verify/math/matrix/bit_matrix_linear_system.test.cpp

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• Last update: 2026-10-06 01:57:24+09:00
• Problem: https://judge.yosupo.jp/problem/system_of_linear_equations_mod_2

Depends on

• Bit Matrix (math/matrix/bit_matrix.hpp)
• Fast IO (utilities/fast_io.hpp)

Code

#define PROBLEM "https://judge.yosupo.jp/problem/system_of_linear_equations_mod_2"

#include "../../../utilities/fast_io.hpp"
#include <string>
#include <vector>

#include "../../../math/matrix/bit_matrix.hpp"

namespace {

void print_vector(
    const std::vector<bool>& values,
    m1une::utilities::FastOutput& fast_output
) {
    for (bool value : values) fast_output << int(value);
    fast_output << '\n';
}

}  // namespace

int main() {
    m1une::utilities::FastInput fast_input;
    m1une::utilities::FastOutput fast_output;

    int equation_count, variable_count;
    fast_input >> equation_count >> variable_count;
    m1une::matrix::BitMatrix coefficients(equation_count, variable_count);
    std::string bits;
    for (int row = 0; row < equation_count; row++) {
        fast_input >> bits;
        coefficients.set_row(row, bits);
    }
    fast_input >> bits;
    std::vector<bool> constants(std::size_t(equation_count), false);
    for (int row = 0; row < equation_count; row++) {
        constants[std::size_t(row)] = bits[std::size_t(row)] == '1';
    }

    m1une::matrix::BitLinearSystemResult result =
        m1une::matrix::solve_linear_system(coefficients, constants);
    if (!result.consistent) {
        fast_output << -1 << '\n';
        return 0;
    }

    fast_output << result.nullity() << '\n';
    print_vector(result.particular_solution, fast_output);
    for (const std::vector<bool>& direction : result.nullspace_basis) {
        print_vector(direction, fast_output);
    }
}

#line 1 "verify/math/matrix/bit_matrix_linear_system.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/system_of_linear_equations_mod_2"

#line 1 "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>
#include <vector>

namespace m1une {
namespace utilities {

struct FastOutput;

namespace internal {

// Shared with the convenience helpers in template.hpp.
inline FastOutput* standard_output_instance = nullptr;

// 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;
    std::string* _capture = nullptr;

    template <class T>
    std::string format_cell(const T& value) {
        std::string result;
        struct CaptureGuard {
            std::string*& target;
            std::string* previous;
            ~CaptureGuard() { target = previous; }
        } guard{_capture, _capture};
        _capture = &result;
        write(value);
        return result;
    }

    template <class Matrix>
    void write_aligned_matrix(const Matrix& matrix) {
        std::vector<std::vector<std::string>> rows;
        std::vector<std::size_t> widths;
        for (const auto& row : matrix) {
            auto& cells = rows.emplace_back();
            std::size_t column = 0;
            for (const auto& value : row) {
                cells.push_back(format_cell(value));
                if (column == widths.size()) widths.push_back(0);
                widths[column] = std::max(widths[column], cells.back().size());
                ++column;
            }
        }
        bool first = true;
        for (const auto& row : rows) {
            if (!first) write_char('\n');
            first = false;
            for (std::size_t column = 0; column < row.size(); ++column) {
                if (column != 0) write_char(_range_separator);
                for (std::size_t padding = row[column].size();
                     padding < widths[column]; ++padding) {
                    write_char(' ');
                }
                write(row[column]);
            }
        }
    }

   public:
    explicit FastOutput(std::FILE* stream = stdout)
        : _stream(stream),
          _position(0),
          _precision(6),
          _float_format(std::chars_format::general),
          _range_separator(' ') {
        if (_stream == stdout
            && internal::standard_output_instance == nullptr) {
            internal::standard_output_instance = this;
        }
    }

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

    ~FastOutput() {
        flush();
        if (internal::standard_output_instance == this) {
            internal::standard_output_instance = nullptr;
        }
    }

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

    void write_char(char c) {
        if (_capture != nullptr) {
            _capture->push_back(c);
            return;
        }
        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) {
        if (_capture != nullptr) {
            _capture->append(s);
            return;
        }
        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 (_capture == nullptr && _position > buffer_size - 64) flush();
        char captured[64];
        char* const begin = _capture != nullptr ? captured : _buffer + _position;
        char* destination = begin;
        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++) *destination++ = first[skip];
        while (count--) {
            const char* digits = digit_quads.data() + 4 * chunks[count];
            std::memcpy(destination, digits, 4);
            destination += 4;
        }
        if (_capture != nullptr) {
            _capture->append(begin, destination - begin);
        } else {
            _position += int(destination - begin);
        }
    }

    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 Matrix>
    void write_aligned(const Matrix& matrix) {
        using Row = internal::range_stored_value_t<const Matrix>;
        using Cell = internal::range_stored_value_t<const Row>;
        static_assert(internal::is_range_v<Row> && !internal::is_string_like_v<Row>,
                      "write_aligned requires a two-dimensional range");
        static_assert(!internal::is_range_v<Cell> || internal::is_string_like_v<Cell>,
                      "write_aligned requires scalar cells");
        write_aligned_matrix(matrix);
    }

    template <class Matrix>
    void println_aligned(const Matrix& matrix) {
        write_aligned(matrix);
        write_char('\n');
    }

    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


#line 6 "verify/math/matrix/bit_matrix_linear_system.test.cpp"

#line 1 "math/matrix/bit_matrix.hpp"



#line 5 "math/matrix/bit_matrix.hpp"
#include <bit>
#include <cassert>
#line 9 "math/matrix/bit_matrix.hpp"
#include <optional>
#line 11 "math/matrix/bit_matrix.hpp"
#include <string_view>
#line 14 "math/matrix/bit_matrix.hpp"

namespace m1une {
namespace matrix {

class BitMatrix {
   private:
    int _rows;
    int _cols;
    int _blocks;
    std::vector<std::uint64_t> _data;

    static int block_count(int cols) {
        assert(cols >= 0);
        return (cols + 63) / 64;
    }

    static std::size_t storage_size(int rows, int blocks) {
        assert(rows >= 0);
        return std::size_t(rows) * std::size_t(blocks);
    }

    std::size_t word_index(int row, int col) const {
        assert(0 <= row && row < _rows);
        assert(0 <= col && col < _cols);
        return std::size_t(row) * std::size_t(_blocks) +
               std::size_t(col / 64);
    }

    std::uint64_t trailing_mask() const {
        if ((_cols & 63) == 0) return ~std::uint64_t(0);
        return (std::uint64_t(1) << (_cols & 63)) - 1;
    }

   public:
    class BitReference {
       private:
        std::uint64_t* word;
        std::uint64_t mask;

       public:
        BitReference(std::uint64_t& word_value, std::uint64_t mask_value)
            : word(&word_value), mask(mask_value) {}

        operator bool() const {
            return (*word & mask) != 0;
        }

        BitReference& operator=(bool value) {
            if (value) {
                *word |= mask;
            } else {
                *word &= ~mask;
            }
            return *this;
        }

        BitReference& operator=(const BitReference& other) {
            return *this = bool(other);
        }

        void flip() {
            *word ^= mask;
        }
    };

    class RowReference {
       private:
        BitMatrix* matrix;
        int row;

       public:
        RowReference(BitMatrix& matrix_value, int row_value)
            : matrix(&matrix_value), row(row_value) {}

        BitReference operator[](int col) const {
            return (*matrix)(row, col);
        }
    };

    class ConstRowReference {
       private:
        const BitMatrix* matrix;
        int row;

       public:
        ConstRowReference(const BitMatrix& matrix_value, int row_value)
            : matrix(&matrix_value), row(row_value) {}

        bool operator[](int col) const {
            return (*matrix)(row, col);
        }
    };

    BitMatrix() : _rows(0), _cols(0), _blocks(0) {}

    BitMatrix(int rows, int cols, bool value = false)
        : _rows(rows),
          _cols(cols),
          _blocks(block_count(cols)),
          _data(
              storage_size(rows, _blocks),
              value ? ~std::uint64_t(0) : std::uint64_t(0)
          ) {
        assert(rows >= 0);
        if (value && _blocks > 0) {
            const std::uint64_t mask = trailing_mask();
            for (int row = 0; row < _rows; row++) {
                _data[
                    std::size_t(row + 1) * std::size_t(_blocks) - 1
                ] &= mask;
            }
        }
    }

    int rows() const {
        return _rows;
    }

    int cols() const {
        return _cols;
    }

    int blocks_per_row() const {
        return _blocks;
    }

    bool empty() const {
        return _rows == 0 || _cols == 0;
    }

    RowReference operator[](int row) {
        assert(0 <= row && row < _rows);
        return RowReference(*this, row);
    }

    ConstRowReference operator[](int row) const {
        assert(0 <= row && row < _rows);
        return ConstRowReference(*this, row);
    }

    BitReference operator()(int row, int col) {
        const std::size_t index = word_index(row, col);
        return BitReference(_data[index], std::uint64_t(1) << (col & 63));
    }

    bool operator()(int row, int col) const {
        const std::size_t index = word_index(row, col);
        return (_data[index] >> (col & 63)) & 1;
    }

    bool get(int row, int col) const {
        return (*this)(row, col);
    }

    void set(int row, int col, bool value = true) {
        (*this)(row, col) = value;
    }

    void reset(int row, int col) {
        set(row, col, false);
    }

    void flip(int row, int col) {
        (*this)(row, col).flip();
    }

    void clear() {
        std::fill(_data.begin(), _data.end(), std::uint64_t(0));
    }

    void set_row(int row, std::string_view bits) {
        assert(0 <= row && row < _rows);
        assert(int(bits.size()) == _cols);
        const std::size_t offset =
            std::size_t(row) * std::size_t(_blocks);
        std::fill(
            _data.begin() + std::ptrdiff_t(offset),
            _data.begin() + std::ptrdiff_t(offset + std::size_t(_blocks)),
            std::uint64_t(0)
        );
        for (int col = 0; col < _cols; col++) {
            assert(bits[std::size_t(col)] == '0' || bits[std::size_t(col)] == '1');
            if (bits[std::size_t(col)] == '1') set(row, col);
        }
    }

    std::string row_string(int row) const {
        assert(0 <= row && row < _rows);
        std::string result(std::size_t(_cols), '0');
        for (int col = 0; col < _cols; col++) {
            if (get(row, col)) result[std::size_t(col)] = '1';
        }
        return result;
    }

    static BitMatrix identity(int size) {
        assert(size >= 0);
        BitMatrix result(size, size);
        for (int index = 0; index < size; index++) result.set(index, index);
        return result;
    }

    BitMatrix transposed() const {
        BitMatrix result(_cols, _rows);
        for (int row = 0; row < _rows; row++) {
            for (int col = 0; col < _cols; col++) {
                if (get(row, col)) result.set(col, row);
            }
        }
        return result;
    }

    void swap_rows(int first, int second) {
        assert(0 <= first && first < _rows);
        assert(0 <= second && second < _rows);
        if (first == second) return;
        const std::size_t first_offset =
            std::size_t(first) * std::size_t(_blocks);
        const std::size_t second_offset =
            std::size_t(second) * std::size_t(_blocks);
        for (int block = 0; block < _blocks; block++) {
            std::swap(
                _data[first_offset + std::size_t(block)],
                _data[second_offset + std::size_t(block)]
            );
        }
    }

    void xor_rows(int target, int source, int first_col = 0) {
        assert(0 <= target && target < _rows);
        assert(0 <= source && source < _rows);
        assert(0 <= first_col && first_col <= _cols);
        if (first_col == _cols) return;
        const std::size_t target_offset =
            std::size_t(target) * std::size_t(_blocks);
        const std::size_t source_offset =
            std::size_t(source) * std::size_t(_blocks);
        const int first_block = first_col / 64;
        const int first_bit = first_col & 63;
        if (first_bit != 0) {
            const std::uint64_t mask = ~std::uint64_t(0) << first_bit;
            _data[target_offset + std::size_t(first_block)] ^=
                _data[source_offset + std::size_t(first_block)] & mask;
        } else {
            _data[target_offset + std::size_t(first_block)] ^=
                _data[source_offset + std::size_t(first_block)];
        }
        for (int block = first_block + 1; block < _blocks; block++) {
            _data[target_offset + std::size_t(block)] ^=
                _data[source_offset + std::size_t(block)];
        }
    }

    BitMatrix& operator^=(const BitMatrix& rhs) {
        assert(_rows == rhs._rows && _cols == rhs._cols);
        for (std::size_t index = 0; index < _data.size(); index++) {
            _data[index] ^= rhs._data[index];
        }
        return *this;
    }

    BitMatrix& operator+=(const BitMatrix& rhs) {
        return *this ^= rhs;
    }

    BitMatrix& operator-=(const BitMatrix& rhs) {
        return *this ^= rhs;
    }

    BitMatrix& operator*=(const BitMatrix& rhs) {
        return *this = *this * rhs;
    }

    friend BitMatrix operator^(BitMatrix lhs, const BitMatrix& rhs) {
        return lhs ^= rhs;
    }

    friend BitMatrix operator+(BitMatrix lhs, const BitMatrix& rhs) {
        return lhs += rhs;
    }

    friend BitMatrix operator-(BitMatrix lhs, const BitMatrix& rhs) {
        return lhs -= rhs;
    }

    friend BitMatrix operator*(const BitMatrix& lhs, const BitMatrix& rhs) {
        assert(lhs._cols == rhs._rows);
        BitMatrix result(lhs._rows, rhs._cols);
        for (int row = 0; row < lhs._rows; row++) {
            const std::size_t lhs_offset =
                std::size_t(row) * std::size_t(lhs._blocks);
            const std::size_t result_offset =
                std::size_t(row) * std::size_t(result._blocks);
            for (int lhs_block = 0; lhs_block < lhs._blocks; lhs_block++) {
                std::uint64_t word =
                    lhs._data[lhs_offset + std::size_t(lhs_block)];
                while (word != 0) {
                    const int bit = std::countr_zero(word);
                    const int middle = lhs_block * 64 + bit;
                    const std::size_t rhs_offset =
                        std::size_t(middle) * std::size_t(rhs._blocks);
                    for (int block = 0; block < rhs._blocks; block++) {
                        result._data[result_offset + std::size_t(block)] ^=
                            rhs._data[rhs_offset + std::size_t(block)];
                    }
                    word &= word - 1;
                }
            }
        }
        return result;
    }

    bool operator==(const BitMatrix& rhs) const {
        return
            _rows == rhs._rows && _cols == rhs._cols && _data == rhs._data;
    }

    bool operator!=(const BitMatrix& rhs) const {
        return !(*this == rhs);
    }

    BitMatrix pow(std::uint64_t exponent) const {
        assert(_rows == _cols);
        BitMatrix result = identity(_rows);
        BitMatrix base = *this;
        while (exponent > 0) {
            if (exponent & 1) result *= base;
            exponent >>= 1;
            if (exponent > 0) base *= base;
        }
        return result;
    }
};

namespace bit_matrix_detail {

inline std::vector<int> row_reduce(
    BitMatrix& matrix,
    int pivot_col_limit,
    bool reduced
) {
    assert(0 <= pivot_col_limit && pivot_col_limit <= matrix.cols());
    std::vector<int> pivot_columns;
    int pivot_row = 0;
    for (
        int col = 0;
        col < pivot_col_limit && pivot_row < matrix.rows();
        col++
    ) {
        int pivot = -1;
        for (int row = pivot_row; row < matrix.rows(); row++) {
            if (matrix.get(row, col)) {
                pivot = row;
                break;
            }
        }
        if (pivot == -1) continue;
        matrix.swap_rows(pivot_row, pivot);

        const int first_row = reduced ? 0 : pivot_row + 1;
        for (int row = first_row; row < matrix.rows(); row++) {
            if (row != pivot_row && matrix.get(row, col)) {
                matrix.xor_rows(row, pivot_row, col);
            }
        }
        pivot_columns.push_back(col);
        pivot_row++;
    }
    return pivot_columns;
}

}  // namespace bit_matrix_detail

struct BitRowReduction {
    BitMatrix matrix;
    std::vector<int> pivot_columns;

    int rank() const {
        return int(pivot_columns.size());
    }
};

inline BitRowReduction reduced_row_echelon_form(BitMatrix matrix) {
    BitRowReduction result;
    result.pivot_columns = bit_matrix_detail::row_reduce(
        matrix,
        matrix.cols(),
        true
    );
    result.matrix = std::move(matrix);
    return result;
}

inline int matrix_rank(BitMatrix matrix) {
    if (matrix.rows() > matrix.cols()) matrix = matrix.transposed();
    return int(bit_matrix_detail::row_reduce(
        matrix,
        matrix.cols(),
        false
    ).size());
}

inline bool determinant(const BitMatrix& matrix) {
    assert(matrix.rows() == matrix.cols());
    return matrix_rank(matrix) == matrix.rows();
}

inline std::optional<BitMatrix> inverse(const BitMatrix& matrix) {
    assert(matrix.rows() == matrix.cols());
    const int size = matrix.rows();
    BitMatrix augmented(size, 2 * size);
    for (int row = 0; row < size; row++) {
        for (int col = 0; col < size; col++) {
            if (matrix.get(row, col)) augmented.set(row, col);
        }
        augmented.set(row, size + row);
    }

    const std::vector<int> pivots = bit_matrix_detail::row_reduce(
        augmented,
        size,
        true
    );
    if (int(pivots.size()) != size) return std::nullopt;

    BitMatrix result(size, size);
    for (int row = 0; row < size; row++) {
        for (int col = 0; col < size; col++) {
            if (augmented.get(row, size + col)) result.set(row, col);
        }
    }
    return result;
}

struct BitLinearSystemResult {
    bool consistent = false;
    std::vector<bool> particular_solution;
    std::vector<std::vector<bool>> nullspace_basis;
    std::vector<int> pivot_columns;

    int rank() const {
        return int(pivot_columns.size());
    }

    int nullity() const {
        return consistent ? int(nullspace_basis.size()) : 0;
    }

    bool has_unique_solution() const {
        return consistent && nullspace_basis.empty();
    }
};

inline BitLinearSystemResult solve_linear_system(
    const BitMatrix& coefficients,
    const std::vector<bool>& constants
) {
    assert(coefficients.rows() == int(constants.size()));
    const int equation_count = coefficients.rows();
    const int variable_count = coefficients.cols();
    BitMatrix augmented(equation_count, variable_count + 1);
    for (int row = 0; row < equation_count; row++) {
        for (int col = 0; col < variable_count; col++) {
            if (coefficients.get(row, col)) augmented.set(row, col);
        }
        if (constants[std::size_t(row)]) augmented.set(row, variable_count);
    }

    BitLinearSystemResult result;
    result.pivot_columns = bit_matrix_detail::row_reduce(
        augmented,
        variable_count,
        true
    );
    for (int row = result.rank(); row < equation_count; row++) {
        if (augmented.get(row, variable_count)) return result;
    }

    result.consistent = true;
    result.particular_solution.assign(std::size_t(variable_count), false);
    std::vector<bool> is_pivot(std::size_t(variable_count), false);
    for (int row = 0; row < result.rank(); row++) {
        const int col = result.pivot_columns[std::size_t(row)];
        is_pivot[std::size_t(col)] = true;
        result.particular_solution[std::size_t(col)] =
            augmented.get(row, variable_count);
    }

    for (int free_col = 0; free_col < variable_count; free_col++) {
        if (is_pivot[std::size_t(free_col)]) continue;
        std::vector<bool> direction(std::size_t(variable_count), false);
        direction[std::size_t(free_col)] = true;
        for (int row = 0; row < result.rank(); row++) {
            const int pivot_col = result.pivot_columns[std::size_t(row)];
            direction[std::size_t(pivot_col)] = augmented.get(row, free_col);
        }
        result.nullspace_basis.push_back(std::move(direction));
    }
    return result;
}

}  // namespace matrix
}  // namespace m1une


#line 8 "verify/math/matrix/bit_matrix_linear_system.test.cpp"

namespace {

void print_vector(
    const std::vector<bool>& values,
    m1une::utilities::FastOutput& fast_output
) {
    for (bool value : values) fast_output << int(value);
    fast_output << '\n';
}

}  // namespace

int main() {
    m1une::utilities::FastInput fast_input;
    m1une::utilities::FastOutput fast_output;

    int equation_count, variable_count;
    fast_input >> equation_count >> variable_count;
    m1une::matrix::BitMatrix coefficients(equation_count, variable_count);
    std::string bits;
    for (int row = 0; row < equation_count; row++) {
        fast_input >> bits;
        coefficients.set_row(row, bits);
    }
    fast_input >> bits;
    std::vector<bool> constants(std::size_t(equation_count), false);
    for (int row = 0; row < equation_count; row++) {
        constants[std::size_t(row)] = bits[std::size_t(row)] == '1';
    }

    m1une::matrix::BitLinearSystemResult result =
        m1une::matrix::solve_linear_system(coefficients, constants);
    if (!result.consistent) {
        fast_output << -1 << '\n';
        return 0;
    }

    fast_output << result.nullity() << '\n';
    print_vector(result.particular_solution, fast_output);
    for (const std::vector<bool>& direction : result.nullspace_basis) {
        print_vector(direction, fast_output);
    }
}

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