::true_type {};

template <class T>
inline constexpr bool has_static_mod_raw_v = has_static_mod_raw<T>::value;

// libstdc++ before GCC 16 does not classify __int128 as an integral type in
// strict ISO modes such as -std=c++23. Keep the fast-I/O interface independent
// of that implementation detail.
template <class T>
inline constexpr bool is_integral_v =
    std::is_integral_v<T>
    || std::is_same_v<std::remove_cv_t<T>, __int128_t>
    || std::is_same_v<std::remove_cv_t<T>, __uint128_t>;

template <class T>
inline constexpr bool is_signed_v =
    std::is_signed_v<T>
    || std::is_same_v<std::remove_cv_t<T>, __int128_t>;

template <class T>
struct make_unsigned {
    using type = std::make_unsigned_t<T>;
};

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

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

template <class T>
using make_unsigned_t = typename make_unsigned<std::remove_cv_t<T>>::type;

}  // namespace internal

struct FastInput {
    static constexpr int buffer_size = 1 << 20;

   private:
    std::FILE* _stream;
    char _buffer[buffer_size];
    int _position;
    int _length;
    int _file_descriptor;
    bool _streaming;

    bool refill() {
        _position = 0;
        if (_streaming) {
            ssize_t length;
            do {
                length = ::read(_file_descriptor, _buffer, buffer_size);
            } while (length < 0 && errno == EINTR);
            if (length <= 0) {
                _length = 0;
                return false;
            }
            _length = int(length);
        } else {
            _length = int(std::fread(_buffer, 1, buffer_size, _stream));
        }
        return _length != 0;
    }

    template <class T>
    bool read_integer_from_stream(T& value) {
        if (!skip_spaces()) return false;
        int c = read_char_raw();

        bool negative = false;
        if (c == '-') {
            negative = true;
            c = read_char_raw();
        }

        if constexpr (internal::is_signed_v<T>) {
            T result = 0;
            while ('0' <= c && c <= '9') {
                result = negative ? result * 10 - (c - '0')
                                  : result * 10 + (c - '0');
                c = read_char_raw();
            }
            value = result;
        } else {
            T result = 0;
            while ('0' <= c && c <= '9') {
                result = result * 10 + T(c - '0');
                c = read_char_raw();
            }
            value = negative ? T(0) - result : result;
        }
        return true;
    }

    bool prepare_number() {
        if (_length - _position >= 64) return true;
        const int remaining = _length - _position;
        if (remaining > 0) std::memmove(_buffer, _buffer + _position, remaining);
        const int added = int(std::fread(_buffer + remaining, 1, buffer_size - remaining, _stream));
        _position = 0;
        _length = remaining + added;
        if (_length < buffer_size) _buffer[_length] = '\0';
        return _length != 0;
    }

   public:
    explicit FastInput(std::FILE* stream = stdin)
        : _stream(stream),
          _position(0),
          _length(0),
          _file_descriptor(::fileno(stream)),
          _streaming([&] {
              struct stat status;
              return _file_descriptor >= 0
                     && ::fstat(_file_descriptor, &status) == 0
                     && !S_ISREG(status.st_mode);
          }()) {}

    FastInput(const FastInput&) = delete;
    FastInput& operator=(const FastInput&) = delete;

    int read_char_raw() {
        if (_position == _length && !refill()) return EOF;
        return _buffer[_position++];
    }

    bool skip_spaces() {
        int c = read_char_raw();
        while (c != EOF && c <= ' ') c = read_char_raw();
        if (c == EOF) return false;
        --_position;
        return true;
    }

    bool read(char& value) {
        if (!skip_spaces()) return false;
        value = char(read_char_raw());
        return true;
    }

    bool read(std::string& value) {
        if (!skip_spaces()) return false;
        value.clear();
        while (true) {
            const int begin = _position;
            while (_position < _length &&
                   static_cast<unsigned char>(_buffer[_position]) > ' ') {
                ++_position;
            }
            value.append(_buffer + begin, _position - begin);
            if (_position < _length) {
                ++_position;
                return true;
            }
            if (!refill()) return true;
        }
    }

    bool read(bool& value) {
        int x;
        if (!read(x)) return false;
        value = x != 0;
        return true;
    }

    template <class T>
    std::enable_if_t<
        internal::is_integral_v<T>
            && !std::is_same_v<std::remove_cv_t<T>, bool>
            && !std::is_same_v<std::remove_cv_t<T>, char>,
        bool
    >
    read(T& value) {
        if (_streaming) return read_integer_from_stream(value);
        if (!prepare_number()) return false;
        int c = static_cast<unsigned char>(_buffer[_position++]);
        while (c <= ' ') c = static_cast<unsigned char>(_buffer[_position++]);

        bool negative = false;
        if (c == '-') {
            negative = true;
            c = static_cast<unsigned char>(_buffer[_position++]);
        }

        if constexpr (internal::is_signed_v<T>) {
            T result = 0;
            while ('0' <= c && c <= '9') {
                const int first = c - '0';
                const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
                if (0 <= second && second <= 9) {
                    result = negative ? result * 100 - (first * 10 + second)
                                      : result * 100 + (first * 10 + second);
                    ++_position;
                } else {
                    result = negative ? result * 10 - first : result * 10 + first;
                }
                c = static_cast<unsigned char>(_buffer[_position++]);
            }
            value = result;
        } else {
            T result = 0;
            while ('0' <= c && c <= '9') {
                const unsigned first = unsigned(c - '0');
                const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
                if (0 <= second && second <= 9) {
                    result = result * 100 + T(first * 10 + unsigned(second));
                    ++_position;
                } else {
                    result = result * 10 + T(first);
                }
                c = static_cast<unsigned char>(_buffer[_position++]);
            }
            value = negative ? T(0) - result : result;
        }
        if (_position > _length) _position = _length;
        return true;
    }

    template <class T>
    std::enable_if_t<std::is_floating_point_v<T>, bool>
    read(T& value) {
        if (!skip_spaces()) return false;
        int c = read_char_raw();
        bool negative = false;
        if (c == '-' || c == '+') {
            negative = c == '-';
            c = read_char_raw();
        }

        long double result = 0;
        while ('0' <= c && c <= '9') {
            result = result * 10 + (c - '0');
            c = read_char_raw();
        }
        if (c == '.') {
            long double place = 0.1L;
            c = read_char_raw();
            while ('0' <= c && c <= '9') {
                result += (c - '0') * place;
                place *= 0.1L;
                c = read_char_raw();
            }
        }
        if (c == 'e' || c == 'E') {
            c = read_char_raw();
            bool exponent_negative = false;
            if (c == '-' || c == '+') {
                exponent_negative = c == '-';
                c = read_char_raw();
            }
            int exponent = 0;
            while ('0' <= c && c <= '9') {
                exponent = exponent * 10 + (c - '0');
                c = read_char_raw();
            }
            long double scale = 1;
            long double power = 10;
            while (exponent > 0) {
                if (exponent & 1) scale *= power;
                power *= power;
                exponent >>= 1;
            }
            result = exponent_negative ? result / scale : result * scale;
        }
        value = static_cast<T>(negative ? -result : result);
        return true;
    }

    template <class T>
    std::enable_if_t<
        internal::has_val_method_v<T>
            && !internal::is_integral_v<T>
            && !internal::is_range_v<T>,
        bool
    >
    read(T& value) {
        long long x;
        if (!read(x)) return false;
        if constexpr (internal::has_static_mod_raw_v<T>) {
            if (x >= 0 && uint64_t(x) < uint64_t(T::mod())) {
                value = T::raw(uint32_t(x));
            } else {
                value = T(x);
            }
        } else {
            value = T(x);
        }
        return true;
    }

    template <class First, class Second>
    bool read(std::pair<First, Second>& value) {
        if (!read(value.first)) return false;
        return read(value.second);
    }

    template <class Range>
    std::enable_if_t<
        internal::is_range_v<Range>
            && !internal::is_string_like_v<Range>,
        bool
    >
    read(Range& range) {
        using StoredValue = internal::range_stored_value_t<Range>;
        constexpr bool nested = internal::is_range_v<StoredValue>
                                && !internal::is_string_like_v<StoredValue>;

        for (auto&& value : range) {
            if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
                bool x;
                if (!read(x)) return false;
                value = x;
            } else {
                if (!read(value)) return false;
            }
        }
        return true;
    }

    template <class First, class Second, class... Rest>
    bool read(First& first, Second& second, Rest&... rest) {
        if (!read(first)) return false;
        return read(second, rest...);
    }

    template <class T>
    FastInput& operator>>(T& value) {
        if (!read(value)) std::abort();
        return *this;
    }
};

struct FastOutput {
    static constexpr int buffer_size = 1 << 20;

   private:
    inline static const auto digit_quads = [] {
        std::array<char, 40000> result{};
        for (int i = 0; i < 10000; i++) {
            int value = i;
            for (int j = 3; j >= 0; j--) {
                result[4 * i + j] = char('0' + value % 10);
                value /= 10;
            }
        }
        return result;
    }();

    std::FILE* _stream;
    char _buffer[buffer_size];
    int _position;
    int _precision;
    std::chars_format _float_format;
    char _range_separator;

   public:
    explicit FastOutput(std::FILE* stream = stdout)
        : _stream(stream),
          _position(0),
          _precision(6),
          _float_format(std::chars_format::general),
          _range_separator(' ') {}

    FastOutput(const FastOutput&) = delete;
    FastOutput& operator=(const FastOutput&) = delete;

    ~FastOutput() {
        flush();
    }

    void flush() {
        if (_position != 0) {
            std::fwrite(_buffer, 1, _position, _stream);
            _position = 0;
        }
        std::fflush(_stream);
    }

    void write_char(char c) {
        if (_position == buffer_size) flush();
        _buffer[_position++] = c;
    }

    void write(const char* s) {
        while (*s != '\0') write_char(*s++);
    }

    void write(const std::string& s) {
        std::size_t position = 0;
        while (position < s.size()) {
            if (_position == buffer_size) flush();
            const std::size_t copied =
                std::min<std::size_t>(buffer_size - _position, s.size() - position);
            std::memcpy(_buffer + _position, s.data() + position, copied);
            _position += int(copied);
            position += copied;
        }
    }

    void write(char c) {
        write_char(c);
    }

    void write(bool value) {
        write_char(value ? '1' : '0');
    }

    template <class T>
    std::enable_if_t<std::is_floating_point_v<T>>
    write(T value) {
        char digits[128];
        auto [end, error] = std::to_chars(
            digits,
            digits + sizeof(digits),
            value,
            _float_format,
            _precision
        );
        if (error != std::errc()) std::abort();
        for (const char* pointer = digits; pointer != end; pointer++) {
            write_char(*pointer);
        }
    }

    template <class T>
    std::enable_if_t<
        internal::is_integral_v<T>
            && !std::is_same_v<std::remove_cv_t<T>, bool>
            && !std::is_same_v<std::remove_cv_t<T>, char>
    >
    write(T value) {
        using Raw = std::remove_cv_t<T>;
        using Unsigned = internal::make_unsigned_t<Raw>;

        Unsigned magnitude;
        if constexpr (internal::is_signed_v<Raw>) {
            if (value < 0) {
                write_char('-');
                magnitude = Unsigned(0) - Unsigned(value);
            } else {
                magnitude = Unsigned(value);
            }
        } else {
            magnitude = value;
        }

        if (magnitude == 0) {
            write_char('0');
            return;
        }

        unsigned chunks[16];
        int count = 0;
        while (magnitude >= 10000) {
            const Unsigned quotient = magnitude / 10000;
            chunks[count++] = unsigned(magnitude - quotient * 10000);
            magnitude = quotient;
        }
        if (_position > buffer_size - 64) flush();
        const unsigned leading = unsigned(magnitude);
        const char* first = digit_quads.data() + 4 * leading;
        int skip = leading < 10 ? 3 : leading < 100 ? 2 : leading < 1000 ? 1 : 0;
        for (; skip < 4; skip++) _buffer[_position++] = first[skip];
        while (count--) {
            const char* digits = digit_quads.data() + 4 * chunks[count];
            std::memcpy(_buffer + _position, digits, 4);
            _position += 4;
        }
    }

    template <class T>
    std::enable_if_t<
        internal::has_val_method_v<T>
            && !internal::is_integral_v<T>
            && !internal::is_range_v<T>
    >
    write(const T& value) {
        write(value.val());
    }

    template <class First, class Second>
    void write(const std::pair<First, Second>& value) {
        write(value.first);
        write_char(' ');
        write(value.second);
    }

    template <class Range>
    std::enable_if_t<
        internal::is_range_v<Range>
            && !internal::is_string_like_v<Range>
    >
    write(const Range& range) {
        using StoredValue = internal::range_stored_value_t<const Range>;
        constexpr bool nested = internal::is_range_v<StoredValue>
                                && !internal::is_string_like_v<StoredValue>;

        bool first = true;
        for (const auto& value : range) {
            if (!first) write_char(nested ? '\n' : _range_separator);
            first = false;
            if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
                write(static_cast<bool>(value));
            } else {
                write(value);
            }
        }
    }

    template <class First, class... Rest>
    void print(const First& first, const Rest&... rest) {
        write(first);
        ((write_char(' '), write(rest)), ...);
    }

    void println() {
        write_char('\n');
    }

    void set_precision(int precision) {
        _precision = precision;
    }

    void set_fixed(int precision = 6) {
        _float_format = std::chars_format::fixed;
        _precision = precision;
    }

    void set_general(int precision = 6) {
        _float_format = std::chars_format::general;
        _precision = precision;
    }

    void set_range_separator(char separator) {
        _range_separator = separator;
    }

    template <class... Args>
    void println(const Args&... args) {
        print(args...);
        write_char('\n');
    }

    template <class T>
    FastOutput& operator<<(const T& value) {
        write(value);
        return *this;
    }
};

}  // namespace utilities
}  // namespace m1une

// END: utilities/fast_io.hpp
#line 103 "template.hpp"
using namespace std;

namespace m1une {
namespace template_io {

inline utilities::FastInput& input() {
    static utilities::FastInput instance;
    return instance;
}

inline utilities::FastOutput& output() {
    static utilities::FastOutput instance;
    return instance;
}

}  // namespace template_io
}  // namespace m1une

using ll = long long;
using u32 = unsigned int;
using u64 = unsigned long long;
using i128 = __int128;
using u128 = unsigned __int128;
#ifdef __SIZEOF_FLOAT128__
using f128 = __float128;
#endif

template <class T>
constexpr T infty = 0;
template <>
constexpr int infty<int> = 1'000'000'000;
template <>
constexpr ll infty<ll> = ll(infty<int>) * infty<int> * 2;
template <>
constexpr u32 infty<u32> = infty<int>;
template <>
constexpr u64 infty<u64> = infty<ll>;
template <>
constexpr i128 infty<i128> = i128(infty<ll>) * infty<ll>;
template <>
constexpr double infty<double> = infty<ll>;
template <>
constexpr long double infty<long double> = infty<ll>;

using pi = pair<int, int>;
using pl = pair<ll, ll>;
using vi = vector<int>;
using vl = vector<ll>;
template <class T>
using vc = vector<T>;
template <class T>
using vvc = vector<vc<T>>;
using vvi = vvc<int>;
using vvl = vvc<ll>;
template <class T>
using vvvc = vector<vvc<T>>;
template <class T>
using vvvvc = vector<vvvc<T>>;
template <class T>
using vvvvvc = vector<vvvvc<T>>;
template <class T>
using pqg = std::priority_queue<T, vector<T>, greater<T>>;
template <class T, class U>
using umap = unordered_map<T, U>;

// template <typename K>
// using tree = __gnu_pbds::tree<K, __gnu_pbds::null_type, std::less<>,
//                               __gnu_pbds::rb_tree_tag,
//                               __gnu_pbds::tree_order_statistics_node_update>;

#define vv(type, name, h, ...) vector<vector<type>> name(h, vector<type>(__VA_ARGS__))
#define vvv(type, name, h, w, ...) \
    vector<vector<vector<type>>> name(h, vector<vector<type>>(w, vector<type>(__VA_ARGS__)))
#define vvvv(type, name, a, b, c, ...)         \
    vector<vector<vector<vector<type>>>> name( \
        a, vector<vector<vector<type>>>(b, vector<vector<type>>(c, vector<type>(__VA_ARGS__))))

#define overload4(a, b, c, d, e, ...) e
#define overload3(a, b, c, d, ...) d

// FOR(a) :=  for (ll _ = 0; _ < (ll)a; ++_)
// FOR(i, a) := for (ll i = 0; i < (ll)a; ++i)
// FOR(i, a, b) := for (ll i = a; i < (ll)b; ++i)
// FOR(i, a, b, c) := for (ll i = a; i < (ll)b; i += (c))
// FOR_R(a) := for (ll i = (a) - 1; i >= 0; --i)
// FOR_R(i, a) := for (ll i = (a) - 1; i >= 0; --i)
// FOR_R(i, a, b) := for (ll i = (b) - 1; i >= (ll)a; --i)
#define FOR1(a) for (ll _ = 0; _ < (ll)a; ++_)
#define FOR2(i, a) for (ll i = 0; i < (ll)a; ++i)
#define FOR3(i, a, b) for (ll i = a; i < (ll)b; ++i)
#define FOR4(i, a, b, c) for (ll i = a; i < (ll)b; i += (c))
#define FOR1_R(a) for (ll i = (a) - 1; i >= 0; --i)
#define FOR2_R(i, a) for (ll i = (a) - 1; i >= 0; --i)
#define FOR3_R(i, a, b) for (ll i = (b) - 1; i >= (ll)a; --i)
#define FOR(...) overload4(__VA_ARGS__, FOR4, FOR3, FOR2, FOR1)(__VA_ARGS__)
#define FOR_R(...) overload3(__VA_ARGS__, FOR3_R, FOR2_R, FOR1_R)(__VA_ARGS__)

#define FORI1(a) for (int _ = 0; _ < (int)a; ++_)
#define FORI2(i, a) for (int i = 0; i < (int)a; ++i)
#define FORI3(i, a, b) for (int i = a; i < (int)b; ++i)
#define FORI4(i, a, b, c) for (int i = a; i < (int)b; i += (c))
#define FORI1_R(a) for (int i = (a) - 1; i >= 0; --i)
#define FORI2_R(i, a) for (int i = (a) - 1; i >= 0; --i)
#define FORI3_R(i, a, b) for (int i = (b) - 1; i >= (int)a; --i)
#define FORI(...) overload4(__VA_ARGS__, FORI4, FORI3, FORI2, FORI1)(__VA_ARGS__)
#define FORI_R(...) overload3(__VA_ARGS__, FORI3_R, FORI2_R, FORI1_R)(__VA_ARGS__)

#define FOR_subset(t, s) for (int t = (s); t >= 0; t = (t == 0 ? -1 : (t - 1) & (s)))
#define all(x) x.begin(), x.end()
#define rall(x) x.rbegin(), x.rend()

int popcnt(int x) {
    return __builtin_popcount(x);
}
int popcnt(u32 x) {
    return __builtin_popcount(x);
}
int popcnt(ll x) {
    return __builtin_popcountll(x);
}
int popcnt(u64 x) {
    return __builtin_popcountll(x);
}
int popcnt_mod_2(int x) {
    return __builtin_parity(x);
}
int popcnt_mod_2(u32 x) {
    return __builtin_parity(x);
}
int popcnt_mod_2(ll x) {
    return __builtin_parityll(x);
}
int popcnt_mod_2(u64 x) {
    return __builtin_parityll(x);
}
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 1, 2)
int topbit(int x) {
    return (x == 0 ? -1 : 31 - __builtin_clz(x));
}
int topbit(u32 x) {
    return (x == 0 ? -1 : 31 - __builtin_clz(x));
}
int topbit(ll x) {
    return (x == 0 ? -1 : 63 - __builtin_clzll(x));
}
int topbit(u64 x) {
    return (x == 0 ? -1 : 63 - __builtin_clzll(x));
}
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 0, 2)
int lowbit(int x) {
    return (x == 0 ? -1 : __builtin_ctz(x));
}
int lowbit(u32 x) {
    return (x == 0 ? -1 : __builtin_ctz(x));
}
int lowbit(ll x) {
    return (x == 0 ? -1 : __builtin_ctzll(x));
}
int lowbit(u64 x) {
    return (x == 0 ? -1 : __builtin_ctzll(x));
}

template <typename T>
T floor(T a, T b) {
    return a / b - (a % b && (a ^ b) < 0);
}
template <typename T>
T ceil(T x, T y) {
    return floor(x + y - 1, y);
}
template <typename T>
T bmod(T x, T y) {
    return x - y * floor(x, y);
}
template <typename T>
pair<T, T> divmod(T x, T y) {
    T q = floor(x, y);
    return {q, x - q * y};
}

template <typename T, typename U>
T POW(U x_, int n) {
    T x = x_;
    T ret = 1;
    while (n > 0) {
        if (n & 1) ret *= x;
        x *= x;
        n >>= 1;
    }
    return ret;
}

template <typename T, typename U>
T SUM(const vector<U>& A) {
    T sm = 0;
    for (auto&& a : A) sm += a;
    return sm;
}

#define LB(c, x) distance((c).begin(), lower_bound(all(c), (x)))
#define UB(c, x) distance((c).begin(), upper_bound(all(c), (x)))
#define UNIQUE(x) sort(all(x)), x.erase(unique(all(x)), x.end()), x.shrink_to_fit()

template <class T, class S>
inline bool chmax(T& a, const S& b) {
    return (a < b ? a = b, 1 : 0);
}
template <class T, class S>
inline bool chmin(T& a, const S& b) {
    return (a > b ? a = b, 1 : 0);
}

// ? は -1
vc<int> s_to_vi(const string& S, char first_char) {
    vc<int> A(S.size());
    FOR(i, S.size()) {
        A[i] = (S[i] != '?' ? S[i] - first_char : -1);
    }
    return A;
}

template <typename T, typename U>
vector<T> cumsum(vector<U>& A, int off = 1) {
    int N = A.size();
    vector<T> B(N + 1);
    FOR(i, N) {
        B[i + 1] = B[i] + A[i];
    }
    if (off == 0) B.erase(B.begin());
    return B;
}

template <typename T>
vector<int> argsort(const vector<T>& A) {
    vector<int> ids(A.size());
    iota(all(ids), 0);
    sort(all(ids), [&](int i, int j) { return (A[i] == A[j] ? i < j : A[i] < A[j]); });
    return ids;
}

// A[I[0]], A[I[1]], ...
template <typename T>
vc<T> rearrange(const vc<T>& A, const vc<int>& I) {
    vc<T> B(I.size());
    FOR(i, I.size()) B[i] = A[I[i]];
    return B;
}

template <class... T>
constexpr auto min(T... a) {
    return min(initializer_list<common_type_t<T...>>{a...});
}
template <class... T>
constexpr auto max(T... a) {
    return max(initializer_list<common_type_t<T...>>{a...});
}

template <class... Ts>
bool scan(Ts&... values) {
    return m1une::template_io::input().read(values...);
}

template <class... Ts>
void print(const Ts&... values) {
    m1une::template_io::output().println(values...);
}
void YESNO(bool b) {
    m1une::template_io::output().println(b ? "YES" : "NO");
}
void YesNo(bool b) {
    m1une::template_io::output().println(b ? "Yes" : "No");
}
void YES() {
    m1une::template_io::output().println("YES");
}
void NO() {
    m1une::template_io::output().println("NO");
}
void Yes() {
    m1une::template_io::output().println("Yes");
}
void No() {
    m1une::template_io::output().println("No");
}
// END: template.hpp
#line 29 "pch.hpp"

// END: pch.hpp
#line 2 "..::sakumon::gosayurusanai::main.cpp"
// BEGIN: graph/dijkstra.hpp
#line 3 "graph::dijkstra.hpp"

#include <algorithm>
#include <cassert>
#include <queue>
#include <utility>
#include <vector>

// BEGIN: graph.hpp
#line 3 "graph.hpp"

#include <cassert>
#include <utility>
#include <vector>

namespace m1une {
namespace graph {

template <class T = int>
struct Edge {
    using cost_type = T;

    int from;
    int to;
    T cost;
    int id;
    bool alive;

    Edge() : from(-1), to(-1), cost(T()), id(-1), alive(true) {}
    Edge(int from_, int to_, T cost_ = T(1), int id_ = -1, bool alive_ = true)
        : from(from_), to(to_), cost(cost_), id(id_), alive(alive_) {}

    int other(int v) const {
        assert(v == from || v == to);
        return from ^ to ^ v;
    }
};

template <class T = int>
struct Graph {
    using edge_type = Edge<T>;
    using cost_type = T;

   private:
    int _n;
    int _edge_count;
    std::vector<std::vector<edge_type>> _g;
    std::vector<std::vector<std::pair<int, int>>> _edge_positions;

   public:
    Graph() : _n(0), _edge_count(0) {}
    explicit Graph(int n) : _n(n), _edge_count(0), _g(n) {
        assert(0 <= n);
    }

    int size() const {
        return _n;
    }

    bool empty() const {
        return _n == 0;
    }

    int edge_count() const {
        return _edge_count;
    }

    int add_vertex() {
        _g.emplace_back();
        return _n++;
    }

    int add_directed_edge(int from, int to, T cost = T(1)) {
        assert(0 <= from && from < _n);
        assert(0 <= to && to < _n);
        int id = _edge_count++;
        int idx = int(_g[from].size());
        _g[from].push_back(edge_type(from, to, cost, id));
        _edge_positions.emplace_back();
        _edge_positions.back().push_back({from, idx});
        return id;
    }

    int add_edge(int u, int v, T cost = T(1)) {
        assert(0 <= u && u < _n);
        assert(0 <= v && v < _n);
        int id = _edge_count++;
        int u_idx = int(_g[u].size());
        _g[u].push_back(edge_type(u, v, cost, id));
        int v_idx = int(_g[v].size());
        _g[v].push_back(edge_type(v, u, cost, id));
        _edge_positions.emplace_back();
        _edge_positions.back().push_back({u, u_idx});
        _edge_positions.back().push_back({v, v_idx});
        return id;
    }

    void set_edge_alive(int id, bool alive) {
        assert(0 <= id && id < _edge_count);
        for (auto [v, idx] : _edge_positions[id]) {
            _g[v][idx].alive = alive;
        }
    }

    void erase_edge(int id) {
        set_edge_alive(id, false);
    }

    void revive_edge(int id) {
        set_edge_alive(id, true);
    }

    bool is_edge_alive(int id) const {
        assert(0 <= id && id < _edge_count);
        assert(!_edge_positions[id].empty());
        auto [v, idx] = _edge_positions[id][0];
        return _g[v][idx].alive;
    }

    const std::vector<edge_type>& operator[](int v) const {
        assert(0 <= v && v < _n);
        return _g[v];
    }

    std::vector<edge_type>& operator[](int v) {
        assert(0 <= v && v < _n);
        return _g[v];
    }

    const std::vector<std::vector<edge_type>>& adjacency() const {
        return _g;
    }

    std::vector<std::vector<edge_type>>& adjacency() {
        return _g;
    }

    std::vector<edge_type> edges(bool include_inactive = false) const {
        std::vector<edge_type> result;
        result.reserve(_edge_count);
        std::vector<char> used(_edge_count, false);
        for (int v = 0; v < _n; v++) {
            for (const auto& e : _g[v]) {
                if (!include_inactive && !e.alive) continue;
                if (0 <= e.id && e.id < _edge_count) {
                    if (used[e.id]) continue;
                    used[e.id] = true;
                }
                result.push_back(e);
            }
        }
        return result;
    }

    Graph reversed() const {
        Graph result(_n);
        result._edge_count = _edge_count;
        result._edge_positions.assign(_edge_count, {});
        for (int v = 0; v < _n; v++) {
            for (const auto& e : _g[v]) {
                int idx = int(result._g[e.to].size());
                result._g[e.to].push_back(edge_type(e.to, e.from, e.cost, e.id, e.alive));
                if (0 <= e.id && e.id < _edge_count) result._edge_positions[e.id].push_back({e.to, idx});
            }
        }
        return result;
    }
};

}  // namespace graph
}  // namespace m1une

// END: graph.hpp
#line 11 "graph::dijkstra.hpp"

namespace m1une {
namespace graph {

template <class T>
struct DijkstraResult {
    std::vector<T> dist;
    std::vector<char> reached;
    std::vector<int> parent;
    std::vector<int> parent_edge;
    T inf = T();

    bool reachable(int v) const {
        assert(0 <= v && v < int(dist.size()));
        return reached[v];
    }

    std::vector<int> path(int t) const {
        assert(reachable(t));
        std::vector<int> result;
        for (int v = t; v != -1; v = parent[v]) result.push_back(v);
        std::reverse(result.begin(), result.end());
        return result;
    }
};

namespace internal {

template <class T>
struct DijkstraQueueNode {
    T dist;
    int vertex;
};

template <class T>
struct DijkstraQueueCompare {
    bool operator()(const DijkstraQueueNode<T>& first,
                    const DijkstraQueueNode<T>& second) const {
        return second.dist < first.dist;
    }
};

}  // namespace internal

template <class T>
DijkstraResult<T> dijkstra(const Graph<T>& g,
                           const std::vector<int>& sources) {
    int n = g.size();
    DijkstraResult<T> result;
    result.dist.resize(n);
    result.reached.assign(n, false);
    result.parent.assign(n, -1);
    result.parent_edge.assign(n, -1);

    using Node = internal::DijkstraQueueNode<T>;
    using Compare = internal::DijkstraQueueCompare<T>;
    std::priority_queue<Node, std::vector<Node>, Compare> que;
    for (int s : sources) {
        assert(0 <= s && s < n);
        if (result.reached[s]) continue;
        result.reached[s] = true;
        result.dist[s] = T();
        que.push(Node{T(), s});
    }

    while (!que.empty()) {
        Node current = que.top();
        que.pop();
        if (result.dist[current.vertex] < current.dist) continue;
        for (const auto& e : g[current.vertex]) {
            if (!e.alive) continue;
            T nd = current.dist + e.cost;
            if (result.reached[e.to] && !(nd < result.dist[e.to])) continue;
            result.reached[e.to] = true;
            result.dist[e.to] = nd;
            result.parent[e.to] = current.vertex;
            result.parent_edge[e.to] = e.id;
            que.push(Node{std::move(nd), e.to});
        }
    }

    return result;
}

template <class T>
DijkstraResult<T> dijkstra(const Graph<T>& g, int s) {
    return dijkstra(g, std::vector<int>{s});
}

// Compatibility overload: unreachable distances are replaced by inf after the
// search. Reachability itself never depends on this sentinel.
template <class T>
DijkstraResult<T> dijkstra(const Graph<T>& g,
                           const std::vector<int>& sources, const T& inf) {
    DijkstraResult<T> result = dijkstra(g, sources);
    result.inf = inf;
    for (int v = 0; v < int(result.dist.size()); v++) {
        if (!result.reachable(v)) result.dist[v] = inf;
    }
    return result;
}

template <class T>
DijkstraResult<T> dijkstra(const Graph<T>& g, int s, const T& inf) {
    return dijkstra(g, std::vector<int>{s}, inf);
}

}  // namespace graph
}  // namespace m1une

// END: graph/dijkstra.hpp
#line 3 "..::sakumon::gosayurusanai::main.cpp"
// BEGIN: utilities/int512.hpp
#line 3 "utilities::int512.hpp"

#include <string>
#include <string_view>

// BEGIN: detail/fixed_int.hpp
#line 3 "detail::fixed_int.hpp"

#include <algorithm>
#include <array>
#include <concepts>
#include <cstddef>
#include <cstdint>
#include <istream>
#include <ostream>
#include <stdexcept>
#include <string>
#include <string_view>
#include <type_traits>
#include <utility>

namespace m1une {
namespace utilities {
namespace detail {

// A signed two's-complement integer whose arithmetic wraps modulo 2^Bits.
// Public aliases select contest-friendly fixed widths in int*.hpp.
template <std::size_t Bits>
class FixedInt {
    static_assert(Bits >= 64);
    static_assert(Bits % 64 == 0);

   private:
    static constexpr std::size_t limb_count = Bits / 64;
    using LimbArray = std::array<std::uint64_t, limb_count>;

   public:
    static constexpr std::size_t bit_width = Bits;

    constexpr FixedInt() = default;

    template <std::integral Integer>
    constexpr FixedInt(Integer value) {
        static_assert(sizeof(Integer) <= sizeof(std::uint64_t));
        if constexpr (std::signed_integral<Integer>) {
            const std::uint64_t extension =
                value < 0 ? ~std::uint64_t(0) : std::uint64_t(0);
            limbs_.fill(extension);
            limbs_[0] = static_cast<std::uint64_t>(
                static_cast<std::int64_t>(value)
            );
        } else {
            limbs_[0] = static_cast<std::uint64_t>(value);
        }
    }

    explicit FixedInt(std::string_view text) { read(text); }

    FixedInt& operator=(std::string_view text) {
        read(text);
        return *this;
    }

    void read(std::string_view text) {
        if (text.empty()) {
            throw std::invalid_argument("empty fixed-width integer");
        }
        const bool negative = text.front() == '-';
        std::size_t position =
            (text.front() == '-' || text.front() == '+') ? 1 : 0;
        if (position == text.size()) {
            throw std::invalid_argument("invalid fixed-width integer");
        }

        FixedInt result;
        for (; position < text.size(); ++position) {
            const char digit = text[position];
            if (digit < '0' || digit > '9') {
                throw std::invalid_argument("invalid fixed-width integer");
            }
            result.multiply_unsigned_small(10);
            result += FixedInt(static_cast<unsigned>(digit - '0'));
        }
        *this = negative ? -result : result;
    }

    constexpr bool is_zero() const {
        for (const std::uint64_t limb : limbs_) {
            if (limb != 0) return false;
        }
        return true;
    }

    constexpr bool is_negative() const {
        return (limbs_.back() >> 63) != 0;
    }

    constexpr int sign() const {
        if (is_zero()) return 0;
        return is_negative() ? -1 : 1;
    }

    constexpr FixedInt operator+() const { return *this; }

    constexpr FixedInt operator-() const {
        FixedInt result;
        result.limbs_ = limbs_;
        negate_unsigned(result.limbs_);
        return result;
    }

    constexpr FixedInt& operator+=(const FixedInt& other) {
        __uint128_t carry = 0;
        for (std::size_t index = 0; index < limb_count; ++index) {
            const __uint128_t current =
                __uint128_t(limbs_[index]) + other.limbs_[index] + carry;
            limbs_[index] = static_cast<std::uint64_t>(current);
            carry = current >> 64;
        }
        return *this;
    }

    constexpr FixedInt& operator-=(const FixedInt& other) {
        return *this += -other;
    }

    constexpr FixedInt& operator*=(const FixedInt& other) {
        LimbArray product{};
        for (std::size_t first = 0; first < limb_count; ++first) {
            __uint128_t carry = 0;
            for (
                std::size_t second = 0;
                first + second < limb_count;
                ++second
            ) {
                const std::size_t position = first + second;
                const __uint128_t current =
                    __uint128_t(limbs_[first]) * other.limbs_[second] +
                    product[position] + carry;
                product[position] = static_cast<std::uint64_t>(current);
                carry = current >> 64;
            }
        }
        limbs_ = product;
        return *this;
    }

    constexpr FixedInt& multiply_small(std::uint64_t value) {
        multiply_unsigned_small(value);
        return *this;
    }

    constexpr FixedInt& operator/=(const FixedInt& other) {
        return *this = divmod(*this, other).first;
    }

    constexpr FixedInt& operator%=(const FixedInt& other) {
        return *this = divmod(*this, other).second;
    }

    std::string to_string() const {
        if (is_zero()) return "0";
        const bool negative = is_negative();
        LimbArray magnitude = unsigned_magnitude();

        std::string result;
        while (!magnitude_is_zero(magnitude)) {
            const unsigned digit = divide_unsigned_by_ten(magnitude);
            result.push_back(static_cast<char>('0' + digit));
        }
        if (negative) result.push_back('-');
        std::reverse(result.begin(), result.end());
        return result;
    }

    friend constexpr std::pair<FixedInt, FixedInt> divmod(
        const FixedInt& dividend,
        const FixedInt& divisor
    ) {
        if (divisor.is_zero()) {
            throw std::domain_error("fixed-width integer division by zero");
        }

        const bool quotient_negative =
            dividend.is_negative() != divisor.is_negative();
        const bool remainder_negative = dividend.is_negative();
        auto [quotient_limbs, remainder_limbs] = divide_unsigned(
            dividend.unsigned_magnitude(), divisor.unsigned_magnitude()
        );

        FixedInt quotient;
        FixedInt remainder;
        quotient.limbs_ = quotient_limbs;
        remainder.limbs_ = remainder_limbs;
        if (quotient_negative) quotient = -quotient;
        if (remainder_negative) remainder = -remainder;
        return std::make_pair(quotient, remainder);
    }

    friend constexpr std::pair<FixedInt, std::int64_t> divmod_small(
        const FixedInt& dividend,
        std::uint32_t divisor
    ) {
        if (divisor == 0) {
            throw std::domain_error("fixed-width integer division by zero");
        }

        LimbArray quotient_limbs = dividend.unsigned_magnitude();
        const std::uint64_t unsigned_remainder =
            divide_unsigned_by_small(quotient_limbs, divisor);
        FixedInt quotient;
        quotient.limbs_ = quotient_limbs;
        if (dividend.is_negative()) quotient = -quotient;
        const std::int64_t remainder = dividend.is_negative()
                                           ? -std::int64_t(unsigned_remainder)
                                           : std::int64_t(unsigned_remainder);
        return std::make_pair(quotient, remainder);
    }

    friend constexpr FixedInt operator+(
        FixedInt first,
        const FixedInt& second
    ) {
        return first += second;
    }

    friend constexpr FixedInt operator-(
        FixedInt first,
        const FixedInt& second
    ) {
        return first -= second;
    }

    friend constexpr FixedInt operator*(
        FixedInt first,
        const FixedInt& second
    ) {
        return first *= second;
    }

    friend constexpr FixedInt operator/(
        FixedInt first,
        const FixedInt& second
    ) {
        return first /= second;
    }

    friend constexpr FixedInt operator%(
        FixedInt first,
        const FixedInt& second
    ) {
        return first %= second;
    }

    friend constexpr bool operator==(
        const FixedInt& first,
        const FixedInt& second
    ) = default;

    friend constexpr bool operator<(
        const FixedInt& first,
        const FixedInt& second
    ) {
        const bool first_negative = first.is_negative();
        const bool second_negative = second.is_negative();
        if (first_negative != second_negative) return first_negative;
        return compare_unsigned(first.limbs_, second.limbs_) < 0;
    }

    friend constexpr bool operator!=(
        const FixedInt& first,
        const FixedInt& second
    ) {
        return !(first == second);
    }

    friend constexpr bool operator>(
        const FixedInt& first,
        const FixedInt& second
    ) {
        return second < first;
    }

    friend constexpr bool operator<=(
        const FixedInt& first,
        const FixedInt& second
    ) {
        return !(second < first);
    }

    friend constexpr bool operator>=(
        const FixedInt& first,
        const FixedInt& second
    ) {
        return !(first < second);
    }

    friend std::ostream& operator<<(
        std::ostream& output,
        const FixedInt& value
    ) {
        return output << value.to_string();
    }

    friend std::istream& operator>>(
        std::istream& input,
        FixedInt& value
    ) {
        std::string text;
        if (input >> text) value.read(text);
        return input;
    }

   private:
    LimbArray limbs_{};

    constexpr LimbArray unsigned_magnitude() const {
        LimbArray result = limbs_;
        if (is_negative()) negate_unsigned(result);
        return result;
    }

    constexpr void multiply_unsigned_small(std::uint64_t value) {
        __uint128_t carry = 0;
        for (std::size_t index = 0; index < limb_count; ++index) {
            const __uint128_t current =
                __uint128_t(limbs_[index]) * value + carry;
            limbs_[index] = static_cast<std::uint64_t>(current);
            carry = current >> 64;
        }
    }

    static constexpr void negate_unsigned(LimbArray& value) {
        for (std::uint64_t& limb : value) limb = ~limb;
        for (std::size_t index = 0; index < limb_count; ++index) {
            if (++value[index] != 0) break;
        }
    }

    static constexpr int compare_unsigned(
        const LimbArray& first,
        const LimbArray& second
    ) {
        for (std::size_t offset = 0; offset < limb_count; ++offset) {
            const std::size_t index = limb_count - 1 - offset;
            if (first[index] != second[index]) {
                return first[index] < second[index] ? -1 : 1;
            }
        }
        return 0;
    }

    static constexpr void subtract_unsigned(
        LimbArray& first,
        const LimbArray& second
    ) {
        std::uint64_t borrow = 0;
        for (std::size_t index = 0; index < limb_count; ++index) {
            const std::uint64_t previous = first[index];
            first[index] -= second[index] + borrow;
            const bool addition_overflow =
                borrow != 0 && second[index] == ~std::uint64_t(0);
            borrow = addition_overflow ||
                     previous < second[index] + borrow;
        }
    }

    static constexpr void shift_left_one(LimbArray& value) {
        std::uint64_t carry = 0;
        for (std::size_t index = 0; index < limb_count; ++index) {
            const std::uint64_t next_carry = value[index] >> 63;
            value[index] = (value[index] << 1) | carry;
            carry = next_carry;
        }
    }

    static constexpr std::pair<LimbArray, LimbArray> divide_unsigned(
        const LimbArray& dividend,
        const LimbArray& divisor
    ) {
        LimbArray quotient{};
        LimbArray remainder{};
        for (std::size_t offset = 0; offset < Bits; ++offset) {
            const std::size_t bit = Bits - 1 - offset;
            shift_left_one(remainder);
            remainder[0] |=
                (dividend[bit / 64] >> (bit % 64)) & std::uint64_t(1);
            if (compare_unsigned(remainder, divisor) >= 0) {
                subtract_unsigned(remainder, divisor);
                quotient[bit / 64] |= std::uint64_t(1) << (bit % 64);
            }
        }
        return std::make_pair(quotient, remainder);
    }

    static bool magnitude_is_zero(const LimbArray& value) {
        for (const std::uint64_t limb : value) {
            if (limb != 0) return false;
        }
        return true;
    }

    static constexpr std::uint64_t divide_unsigned_by_small(
        LimbArray& value,
        std::uint64_t divisor
    ) {
        __uint128_t remainder = 0;
        for (std::size_t offset = 0; offset < limb_count; ++offset) {
            const std::size_t index = limb_count - 1 - offset;
            const __uint128_t current =
                (remainder << 64) | value[index];
            value[index] = static_cast<std::uint64_t>(current / divisor);
            remainder = current % divisor;
        }
        return static_cast<std::uint64_t>(remainder);
    }

    static unsigned divide_unsigned_by_ten(LimbArray& value) {
        return static_cast<unsigned>(divide_unsigned_by_small(value, 10));
    }
};

}  // namespace detail
}  // namespace utilities
}  // namespace m1une

// END: detail/fixed_int.hpp
#line 8 "utilities::int512.hpp"

namespace m1une {
namespace utilities {

using Int512 = detail::FixedInt<512>;
using i512 = Int512;

inline Int512 parse_int512(std::string_view text) {
    return Int512(text);
}

inline std::string to_string(const Int512& value) {
    return value.to_string();
}

}  // namespace utilities
}  // namespace m1une

// END: utilities/int512.hpp
#line 4 "..::sakumon::gosayurusanai::main.cpp"

auto& fastin = m1une::template_io::input();
auto& fastout = m1une::template_io::output();

using Int512 = m1une::utilities::Int512;

std::vector<std::pair<int, int>> prime_powers() {
    std::vector<std::pair<int, int>> result;
    std::array<bool, 301> composite{};
    for (int p = 2; p <= 300; ++p) {
        if (composite[p]) continue;
        for (int multiple = p + p; multiple <= 300; multiple += p) {
            composite[multiple] = true;
        }
        int exponent = 0;
        int power = 1;
        while (power <= 300 / p) {
            power *= p;
            ++exponent;
        }
        result.emplace_back(p, exponent);
    }
    return result;
}

void solve() {
    const std::vector<std::pair<int, int>> factors = prime_powers();
    Int512 common_denominator = 1;
    for (auto [prime, exponent] : factors) {
        for (int i = 0; i < exponent; ++i) {
            common_denominator.multiply_small(prime);
        }
    }

    std::array<Int512, 301> scale;
    for (int denominator = 1; denominator <= 300; ++denominator) {
        auto [quotient, remainder] =
            divmod_small(common_denominator, denominator);
        assert(remainder == 0);
        scale[denominator] = quotient;
    }

    int N, M;
    scan(N, M);
    m1une::graph::Graph<Int512> graph(N);
    while (M--) {
        int u, v, a, b;
        scan(u, v, a, b);
        --u;
        --v;
        Int512 cost = scale[b];
        cost.multiply_small(a);
        graph.add_edge(u, v, cost);
    }

    const auto shortest = m1une::graph::dijkstra(graph, 0);

    for (int v = 1; v < N; ++v) {
        Int512 numerator = shortest.dist[v];
        Int512 denominator = 1;
        for (auto [prime, exponent] : factors) {
            int removed = 0;
            while (removed < exponent) {
                auto [quotient, remainder] = divmod_small(numerator, prime);
                if (remainder != 0) break;
                numerator = quotient;
                ++removed;
            }
            for (int i = removed; i < exponent; ++i) {
                denominator.multiply_small(prime);
            }
        }
        print(numerator.to_string(), denominator.to_string());
    }
}

int main() {
    int T = 1;
    while (T--) solve();
    return 0;
}
// END: ../sakumon/gosayurusanai/main.cpp