// BEGIN: ../contest596/A/main.cpp
#line 1 "..::contest596::A::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 <array>
#include <charconv>
#include <cstddef>
#include <cstdio>
#include <cstdlib>
#include <cstdint>
#include <cstring>
#include <iterator>
#include <string>
#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;
    bool _terminal;

    bool refill() {
        _position = 0;
        if (_terminal) {
            if (std::fgets(_buffer, buffer_size, _stream) == nullptr) {
                _length = 0;
                return false;
            }
            _length = int(std::strlen(_buffer));
        } else {
            _length = int(std::fread(_buffer, 1, buffer_size, _stream));
        }
        return _length != 0;
    }

    template <class T>
    bool read_integer_from_terminal(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),
          _terminal(::isatty(::fileno(stream)) != 0) {}

    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();
        int c = read_char_raw();
        while (c != EOF && c > ' ') {
            value.push_back(char(c));
            c = read_char_raw();
        }
        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 (_terminal) return read_integer_from_terminal(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) return;
        std::fwrite(_buffer, 1, _position, _stream);
        _position = 0;
    }

    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) {
        for (char c : s) write_char(c);
    }

    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 "..::contest596::A::main.cpp"

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

// BEGIN: ds/dynamic_array/dynamic_array.hpp
#line 3 "ds::dynamic_array::dynamic_array.hpp"

#include <cassert>
#include <chrono>
#include <cstdint>
#include <initializer_list>
#include <utility>
#include <vector>

namespace m1une {
namespace ds {

template <typename T>
struct DynamicArray {
   private:
    struct Node {
        T val;
        int priority;
        int count;
        int l, r;
        bool rev;

        Node() : val(T()), priority(0), count(0), l(0), r(0), rev(false) {}
        Node(T value, int node_priority)
            : val(std::move(value)), priority(node_priority), count(1), l(0), r(0), rev(false) {}
    };

    std::vector<Node> pool;
    int root;
    std::uint32_t rng_state;

    int new_node(T val) {
        pool.push_back(Node(std::move(val), next_priority()));
        return pool.size() - 1;
    }

    int next_priority() {
        rng_state ^= rng_state << 13;
        rng_state ^= rng_state >> 17;
        rng_state ^= rng_state << 5;
        return int(rng_state);
    }

    void update(int t) {
        if (t) {
            pool[t].count = 1 + pool[pool[t].l].count + pool[pool[t].r].count;
        }
    }

    void apply_reverse(int t) {
        if (t) {
            pool[t].rev = !pool[t].rev;
        }
    }

    void push(int t) {
        if (!t || !pool[t].rev) return;
        std::swap(pool[t].l, pool[t].r);
        apply_reverse(pool[t].l);
        apply_reverse(pool[t].r);
        pool[t].rev = false;
    }

    void split(int t, int pos, int& l, int& r) {
        if (!t) {
            l = r = 0;
            return;
        }
        if (pos == 0) {
            l = 0;
            r = t;
            return;
        }
        if (pos == pool[t].count) {
            l = t;
            r = 0;
            return;
        }
        push(t);
        int left_count = pool[pool[t].l].count;
        if (pos == left_count) {
            l = pool[t].l;
            pool[t].l = 0;
            update(t);
            r = t;
            return;
        }
        if (pos == left_count + 1) {
            r = pool[t].r;
            pool[t].r = 0;
            update(t);
            l = t;
            return;
        }
        if (pos <= left_count) {
            split(pool[t].l, pos, l, pool[t].l);
            r = t;
        } else {
            split(pool[t].r, pos - left_count - 1, pool[t].r, r);
            l = t;
        }
        update(t);
    }

    int merge(int l, int r) {
        if (!l || !r) return l ? l : r;
        if (pool[l].priority > pool[r].priority) {
            push(l);
            if (pool[l].r) {
                pool[l].r = merge(pool[l].r, r);
            } else {
                pool[l].r = r;
            }
            update(l);
            return l;
        } else {
            push(r);
            if (pool[r].l) {
                pool[r].l = merge(l, pool[r].l);
            } else {
                pool[r].l = l;
            }
            update(r);
            return r;
        }
    }

    int insert_node(int t, int pos, int node) {
        if (!t) return node;
        if (pool[node].priority > pool[t].priority) {
            split(t, pos, pool[node].l, pool[node].r);
            update(node);
            return node;
        }
        push(t);
        int left_count = pool[pool[t].l].count;
        if (pos <= left_count) {
            pool[t].l = insert_node(pool[t].l, pos, node);
        } else {
            pool[t].r = insert_node(pool[t].r, pos - left_count - 1, node);
        }
        update(t);
        return t;
    }

    int erase_node(int t, int pos) {
        push(t);
        int left_count = pool[pool[t].l].count;
        if (pos < left_count) {
            pool[t].l = erase_node(pool[t].l, pos);
            update(t);
            return t;
        }
        if (pos == left_count) {
            return merge(pool[t].l, pool[t].r);
        }
        pool[t].r = erase_node(pool[t].r, pos - left_count - 1);
        update(t);
        return t;
    }

    int find_node(int t, int pos) {
        while (t) {
            push(t);
            int left_count = pool[pool[t].l].count;
            if (pos < left_count) {
                t = pool[t].l;
            } else if (pos == left_count) {
                return t;
            } else {
                pos -= left_count + 1;
                t = pool[t].r;
            }
        }
        return 0;
    }

    int find_node(int t, int pos, bool reversed) const {
        while (t) {
            bool cur_reversed = reversed ^ pool[t].rev;
            int left = cur_reversed ? pool[t].r : pool[t].l;
            int right = cur_reversed ? pool[t].l : pool[t].r;
            int left_count = pool[left].count;
            if (pos < left_count) {
                t = left;
                reversed = cur_reversed;
            } else if (pos == left_count) {
                return t;
            } else {
                pos -= left_count + 1;
                t = right;
                reversed = cur_reversed;
            }
        }
        return 0;
    }

    void dump_dfs(int t, std::vector<T>& res, bool reversed = false) const {
        if (!t) return;
        bool cur_reversed = reversed ^ pool[t].rev;
        int left = cur_reversed ? pool[t].r : pool[t].l;
        int right = cur_reversed ? pool[t].l : pool[t].r;
        dump_dfs(left, res, cur_reversed);
        res.push_back(pool[t].val);
        dump_dfs(right, res, cur_reversed);
    }

    void dump_range_dfs(int t, int ql, int qr, int offset, std::vector<T>& res, bool reversed = false) const {
        if (!t || qr <= offset || offset + pool[t].count <= ql) return;
        bool cur_reversed = reversed ^ pool[t].rev;
        int left = cur_reversed ? pool[t].r : pool[t].l;
        int right = cur_reversed ? pool[t].l : pool[t].r;
        int left_count = pool[left].count;
        int node_pos = offset + left_count;
        dump_range_dfs(left, ql, qr, offset, res, cur_reversed);
        if (ql <= node_pos && node_pos < qr) {
            res.push_back(pool[t].val);
        }
        dump_range_dfs(right, ql, qr, node_pos + 1, res, cur_reversed);
    }

    int clone_subtree_from(const DynamicArray& other, int t) {
        if (!t) return 0;
        int res = static_cast<int>(pool.size());
        pool.push_back(other.pool[t]);
        pool[res].l = clone_subtree_from(other, other.pool[t].l);
        pool[res].r = clone_subtree_from(other, other.pool[t].r);
        return res;
    }

    void update_dfs(int t) {
        if (!t) return;
        update_dfs(pool[t].l);
        update_dfs(pool[t].r);
        update(t);
    }

    int build_cartesian(int first, int last) {
        if (first == last) return 0;
        std::vector<int> stack;
        stack.reserve(last - first);
        for (int i = first; i < last; i++) {
            int left_child = 0;
            while (!stack.empty() && pool[stack.back()].priority < pool[i].priority) {
                left_child = stack.back();
                stack.pop_back();
            }
            pool[i].l = left_child;
            if (!stack.empty()) {
                pool[stack.back()].r = i;
            }
            stack.push_back(i);
        }
        int res = stack.front();
        update_dfs(res);
        return res;
    }

    int build_from_vector(const std::vector<T>& v) {
        int first = int(pool.size());
        pool.reserve(pool.size() + v.size());
        for (const T& x : v) {
            new_node(x);
        }
        return build_cartesian(first, int(pool.size()));
    }

    int build_from_vector(std::vector<T>&& v) {
        int first = int(pool.size());
        pool.reserve(pool.size() + v.size());
        for (T& x : v) {
            new_node(std::move(x));
        }
        return build_cartesian(first, int(pool.size()));
    }

    void reset_to_empty() {
        pool.clear();
        pool.push_back(Node());
        root = 0;
    }

   public:
    DynamicArray() : root(0), rng_state(std::uint32_t(std::chrono::steady_clock::now().time_since_epoch().count())) {
        pool.push_back(Node());
        if (rng_state == 0) rng_state = 1;
    }

    DynamicArray(const DynamicArray& other) : pool(other.pool), root(other.root), rng_state(other.rng_state) {}

    DynamicArray(DynamicArray&& other) noexcept
        : pool(std::move(other.pool)), root(other.root), rng_state(other.rng_state) {
        other.reset_to_empty();
    }

    DynamicArray& operator=(const DynamicArray& other) {
        if (this != &other) {
            pool = other.pool;
            root = other.root;
            rng_state = other.rng_state;
        }
        return *this;
    }

    DynamicArray& operator=(DynamicArray&& other) noexcept {
        if (this != &other) {
            pool = std::move(other.pool);
            root = other.root;
            rng_state = other.rng_state;
            other.reset_to_empty();
        }
        return *this;
    }

    explicit DynamicArray(int n) : DynamicArray(n, T()) {}

    DynamicArray(int n, const T& value) : DynamicArray() {
        assert(0 <= n);
        pool.reserve(n + 1);
        int first = int(pool.size());
        for (int i = 0; i < n; i++) {
            new_node(value);
        }
        root = build_cartesian(first, int(pool.size()));
    }

    explicit DynamicArray(const std::vector<T>& v) : DynamicArray() {
        pool.reserve(v.size() + 1);
        root = build_from_vector(v);
    }

    explicit DynamicArray(std::vector<T>&& v) : DynamicArray() {
        pool.reserve(v.size() + 1);
        root = build_from_vector(std::move(v));
    }

    DynamicArray(std::initializer_list<T> init) : DynamicArray() {
        pool.reserve(init.size() + 1);
        for (const T& x : init) push_back(x);
    }

    int size() const {
        return pool[root].count;
    }

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

    void clear() {
        reset_to_empty();
    }

    void insert(int pos, T val) {
        assert(0 <= pos && pos <= size());
        root = insert_node(root, pos, new_node(std::move(val)));
    }

    void insert(int pos, const std::vector<T>& v) {
        assert(0 <= pos && pos <= size());
        pool.reserve(pool.size() + v.size());
        int mid = build_from_vector(v);
        int l, r;
        split(root, pos, l, r);
        root = merge(merge(l, mid), r);
    }

    void insert(int pos, std::vector<T>&& v) {
        assert(0 <= pos && pos <= size());
        pool.reserve(pool.size() + v.size());
        int mid = build_from_vector(std::move(v));
        int l, r;
        split(root, pos, l, r);
        root = merge(merge(l, mid), r);
    }

    void insert(int pos, std::initializer_list<T> init) {
        insert(pos, std::vector<T>(init));
    }

    void insert(int pos, const DynamicArray& other) {
        assert(0 <= pos && pos <= size());
        if (other.empty()) return;
        pool.reserve(pool.size() + other.size());
        int mid = clone_subtree_from(other, other.root);
        int l, r;
        split(root, pos, l, r);
        root = merge(merge(l, mid), r);
    }

    void push_back(T val) {
        insert(size(), std::move(val));
    }

    void push_front(T val) {
        insert(0, std::move(val));
    }

    void append(const std::vector<T>& v) {
        insert(size(), v);
    }

    void append(std::vector<T>&& v) {
        insert(size(), std::move(v));
    }

    void append(const DynamicArray& other) {
        insert(size(), other);
    }

    void erase(int pos) {
        assert(0 <= pos && pos < size());
        root = erase_node(root, pos);
    }

    void erase(int l, int r) {
        assert(0 <= l && l <= r && r <= size());
        if (l == r) return;
        int a, b, c;
        split(root, l, a, b);
        split(b, r - l, b, c);
        root = merge(a, c);
    }

    void pop_back() {
        assert(!empty());
        erase(size() - 1);
    }

    void pop_front() {
        assert(!empty());
        erase(0);
    }

    T& at(int pos) {
        assert(0 <= pos && pos < size());
        return pool[find_node(root, pos)].val;
    }

    const T& at(int pos) const {
        assert(0 <= pos && pos < size());
        return pool[find_node(root, pos, false)].val;
    }

    T& operator[](int pos) {
        return at(pos);
    }

    const T& operator[](int pos) const {
        return at(pos);
    }

    T& front() {
        assert(!empty());
        return at(0);
    }

    const T& front() const {
        assert(!empty());
        return at(0);
    }

    T& back() {
        assert(!empty());
        return at(size() - 1);
    }

    const T& back() const {
        assert(!empty());
        return at(size() - 1);
    }

    void reverse(int l, int r) {
        assert(0 <= l && l <= r && r <= size());
        if (l == r) return;
        int a, b, c;
        split(root, l, a, b);
        split(b, r - l, b, c);
        apply_reverse(b);
        root = merge(merge(a, b), c);
    }

    void reverse() {
        apply_reverse(root);
    }

    void rotate(int l, int m, int r) {
        assert(0 <= l && l <= m && m <= r && r <= size());
        if (l == m || m == r) return;
        int a, b, c, d;
        split(root, l, a, b);
        split(b, m - l, b, c);
        split(c, r - m, c, d);
        root = merge(merge(a, c), merge(b, d));
    }

    T get(int pos) const {
        return at(pos);
    }

    void set(int pos, T val) {
        at(pos) = std::move(val);
    }

    std::vector<T> to_vector() const {
        std::vector<T> res;
        res.reserve(size());
        dump_dfs(root, res);
        return res;
    }

    std::vector<T> to_vector(int l, int r) const {
        assert(0 <= l && l <= r && r <= size());
        std::vector<T> res;
        res.reserve(r - l);
        dump_range_dfs(root, l, r, 0, res);
        return res;
    }

    DynamicArray split_off(int pos) {
        assert(0 <= pos && pos <= size());
        int l, r;
        split(root, pos, l, r);
        root = l;

        DynamicArray res;
        res.pool.reserve(pool[r].count + 1);
        res.root = res.clone_subtree_from(*this, r);
        return res;
    }
};

}  // namespace ds
}  // namespace m1une

// END: ds/dynamic_array/dynamic_array.hpp
#line 7 "..::contest596::A::main.cpp"

void solve() {
    ll N;
    scan(N);
    m1une::ds::DynamicArray<ll> arr(1, 0);

    ll inv = 0;
    FOR(i, 1, N) {
        print('?', 1, i + 1);
        fastout.flush();
        ll X;
        scan(X);
        ll p = X - inv;
        arr.insert(p, i);
        inv = X;
    }
    arr.reverse();
    vl ans(N);
    FOR(i, N) {
        ans[arr[i]] = i + 1;
    }
    print(ans);
    fastout.flush();
}

int main() {
    CPP_DUMP_SET_OPTION(max_line_width, 80);
    CPP_DUMP_SET_OPTION(log_label_func, cpp_dump::log_label::filename());
    CPP_DUMP_SET_OPTION(enable_asterisk, true);
    int T = 1;
    // cin >> T;
    while (T--) solve();
    return 0;
}
// END: ../contest596/A/main.cpp