include/random

// -*- C++ -*-
//===--------------------------- random -----------------------------------===//
//
//                     The LLVM Compiler Infrastructure
//
// This file is distributed under the University of Illinois Open Source
// License. See LICENSE.TXT for details.
//
//===----------------------------------------------------------------------===//

#ifndef _LIBCPP_RANDOM
#define _LIBCPP_RANDOM

/*
    random synopsis

#include <initializer_list>

namespace std
{

// Engines

template <class UIntType, UIntType a, UIntType c, UIntType m>
class linear_congruential_engine
{
public:
    // types
    typedef UIntType result_type;

    // engine characteristics
    static constexpr result_type multiplier = a;
    static constexpr result_type increment = c;
    static constexpr result_type modulus = m;
    static constexpr result_type min() { return c == 0u ? 1u: 0u;}
    static constexpr result_type max() { return m - 1u;}
    static constexpr result_type default_seed = 1u;

    // constructors and seeding functions
    explicit linear_congruential_engine(result_type s = default_seed);
    template<class Sseq> explicit linear_congruential_engine(Sseq& q);
    void seed(result_type s = default_seed);
    template<class Sseq> void seed(Sseq& q);

    // generating functions
    result_type operator()();
    void discard(unsigned long long z);
};

template <class UIntType, UIntType a, UIntType c, UIntType m>
bool
operator==(const linear_congruential_engine<UIntType, a, c, m>& x,
           const linear_congruential_engine<UIntType, a, c, m>& y);

template <class UIntType, UIntType a, UIntType c, UIntType m>
bool
operator!=(const linear_congruential_engine<UIntType, a, c, m>& x,
           const linear_congruential_engine<UIntType, a, c, m>& y);

template <class charT, class traits,
          class UIntType, UIntType a, UIntType c, UIntType m>
basic_ostream<charT, traits>&
operator<<(basic_ostream<charT, traits>& os,
           const linear_congruential_engine<UIntType, a, c, m>& x);

template <class charT, class traits,
          class UIntType, UIntType a, UIntType c, UIntType m>
basic_istream<charT, traits>&
operator>>(basic_istream<charT, traits>& is,
           linear_congruential_engine<UIntType, a, c, m>& x);

template <class UIntType, size_t w, size_t n, size_t m, size_t r,
          UIntType a, size_t u, UIntType d, size_t s,
          UIntType b, size_t t, UIntType c, size_t l, UIntType f>
class mersenne_twister_engine
{
public:
    // types
    typedef UIntType result_type;

    // engine characteristics
    static constexpr size_t word_size = w;
    static constexpr size_t state_size = n;
    static constexpr size_t shift_size = m;
    static constexpr size_t mask_bits = r;
    static constexpr result_type xor_mask = a;
    static constexpr size_t tempering_u = u;
    static constexpr result_type tempering_d = d;
    static constexpr size_t tempering_s = s;
    static constexpr result_type tempering_b = b;
    static constexpr size_t tempering_t = t;
    static constexpr result_type tempering_c = c;
    static constexpr size_t tempering_l = l;
    static constexpr result_type initialization_multiplier = f;
    static constexpr result_type min () { return 0; }
    static constexpr result_type max() { return 2^w - 1; }
    static constexpr result_type default_seed = 5489u;

    // constructors and seeding functions
    explicit mersenne_twister_engine(result_type value = default_seed);
    template<class Sseq> explicit mersenne_twister_engine(Sseq& q);
    void seed(result_type value = default_seed);
    template<class Sseq> void seed(Sseq& q);

    // generating functions
    result_type operator()();
    void discard(unsigned long long z);
};

template <class UIntType, size_t w, size_t n, size_t m, size_t r,
          UIntType a, size_t u, UIntType d, size_t s,
          UIntType b, size_t t, UIntType c, size_t l, UIntType f>
bool
operator==(
    const mersenne_twister_engine<UIntType, w, n, m, r, a, u, d, s, b, t, c, l, f>& x,
    const mersenne_twister_engine<UIntType, w, n, m, r, a, u, d, s, b, t, c, l, f>& y);

template <class UIntType, size_t w, size_t n, size_t m, size_t r,
          UIntType a, size_t u, UIntType d, size_t s,
          UIntType b, size_t t, UIntType c, size_t l, UIntType f>
bool
operator!=(
    const mersenne_twister_engine<UIntType, w, n, m, r, a, u, d, s, b, t, c, l, f>& x,
    const mersenne_twister_engine<UIntType, w, n, m, r, a, u, d, s, b, t, c, l, f>& y);

template <class charT, class traits,
          class UIntType, size_t w, size_t n, size_t m, size_t r,
          UIntType a, size_t u, UIntType d, size_t s,
          UIntType b, size_t t, UIntType c, size_t l, UIntType f>
basic_ostream<charT, traits>&
operator<<(basic_ostream<charT, traits>& os,
           const mersenne_twister_engine<UIntType, w, n, m, r, a, u, d, s, b, t, c, l, f>& x);

template <class charT, class traits,
          class UIntType, size_t w, size_t n, size_t m, size_t r,
          UIntType a, size_t u, UIntType d, size_t s,
          UIntType b, size_t t, UIntType c, size_t l, UIntType f>
basic_istream<charT, traits>&
operator>>(basic_istream<charT, traits>& is,
           mersenne_twister_engine<UIntType, w, n, m, r, a, u, d, s, b, t, c, l, f>& x);

template<class UIntType, size_t w, size_t s, size_t r>
class subtract_with_carry_engine
{
public:
    // types
    typedef UIntType result_type;

    // engine characteristics
    static constexpr size_t word_size = w;
    static constexpr size_t short_lag = s;
    static constexpr size_t long_lag = r;
    static constexpr result_type min() { return 0; }
    static constexpr result_type max() { return m-1; }
    static constexpr result_type default_seed = 19780503u;

    // constructors and seeding functions
    explicit subtract_with_carry_engine(result_type value = default_seed);
    template<class Sseq> explicit subtract_with_carry_engine(Sseq& q);
    void seed(result_type value = default_seed);
    template<class Sseq> void seed(Sseq& q);

    // generating functions
    result_type operator()();
    void discard(unsigned long long z);
};

template<class UIntType, size_t w, size_t s, size_t r>
bool
operator==(
    const subtract_with_carry_engine<UIntType, w, s, r>& x,
    const subtract_with_carry_engine<UIntType, w, s, r>& y);

template<class UIntType, size_t w, size_t s, size_t r>
bool
operator!=(
    const subtract_with_carry_engine<UIntType, w, s, r>& x,
    const subtract_with_carry_engine<UIntType, w, s, r>& y);

template <class charT, class traits,
          class UIntType, size_t w, size_t s, size_t r>
basic_ostream<charT, traits>&
operator<<(basic_ostream<charT, traits>& os,
           const subtract_with_carry_engine<UIntType, w, s, r>& x);

template <class charT, class traits,
          class UIntType, size_t w, size_t s, size_t r>
basic_istream<charT, traits>&
operator>>(basic_istream<charT, traits>& is,
           subtract_with_carry_engine<UIntType, w, s, r>& x);

template<class Engine, size_t p, size_t r>
class discard_block_engine
{
public:
    // types
    typedef typename Engine::result_type result_type;

    // engine characteristics
    static constexpr size_t block_size = p;
    static constexpr size_t used_block = r;
    static constexpr result_type min() { return Engine::min(); }
    static constexpr result_type max() { return Engine::max(); }

    // constructors and seeding functions
    discard_block_engine();
    explicit discard_block_engine(const Engine& e);
    explicit discard_block_engine(Engine&& e);
    explicit discard_block_engine(result_type s);
    template<class Sseq> explicit discard_block_engine(Sseq& q);
    void seed();
    void seed(result_type s);
    template<class Sseq> void seed(Sseq& q);

    // generating functions
    result_type operator()();
    void discard(unsigned long long z);

    // property functions
    const Engine& base() const;
};

template<class Engine, size_t p, size_t r>
bool
operator==(
    const discard_block_engine<Engine, p, r>& x,
    const discard_block_engine<Engine, p, r>& y);

template<class Engine, size_t p, size_t r>
bool
operator!=(
    const discard_block_engine<Engine, p, r>& x,
    const discard_block_engine<Engine, p, r>& y);

template <class charT, class traits,
          class Engine, size_t p, size_t r>
basic_ostream<charT, traits>&
operator<<(basic_ostream<charT, traits>& os,
           const discard_block_engine<Engine, p, r>& x);

template <class charT, class traits,
          class Engine, size_t p, size_t r>
basic_istream<charT, traits>&
operator>>(basic_istream<charT, traits>& is,
           discard_block_engine<Engine, p, r>& x);

template<class Engine, size_t w, class UIntType>
class independent_bits_engine
{
public:
    // types
    typedef UIntType result_type;

    // engine characteristics
    static constexpr result_type min() { return 0; }
    static constexpr result_type max() { return 2^w - 1; }

    // constructors and seeding functions
    independent_bits_engine();
    explicit independent_bits_engine(const Engine& e);
    explicit independent_bits_engine(Engine&& e);
    explicit independent_bits_engine(result_type s);
    template<class Sseq> explicit independent_bits_engine(Sseq& q);
    void seed();
    void seed(result_type s);
    template<class Sseq> void seed(Sseq& q);

    // generating functions
    result_type operator()(); void discard(unsigned long long z);

    // property functions
    const Engine& base() const;
};

template<class Engine, size_t w, class UIntType>
bool
operator==(
    const independent_bits_engine<Engine, w, UIntType>& x,
    const independent_bits_engine<Engine, w, UIntType>& y);

template<class Engine, size_t w, class UIntType>
bool
operator!=(
    const independent_bits_engine<Engine, w, UIntType>& x,
    const independent_bits_engine<Engine, w, UIntType>& y);

template <class charT, class traits,
          class Engine, size_t w, class UIntType>
basic_ostream<charT, traits>&
operator<<(basic_ostream<charT, traits>& os,
           const independent_bits_engine<Engine, w, UIntType>& x);

template <class charT, class traits,
          class Engine, size_t w, class UIntType>
basic_istream<charT, traits>&
operator>>(basic_istream<charT, traits>& is,
           independent_bits_engine<Engine, w, UIntType>& x);

template<class Engine, size_t k>
class shuffle_order_engine
{
public:
    // types
    typedef typename Engine::result_type result_type;

    // engine characteristics
    static constexpr size_t table_size = k;
    static constexpr result_type min() { return Engine::min; }
    static constexpr result_type max() { return Engine::max; }

    // constructors and seeding functions
    shuffle_order_engine();
    explicit shuffle_order_engine(const Engine& e);
    explicit shuffle_order_engine(Engine&& e);
    explicit shuffle_order_engine(result_type s);
    template<class Sseq> explicit shuffle_order_engine(Sseq& q);
    void seed();
    void seed(result_type s);
    template<class Sseq> void seed(Sseq& q);

    // generating functions
    result_type operator()();
    void discard(unsigned long long z);

    // property functions
    const Engine& base() const;
};

template<class Engine, size_t k>
bool
operator==(
    const shuffle_order_engine<Engine, k>& x,
    const shuffle_order_engine<Engine, k>& y);

template<class Engine, size_t k>
bool
operator!=(
    const shuffle_order_engine<Engine, k>& x,
    const shuffle_order_engine<Engine, k>& y);

template <class charT, class traits,
          class Engine, size_t k>
basic_ostream<charT, traits>&
operator<<(basic_ostream<charT, traits>& os,
           const shuffle_order_engine<Engine, k>& x);

template <class charT, class traits,
          class Engine, size_t k>
basic_istream<charT, traits>&
operator>>(basic_istream<charT, traits>& is,
           shuffle_order_engine<Engine, k>& x);

typedef linear_congruential_engine<uint_fast32_t, 16807, 0, 2147483647>
                                                                   minstd_rand0;
typedef linear_congruential_engine<uint_fast32_t, 48271, 0, 2147483647>
                                                                    minstd_rand;
typedef mersenne_twister_engine<uint_fast32_t, 32, 624, 397, 31,
                                0x9908b0df,
                                11, 0xffffffff,
                                7,  0x9d2c5680,
                                15, 0xefc60000,
                                18, 1812433253>                         mt19937;
typedef mersenne_twister_engine<uint_fast64_t, 64, 312, 156, 31,
                                0xb5026f5aa96619e9,
                                29, 0x5555555555555555,
                                17, 0x71d67fffeda60000,
                                37, 0xfff7eee000000000,
                                43, 6364136223846793005>             mt19937_64;
typedef subtract_with_carry_engine<uint_fast32_t, 24, 10, 24>     ranlux24_base;
typedef subtract_with_carry_engine<uint_fast64_t, 48,  5, 12>     ranlux48_base;
typedef discard_block_engine<ranlux24_base, 223, 23>                   ranlux24;
typedef discard_block_engine<ranlux48_base, 389, 11>                   ranlux48;
typedef shuffle_order_engine<minstd_rand0, 256>                         knuth_b;
typedef minstd_rand0                                      default_random_engine;

// Generators

class random_device
{
public:
    // types
    typedef unsigned int result_type;

    // generator characteristics
    static constexpr result_type min() { return numeric_limits<result_type>::min(); }
    static constexpr result_type max() { return numeric_limits<result_type>::max(); }

    // constructors
    explicit random_device(const string& token = "/dev/urandom");

    // generating functions
    result_type operator()();

    // property functions
    double entropy() const;

    // no copy functions
    random_device(const random_device& ) = delete;
    void operator=(const random_device& ) = delete;
};

// Utilities

class seed_seq
{
public:
    // types
    typedef uint_least32_t result_type;

    // constructors
    seed_seq();
    template<class T>
        seed_seq(initializer_list<T> il);
    template<class InputIterator>
        seed_seq(InputIterator begin, InputIterator end);

    // generating functions
    template<class RandomAccessIterator>
        void generate(RandomAccessIterator begin, RandomAccessIterator end);

    // property functions
    size_t size() const;
    template<class OutputIterator>
        void param(OutputIterator dest) const;

    // no copy functions
    seed_seq(const seed_seq&) = delete;
    void operator=(const seed_seq& ) = delete;
};

template<class RealType, size_t bits, class URNG>
    RealType generate_canonical(URNG& g);

// Distributions

template<class IntType = int>
class uniform_int_distribution
{
public:
    // types
    typedef IntType result_type;

    class param_type
    {
    public:
        typedef uniform_int_distribution distribution_type;

        explicit param_type(IntType a = 0,
                                    IntType b = numeric_limits<IntType>::max());

        result_type a() const;
        result_type b() const;

        friend bool operator==(const param_type& x, const param_type& y);
        friend bool operator!=(const param_type& x, const param_type& y);
    };

    // constructors and reset functions
    explicit uniform_int_distribution(IntType a = 0,
                                    IntType b = numeric_limits<IntType>::max());
    explicit uniform_int_distribution(const param_type& parm);
    void reset();

    // generating functions
    template<class URNG> result_type operator()(URNG& g);
    template<class URNG> result_type operator()(URNG& g, const param_type& parm);

    // property functions
    result_type a() const;
    result_type b() const;

    param_type param() const;
    void param(const param_type& parm);

    result_type min() const;
    result_type max() const;

    friend bool operator==(const uniform_int_distribution& x,
                           const uniform_int_distribution& y);
    friend bool operator!=(const uniform_int_distribution& x,
                           const uniform_int_distribution& y);

    template <class charT, class traits>
    friend
    basic_ostream<charT, traits>&
    operator<<(basic_ostream<charT, traits>& os,
               const uniform_int_distribution& x);
    
    template <class charT, class traits>
    friend
    basic_istream<charT, traits>&
    operator>>(basic_istream<charT, traits>& is,
               uniform_int_distribution& x);
};

template<class RealType = double>
class uniform_real_distribution
{
public:
    // types
    typedef RealType result_type;

    class param_type
    {
    public:
        typedef uniform_real_distribution distribution_type;

        explicit param_type(RealType a = 0,
                            RealType b = 1);

        result_type a() const;
        result_type b() const;

        friend bool operator==(const param_type& x, const param_type& y);
        friend bool operator!=(const param_type& x, const param_type& y);
    };

    // constructors and reset functions
    explicit uniform_real_distribution(RealType a = 0.0, RealType b = 1.0);
    explicit uniform_real_distribution(const param_type& parm);
    void reset();

    // generating functions
    template<class URNG> result_type operator()(URNG& g);
    template<class URNG> result_type operator()(URNG& g, const param_type& parm);

    // property functions
    result_type a() const;
    result_type b() const;

    param_type param() const;
    void param(const param_type& parm);

    result_type min() const;
    result_type max() const;

    friend bool operator==(const uniform_real_distribution& x,
                           const uniform_real_distribution& y);
    friend bool operator!=(const uniform_real_distribution& x,
                           const uniform_real_distribution& y);

    template <class charT, class traits>
    friend
    basic_ostream<charT, traits>&
    operator<<(basic_ostream<charT, traits>& os,
               const uniform_real_distribution& x);
    
    template <class charT, class traits>
    friend
    basic_istream<charT, traits>&
    operator>>(basic_istream<charT, traits>& is,
               uniform_real_distribution& x);
};

class bernoulli_distribution
{
public:
    // types
    typedef bool result_type;

    class param_type
    {
    public:
        typedef bernoulli_distribution distribution_type;

        explicit param_type(double p = 0.5);

        double p() const;

        friend bool operator==(const param_type& x, const param_type& y);
        friend bool operator!=(const param_type& x, const param_type& y);
    };

    // constructors and reset functions
    explicit bernoulli_distribution(double p = 0.5);
    explicit bernoulli_distribution(const param_type& parm);
    void reset();

    // generating functions
    template<class URNG> result_type operator()(URNG& g);
    template<class URNG> result_type operator()(URNG& g, const param_type& parm);

    // property functions
    double p() const;

    param_type param() const;
    void param(const param_type& parm);

    result_type min() const;
    result_type max() const;

    friend bool operator==(const bernoulli_distribution& x,
                           const bernoulli_distribution& y);
    friend bool operator!=(const bernoulli_distribution& x,
                           const bernoulli_distribution& y);

    template <class charT, class traits>
    friend
    basic_ostream<charT, traits>&
    operator<<(basic_ostream<charT, traits>& os,
               const bernoulli_distribution& x);
    
    template <class charT, class traits>
    friend
    basic_istream<charT, traits>&
    operator>>(basic_istream<charT, traits>& is,
               bernoulli_distribution& x);
};

template<class IntType = int>
    class binomial_distribution;

template<class IntType = int>
    class geometric_distribution;

template<class IntType = int>
    class negative_binomial_distribution;

template<class IntType = int>
    class poisson_distribution;

template<class RealType = double>
    class exponential_distribution;

template<class RealType = double>
    class gamma_distribution;

template<class RealType = double>
    class weibull_distribution;

template<class RealType = double>
    class extreme_value_distribution;

template<class RealType = double>
    class normal_distribution;

template<class RealType = double>
    class lognormal_distribution;

template<class RealType = double>
    class chi_squared_distribution;

template<class RealType = double>
    class cauchy_distribution;

template<class RealType = double>
    class fisher_f_distribution;

template<class RealType = double>
    class student_t_distribution;

template<class IntType = int>
    class discrete_distribution;

template<class RealType = double>
    class piecewise_constant_distribution;

template<class RealType = double>
    class piecewise_linear_distribution;

} // std
*/

#include <__config>
#include <cstddef>
#include <type_traits>
#include <initializer_list>
#include <cstdint>
#include <limits>
#include <algorithm>
#include <vector>
#include <string>
#include <istream>
#include <ostream>

#pragma GCC system_header

_LIBCPP_BEGIN_NAMESPACE_STD

// linear_congruential_engine

template <unsigned long long __a, unsigned long long __c,
          unsigned long long __m, unsigned long long _M,
          bool _MightOverflow = (__a != 0 && __m != 0 && __m-1 > (_M-__c)/__a)>
struct __lce_ta;

// 64

template <unsigned long long __a, unsigned long long __c, unsigned long long __m>
struct __lce_ta<__a, __c, __m, (unsigned long long)(~0), true>
{
    typedef unsigned long long result_type;
    static result_type next(result_type __x)
    {
        // Schrage's algorithm
        const result_type __q = __m / __a;
        const result_type __r = __m % __a;
        const result_type __t0 = __a * (__x % __q);
        const result_type __t1 = __r * (__x / __q);
        __x = __t0 + (__t0 < __t1) * __m - __t1;
        __x += __c - (__x >= __m - __c) * __m;
        return __x;
    }
};

template <unsigned long long __a, unsigned long long __m>
struct __lce_ta<__a, 0, __m, (unsigned long long)(~0), true>
{
    typedef unsigned long long result_type;
    static result_type next(result_type __x)
    {
        // Schrage's algorithm
        const result_type __q = __m / __a;
        const result_type __r = __m % __a;
        const result_type __t0 = __a * (__x % __q);
        const result_type __t1 = __r * (__x / __q);
        __x = __t0 + (__t0 < __t1) * __m - __t1;
        return __x;
    }
};

template <unsigned long long __a, unsigned long long __c, unsigned long long __m>
struct __lce_ta<__a, __c, __m, (unsigned long long)(~0), false>
{
    typedef unsigned long long result_type;
    static result_type next(result_type __x)
    {
        return (__a * __x + __c) % __m;
    }
};

template <unsigned long long __a, unsigned long long __c>
struct __lce_ta<__a, __c, 0, (unsigned long long)(~0), false>
{
    typedef unsigned long long result_type;
    static result_type next(result_type __x)
    {
        return __a * __x + __c;
    }
};

// 32

template <unsigned long long _A, unsigned long long _C, unsigned long long _M>
struct __lce_ta<_A, _C, _M, unsigned(~0), true>
{
    typedef unsigned result_type;
    static result_type next(result_type __x)
    {
        const result_type __a = static_cast<result_type>(_A);
        const result_type __c = static_cast<result_type>(_C);
        const result_type __m = static_cast<result_type>(_M);
        // Schrage's algorithm
        const result_type __q = __m / __a;
        const result_type __r = __m % __a;
        const result_type __t0 = __a * (__x % __q);
        const result_type __t1 = __r * (__x / __q);
        __x = __t0 + (__t0 < __t1) * __m - __t1;
        __x += __c - (__x >= __m - __c) * __m;
        return __x;
    }
};

template <unsigned long long _A, unsigned long long _M>
struct __lce_ta<_A, 0, _M, unsigned(~0), true>
{
    typedef unsigned result_type;
    static result_type next(result_type __x)
    {
        const result_type __a = static_cast<result_type>(_A);
        const result_type __m = static_cast<result_type>(_M);
        // Schrage's algorithm
        const result_type __q = __m / __a;
        const result_type __r = __m % __a;
        const result_type __t0 = __a * (__x % __q);
        const result_type __t1 = __r * (__x / __q);
        __x = __t0 + (__t0 < __t1) * __m - __t1;
        return __x;
    }
};

template <unsigned long long _A, unsigned long long _C, unsigned long long _M>
struct __lce_ta<_A, _C, _M, unsigned(~0), false>
{
    typedef unsigned result_type;
    static result_type next(result_type __x)
    {
        const result_type __a = static_cast<result_type>(_A);
        const result_type __c = static_cast<result_type>(_C);
        const result_type __m = static_cast<result_type>(_M);
        return (__a * __x + __c) % __m;
    }
};

template <unsigned long long _A, unsigned long long _C>
struct __lce_ta<_A, _C, 0, unsigned(~0), false>
{
    typedef unsigned result_type;
    static result_type next(result_type __x)
    {
        const result_type __a = static_cast<result_type>(_A);
        const result_type __c = static_cast<result_type>(_C);
        return __a * __x + __c;
    }
};

// 16

template <unsigned long long __a, unsigned long long __c, unsigned long long __m, bool __b>
struct __lce_ta<__a, __c, __m, (unsigned short)(~0), __b>
{
    typedef unsigned short result_type;
    static result_type next(result_type __x)
    {
        return static_cast<result_type>(__lce_ta<__a, __c, __m, unsigned(~0)>::next(__x));
    }
};

template <class _UIntType, _UIntType __a, _UIntType __c, _UIntType __m>
class linear_congruential_engine;

template <class _CharT, class _Traits,
          class _U, _U _A, _U _C, _U _N>
basic_ostream<_CharT, _Traits>&
operator<<(basic_ostream<_CharT, _Traits>& __os,
           const linear_congruential_engine<_U, _A, _C, _N>&);

template <class _CharT, class _Traits,
          class _U, _U _A, _U _C, _U _N>
basic_istream<_CharT, _Traits>&
operator>>(basic_istream<_CharT, _Traits>& __is,
           linear_congruential_engine<_U, _A, _C, _N>& __x);

template <class _UIntType, _UIntType __a, _UIntType __c, _UIntType __m>
class linear_congruential_engine
{
public:
    // types
    typedef _UIntType result_type;

private:
    result_type __x_;

    static const result_type _M = result_type(~0);

    static_assert(__m == 0 || __a < __m, "linear_congruential_engine invalid parameters");
    static_assert(__m == 0 || __c < __m, "linear_congruential_engine invalid parameters");
public:
    static const result_type _Min = __c == 0u ? 1u: 0u;
    static const result_type _Max = __m - 1u;
    static_assert(_Min < _Max,           "linear_congruential_engine invalid parameters");

    // engine characteristics
    static const/*expr*/ result_type multiplier = __a;
    static const/*expr*/ result_type increment = __c;
    static const/*expr*/ result_type modulus = __m;
    static const/*expr*/ result_type min() {return _Min;}
    static const/*expr*/ result_type max() {return _Max;}
    static const/*expr*/ result_type default_seed = 1u;

    // constructors and seeding functions
    explicit linear_congruential_engine(result_type __s = default_seed)
        {seed(__s);}
    template<class _Sseq> explicit linear_congruential_engine(_Sseq& __q)
        {seed(__q);}
    void seed(result_type __s = default_seed)
        {seed(integral_constant<bool, __m == 0>(),
              integral_constant<bool, __c == 0>(), __s);}
    template<class _Sseq>
        typename enable_if
        <
            !is_convertible<_Sseq, result_type>::value,
            void
        >::type
        seed(_Sseq& __q)
            {__seed(__q, integral_constant<unsigned,
                1 + (__m == 0 ? (sizeof(result_type) * __CHAR_BIT__ - 1)/32
                             :  (__m-1) / 0x100000000ull)>());}

    // generating functions
    result_type operator()()
        {return __x_ = static_cast<result_type>(__lce_ta<__a, __c, __m, _M>::next(__x_));}
    void discard(unsigned long long __z) {for (; __z; --__z) operator()();}

    friend bool operator==(const linear_congruential_engine& __x,
                           const linear_congruential_engine& __y)
        {return __x.__x_ == __y.__x_;}
    friend bool operator!=(const linear_congruential_engine& __x,
                           const linear_congruential_engine& __y)
        {return !(__x == __y);}

private:

    void seed(true_type, true_type, result_type __s) {__x_ = __s == 0 ? 1 : __s;}
    void seed(true_type, false_type, result_type __s) {__x_ = __s;}
    void seed(false_type, true_type, result_type __s) {__x_ = __s % __m == 0 ?
                                                                 1 : __s % __m;}
    void seed(false_type, false_type, result_type __s) {__x_ = __s % __m;}

    template<class _Sseq>
        void __seed(_Sseq& __q, integral_constant<unsigned, 1>);
    template<class _Sseq>
        void __seed(_Sseq& __q, integral_constant<unsigned, 2>);

    template <class _CharT, class _Traits,
              class _U, _U _A, _U _C, _U _N>
    friend
    basic_ostream<_CharT, _Traits>&
    operator<<(basic_ostream<_CharT, _Traits>& __os,
               const linear_congruential_engine<_U, _A, _C, _N>&);
    
    template <class _CharT, class _Traits,
              class _U, _U _A, _U _C, _U _N>
    friend
    basic_istream<_CharT, _Traits>&
    operator>>(basic_istream<_CharT, _Traits>& __is,
               linear_congruential_engine<_U, _A, _C, _N>& __x);
};

template <class _UIntType, _UIntType __a, _UIntType __c, _UIntType __m>
template<class _Sseq>
void
linear_congruential_engine<_UIntType, __a, __c, __m>::__seed(_Sseq& __q,
                                                 integral_constant<unsigned, 1>)
{
    const unsigned __k = 1;
    uint32_t __ar[__k+3];
    __q.generate(__ar, __ar + __k + 3);
    result_type __s = static_cast<result_type>(__ar[3] % __m);
    __x_ = __c == 0 && __s == 0 ? result_type(1) : __s;
}

template <class _UIntType, _UIntType __a, _UIntType __c, _UIntType __m>
template<class _Sseq>
void
linear_congruential_engine<_UIntType, __a, __c, __m>::__seed(_Sseq& __q,
                                                 integral_constant<unsigned, 2>)
{
    const unsigned __k = 2;
    uint32_t __ar[__k+3];
    __q.generate(__ar, __ar + __k + 3);
    result_type __s = static_cast<result_type>((__ar[3] +
                                                (uint64_t)__ar[4] << 32) % __m);
    __x_ = __c == 0 && __s == 0 ? result_type(1) : __s;
}

template <class _CharT, class _Traits>
class __save_flags
{
    typedef basic_ios<_CharT, _Traits> __stream_type;
    typedef typename __stream_type::fmtflags fmtflags;

    __stream_type& __stream_;
    fmtflags       __fmtflags_;
    _CharT         __fill_;

    __save_flags(const __save_flags&);
    __save_flags& operator=(const __save_flags&);
public:
    explicit __save_flags(__stream_type& __stream)
        : __stream_(__stream),
          __fmtflags_(__stream.flags()),
          __fill_(__stream.fill())
        {}
    ~__save_flags()
    {
        __stream_.flags(__fmtflags_);
        __stream_.fill(__fill_);
    }
};

template <class _CharT, class _Traits,
          class _UIntType, _UIntType __a, _UIntType __c, _UIntType __m>
inline
basic_ostream<_CharT, _Traits>&
operator<<(basic_ostream<_CharT, _Traits>& __os,
           const linear_congruential_engine<_UIntType, __a, __c, __m>& __x)
{
    __save_flags<_CharT, _Traits> _(__os);
    __os.flags(ios_base::dec | ios_base::left);
    __os.fill(__os.widen(' '));
    return __os << __x.__x_;
}

template <class _CharT, class _Traits,
          class _UIntType, _UIntType __a, _UIntType __c, _UIntType __m>
basic_istream<_CharT, _Traits>&
operator>>(basic_istream<_CharT, _Traits>& __is,
           linear_congruential_engine<_UIntType, __a, __c, __m>& __x)
{
    __save_flags<_CharT, _Traits> _(__is);
    __is.flags(ios_base::dec | ios_base::skipws);
    _UIntType __t;
    __is >> __t;
    if (!__is.fail())
        __x.__x_ = __t;
    return __is;
}

typedef linear_congruential_engine<uint_fast32_t, 16807, 0, 2147483647>
                                                                   minstd_rand0;
typedef minstd_rand0                                      default_random_engine;
typedef linear_congruential_engine<uint_fast32_t, 48271, 0, 2147483647>
                                                                    minstd_rand;
// mersenne_twister_engine

template <class _UIntType, size_t __w, size_t __n, size_t __m, size_t __r,
          _UIntType __a, size_t __u, _UIntType __d, size_t __s,
          _UIntType __b, size_t __t, _UIntType __c, size_t __l, _UIntType __f>
class mersenne_twister_engine;

template <class _UI, size_t _W, size_t _N, size_t _M, size_t _R,
          _UI _A, size_t _U, _UI _D, size_t _S,
          _UI _B, size_t _T, _UI _C, size_t _L, _UI _F>
bool
operator==(const mersenne_twister_engine<_UI, _W, _N, _M, _R, _A, _U, _D, _S,
                                         _B, _T, _C, _L, _F>& __x,
           const mersenne_twister_engine<_UI, _W, _N, _M, _R, _A, _U, _D, _S,
                                         _B, _T, _C, _L, _F>& __y);

template <class _UI, size_t _W, size_t _N, size_t _M, size_t _R,
          _UI _A, size_t _U, _UI _D, size_t _S,
          _UI _B, size_t _T, _UI _C, size_t _L, _UI _F>
bool
operator!=(const mersenne_twister_engine<_UI, _W, _N, _M, _R, _A, _U, _D, _S,
                                         _B, _T, _C, _L, _F>& __x,
           const mersenne_twister_engine<_UI, _W, _N, _M, _R, _A, _U, _D, _S,
                                         _B, _T, _C, _L, _F>& __y);

template <class _CharT, class _Traits,
          class _UI, size_t _W, size_t _N, size_t _M, size_t _R,
          _UI _A, size_t _U, _UI _D, size_t _S,
          _UI _B, size_t _T, _UI _C, size_t _L, _UI _F>
basic_ostream<_CharT, _Traits>&
operator<<(basic_ostream<_CharT, _Traits>& __os,
           const mersenne_twister_engine<_UI, _W, _N, _M, _R, _A, _U, _D, _S,
                                         _B, _T, _C, _L, _F>& __x);

template <class _CharT, class _Traits,
          class _UI, size_t _W, size_t _N, size_t _M, size_t _R,
          _UI _A, size_t _U, _UI _D, size_t _S,
          _UI _B, size_t _T, _UI _C, size_t _L, _UI _F>
basic_istream<_CharT, _Traits>&
operator>>(basic_istream<_CharT, _Traits>& __is,
           mersenne_twister_engine<_UI, _W, _N, _M, _R, _A, _U, _D, _S,
                                   _B, _T, _C, _L, _F>& __x);

template <class _UIntType, size_t __w, size_t __n, size_t __m, size_t __r,
          _UIntType __a, size_t __u, _UIntType __d, size_t __s,
          _UIntType __b, size_t __t, _UIntType __c, size_t __l, _UIntType __f>
class mersenne_twister_engine
{
public:
    // types
    typedef _UIntType result_type;

private:
    result_type __x_[__n];
    size_t      __i_;

    static_assert(  0 <  __m, "mersenne_twister_engine invalid parameters");
    static_assert(__m <= __n, "mersenne_twister_engine invalid parameters");
    static const result_type _Dt = numeric_limits<result_type>::digits;
    static_assert(__w <= _Dt, "mersenne_twister_engine invalid parameters");
    static_assert(  2 <= __w, "mersenne_twister_engine invalid parameters");
    static_assert(__r <= __w, "mersenne_twister_engine invalid parameters");
    static_assert(__u <= __w, "mersenne_twister_engine invalid parameters");
    static_assert(__s <= __w, "mersenne_twister_engine invalid parameters");
    static_assert(__t <= __w, "mersenne_twister_engine invalid parameters");
    static_assert(__l <= __w, "mersenne_twister_engine invalid parameters");
public:
    static const result_type _Min = 0;
    static const result_type _Max = __w == _Dt ? result_type(~0) :
                                   (result_type(1) << __w) - result_type(1);
    static_assert(_Min < _Max, "mersenne_twister_engine invalid parameters");
    static_assert(__a <= _Max, "mersenne_twister_engine invalid parameters");
    static_assert(__b <= _Max, "mersenne_twister_engine invalid parameters");
    static_assert(__c <= _Max, "mersenne_twister_engine invalid parameters");
    static_assert(__d <= _Max, "mersenne_twister_engine invalid parameters");
    static_assert(__f <= _Max, "mersenne_twister_engine invalid parameters");

    // engine characteristics
    static const/*expr*/ size_t word_size = __w;
    static const/*expr*/ size_t state_size = __n;
    static const/*expr*/ size_t shift_size = __m;
    static const/*expr*/ size_t mask_bits = __r;
    static const/*expr*/ result_type xor_mask = __a;
    static const/*expr*/ size_t tempering_u = __u;
    static const/*expr*/ result_type tempering_d = __d;
    static const/*expr*/ size_t tempering_s = __s;
    static const/*expr*/ result_type tempering_b = __b;
    static const/*expr*/ size_t tempering_t = __t;
    static const/*expr*/ result_type tempering_c = __c;
    static const/*expr*/ size_t tempering_l = __l;
    static const/*expr*/ result_type initialization_multiplier = __f;
    static const/*expr*/ result_type min() { return _Min; }
    static const/*expr*/ result_type max() { return _Max; }
    static const/*expr*/ result_type default_seed = 5489u;

    // constructors and seeding functions
    explicit mersenne_twister_engine(result_type __sd = default_seed)
        {seed(__sd);}
    template<class _Sseq> explicit mersenne_twister_engine(_Sseq& __q)
        {seed(__q);}
    void seed(result_type __sd = default_seed);
    template<class _Sseq>
        typename enable_if
        <
            !is_convertible<_Sseq, result_type>::value,
            void
        >::type
        seed(_Sseq& __q)
            {__seed(__q, integral_constant<unsigned, 1 + (__w - 1) / 32>());}

    // generating functions
    result_type operator()();
    void discard(unsigned long long __z) {for (; __z; --__z) operator()();}

    template <class _UI, size_t _W, size_t _N, size_t _M, size_t _R,
              _UI _A, size_t _U, _UI _D, size_t _S,
              _UI _B, size_t _T, _UI _C, size_t _L, _UI _F>
    friend
    bool
    operator==(const mersenne_twister_engine<_UI, _W, _N, _M, _R, _A, _U, _D, _S,
                                             _B, _T, _C, _L, _F>& __x,
               const mersenne_twister_engine<_UI, _W, _N, _M, _R, _A, _U, _D, _S,
                                             _B, _T, _C, _L, _F>& __y);
    
    template <class _UI, size_t _W, size_t _N, size_t _M, size_t _R,
              _UI _A, size_t _U, _UI _D, size_t _S,
              _UI _B, size_t _T, _UI _C, size_t _L, _UI _F>
    friend
    bool
    operator!=(const mersenne_twister_engine<_UI, _W, _N, _M, _R, _A, _U, _D, _S,
                                             _B, _T, _C, _L, _F>& __x,
               const mersenne_twister_engine<_UI, _W, _N, _M, _R, _A, _U, _D, _S,
                                             _B, _T, _C, _L, _F>& __y);

    template <class _CharT, class _Traits,
              class _UI, size_t _W, size_t _N, size_t _M, size_t _R,
              _UI _A, size_t _U, _UI _D, size_t _S,
              _UI _B, size_t _T, _UI _C, size_t _L, _UI _F>
    friend
    basic_ostream<_CharT, _Traits>&
    operator<<(basic_ostream<_CharT, _Traits>& __os,
               const mersenne_twister_engine<_UI, _W, _N, _M, _R, _A, _U, _D, _S,
                                             _B, _T, _C, _L, _F>& __x);

    template <class _CharT, class _Traits,
              class _UI, size_t _W, size_t _N, size_t _M, size_t _R,
              _UI _A, size_t _U, _UI _D, size_t _S,
              _UI _B, size_t _T, _UI _C, size_t _L, _UI _F>
    friend
    basic_istream<_CharT, _Traits>&
    operator>>(basic_istream<_CharT, _Traits>& __is,
               mersenne_twister_engine<_UI, _W, _N, _M, _R, _A, _U, _D, _S,
                                       _B, _T, _C, _L, _F>& __x);
private:

    template<class _Sseq>
        void __seed(_Sseq& __q, integral_constant<unsigned, 1>);
    template<class _Sseq>
        void __seed(_Sseq& __q, integral_constant<unsigned, 2>);

    template <size_t __count>
        static
        typename enable_if
        <
            __count < __w,
            result_type
        >::type
        __lshift(result_type __x) {return (__x << __count) & _Max;}

    template <size_t __count>
        static
        typename enable_if
        <
            (__count >= __w),
            result_type
        >::type
        __lshift(result_type __x) {return result_type(0);}

    template <size_t __count>
        static
        typename enable_if
        <
            __count < _Dt,
            result_type
        >::type
        __rshift(result_type __x) {return __x >> __count;}

    template <size_t __count>
        static
        typename enable_if
        <
            (__count >= _Dt),
            result_type
        >::type
        __rshift(result_type __x) {return result_type(0);}
};

template <class _UIntType, size_t __w, size_t __n, size_t __m, size_t __r,
          _UIntType __a, size_t __u, _UIntType __d, size_t __s,
          _UIntType __b, size_t __t, _UIntType __c, size_t __l, _UIntType __f>
void
mersenne_twister_engine<_UIntType, __w, __n, __m, __r, __a, __u, __d, __s, __b,
    __t, __c, __l, __f>::seed(result_type __sd)
{   // __w >= 2
    __x_[0] = __sd & _Max;
    for (size_t __i = 1; __i < __n; ++__i)
        __x_[__i] = (__f * (__x_[__i-1] ^ __rshift<__w - 2>(__x_[__i-1])) + __i) & _Max;
    __i_ = 0;
}

template <class _UIntType, size_t __w, size_t __n, size_t __m, size_t __r,
          _UIntType __a, size_t __u, _UIntType __d, size_t __s,
          _UIntType __b, size_t __t, _UIntType __c, size_t __l, _UIntType __f>
template<class _Sseq>
void
mersenne_twister_engine<_UIntType, __w, __n, __m, __r, __a, __u, __d, __s, __b,
    __t, __c, __l, __f>::__seed(_Sseq& __q, integral_constant<unsigned, 1>)
{
    const unsigned __k = 1;
    uint32_t __ar[__n * __k];
    __q.generate(__ar, __ar + __n * __k);
    for (size_t __i = 0; __i < __n; ++__i)
        __x_[__i] = static_cast<result_type>(__ar[__i] & _Max);
    const result_type __mask = __r == _Dt ? result_type(~0) :
                                       (result_type(1) << __r) - result_type(1);
    __i_ = 0;
    if ((__x_[0] & ~__mask) == 0)
    {
        for (size_t __i = 1; __i < __n; ++__i)
            if (__x_[__i] != 0)
                return;
        __x_[0] = _Max;
    }
}

template <class _UIntType, size_t __w, size_t __n, size_t __m, size_t __r,
          _UIntType __a, size_t __u, _UIntType __d, size_t __s,
          _UIntType __b, size_t __t, _UIntType __c, size_t __l, _UIntType __f>
template<class _Sseq>
void
mersenne_twister_engine<_UIntType, __w, __n, __m, __r, __a, __u, __d, __s, __b,
    __t, __c, __l, __f>::__seed(_Sseq& __q, integral_constant<unsigned, 2>)
{
    const unsigned __k = 2;
    uint32_t __ar[__n * __k];
    __q.generate(__ar, __ar + __n * __k);
    for (size_t __i = 0; __i < __n; ++__i)
        __x_[__i] = static_cast<result_type>(
            (__ar[2 * __i] + ((uint64_t)__ar[2 * __i + 1] << 32)) & _Max);
    const result_type __mask = __r == _Dt ? result_type(~0) :
                                       (result_type(1) << __r) - result_type(1);
    __i_ = 0;
    if ((__x_[0] & ~__mask) == 0)
    {
        for (size_t __i = 1; __i < __n; ++__i)
            if (__x_[__i] != 0)
                return;
        __x_[0] = _Max;
    }
}

template <class _UIntType, size_t __w, size_t __n, size_t __m, size_t __r,
          _UIntType __a, size_t __u, _UIntType __d, size_t __s,
          _UIntType __b, size_t __t, _UIntType __c, size_t __l, _UIntType __f>
_UIntType
mersenne_twister_engine<_UIntType, __w, __n, __m, __r, __a, __u, __d, __s, __b,
    __t, __c, __l, __f>::operator()()
{
    const size_t __j = (__i_ + 1) % __n;
    const result_type __mask = __r == _Dt ? result_type(~0) :
                                       (result_type(1) << __r) - result_type(1);
    const result_type _Y = (__x_[__i_] & ~__mask) | (__x_[__j] & __mask);
    const size_t __k = (__i_ + __m) % __n;
    __x_[__i_] = __x_[__k] ^ __rshift<1>(_Y) ^ (__a * (_Y & 1));
    result_type __z = __x_[__i_] ^ (__rshift<__u>(__x_[__i_]) & __d);
    __i_ = __j;
    __z ^= __lshift<__s>(__z) & __b;
    __z ^= __lshift<__t>(__z) & __c;
    return __z ^ __rshift<__l>(__z);
}

template <class _UI, size_t _W, size_t _N, size_t _M, size_t _R,
          _UI _A, size_t _U, _UI _D, size_t _S,
          _UI _B, size_t _T, _UI _C, size_t _L, _UI _F>
bool
operator==(const mersenne_twister_engine<_UI, _W, _N, _M, _R, _A, _U, _D, _S,
                                         _B, _T, _C, _L, _F>& __x,
           const mersenne_twister_engine<_UI, _W, _N, _M, _R, _A, _U, _D, _S,
                                         _B, _T, _C, _L, _F>& __y)
{
    if (__x.__i_ == __y.__i_)
        return _STD::equal(__x.__x_, __x.__x_ + _N, __y.__x_);
    if (__x.__i_ == 0 || __y.__i_ == 0)
    {
        size_t __j = _STD::min(_N - __x.__i_, _N - __y.__i_);
        if (!_STD::equal(__x.__x_ + __x.__i_, __x.__x_ + __x.__i_ + __j,
                         __y.__x_ + __y.__i_))
            return false;
        if (__x.__i_ == 0)
            return _STD::equal(__x.__x_ + __j, __x.__x_ + _N, __y.__x_);
        return _STD::equal(__x.__x_, __x.__x_ + (_N - __j), __y.__x_ + __j);
    }
    if (__x.__i_ < __y.__i_)
    {
        size_t __j = _N - __y.__i_;
        if (!_STD::equal(__x.__x_ + __x.__i_, __x.__x_ + (__x.__i_ + __j),
                         __y.__x_ + __y.__i_))
            return false;
        if (!_STD::equal(__x.__x_ + (__x.__i_ + __j), __x.__x_ + _N,
                         __y.__x_))
            return false;
        return _STD::equal(__x.__x_, __x.__x_ + __x.__i_,
                           __y.__x_ + (_N - (__x.__i_ + __j)));
    }
    size_t __j = _N - __x.__i_;
    if (!_STD::equal(__y.__x_ + __y.__i_, __y.__x_ + (__y.__i_ + __j),
                     __x.__x_ + __x.__i_))
        return false;
    if (!_STD::equal(__y.__x_ + (__y.__i_ + __j), __y.__x_ + _N,
                     __x.__x_))
        return false;
    return _STD::equal(__y.__x_, __y.__x_ + __y.__i_,
                       __x.__x_ + (_N - (__y.__i_ + __j)));
}

template <class _UI, size_t _W, size_t _N, size_t _M, size_t _R,
          _UI _A, size_t _U, _UI _D, size_t _S,
          _UI _B, size_t _T, _UI _C, size_t _L, _UI _F>
inline
bool
operator!=(const mersenne_twister_engine<_UI, _W, _N, _M, _R, _A, _U, _D, _S,
                                         _B, _T, _C, _L, _F>& __x,
           const mersenne_twister_engine<_UI, _W, _N, _M, _R, _A, _U, _D, _S,
                                         _B, _T, _C, _L, _F>& __y)
{
    return !(__x == __y);
}

template <class _CharT, class _Traits,
          class _UI, size_t _W, size_t _N, size_t _M, size_t _R,
          _UI _A, size_t _U, _UI _D, size_t _S,
          _UI _B, size_t _T, _UI _C, size_t _L, _UI _F>
basic_ostream<_CharT, _Traits>&
operator<<(basic_ostream<_CharT, _Traits>& __os,
           const mersenne_twister_engine<_UI, _W, _N, _M, _R, _A, _U, _D, _S,
                                         _B, _T, _C, _L, _F>& __x)
{
    __save_flags<_CharT, _Traits> _(__os);
    __os.flags(ios_base::dec | ios_base::left);
    _CharT __sp = __os.widen(' ');
    __os.fill(__sp);
    __os << __x.__x_[__x.__i_];
    for (size_t __j = __x.__i_ + 1; __j < _N; ++__j)
        __os << __sp << __x.__x_[__j];
    for (size_t __j = 0; __j < __x.__i_; ++__j)
        __os << __sp << __x.__x_[__j];
    return __os;
}

template <class _CharT, class _Traits,
          class _UI, size_t _W, size_t _N, size_t _M, size_t _R,
          _UI _A, size_t _U, _UI _D, size_t _S,
          _UI _B, size_t _T, _UI _C, size_t _L, _UI _F>
basic_istream<_CharT, _Traits>&
operator>>(basic_istream<_CharT, _Traits>& __is,
           mersenne_twister_engine<_UI, _W, _N, _M, _R, _A, _U, _D, _S,
                                   _B, _T, _C, _L, _F>& __x)
{
    __save_flags<_CharT, _Traits> _(__is);
    __is.flags(ios_base::dec | ios_base::skipws);
    _UI __t[_N];
    for (size_t __i = 0; __i < _N; ++__i)
        __is >> __t[__i];
    if (!__is.fail())
    {
        for (size_t __i = 0; __i < _N; ++__i)
            __x.__x_[__i] = __t[__i];
        __x.__i_ = 0;
    }
    return __is;
}

typedef mersenne_twister_engine<uint_fast32_t, 32, 624, 397, 31,
                                0x9908b0df, 11, 0xffffffff,
                                7,  0x9d2c5680,
                                15, 0xefc60000,
                                18, 1812433253>                         mt19937;
typedef mersenne_twister_engine<uint_fast64_t, 64, 312, 156, 31,
                                0xb5026f5aa96619e9ULL, 29, 0x5555555555555555ULL,
                                17, 0x71d67fffeda60000ULL,
                                37, 0xfff7eee000000000ULL,
                                43, 6364136223846793005ULL>          mt19937_64;

// subtract_with_carry_engine

template<class _UIntType, size_t __w, size_t __s, size_t __r>
class subtract_with_carry_engine;

template<class _UI, size_t _W, size_t _S, size_t _R>
bool
operator==(
    const subtract_with_carry_engine<_UI, _W, _S, _R>& __x,
    const subtract_with_carry_engine<_UI, _W, _S, _R>& __y);

template<class _UI, size_t _W, size_t _S, size_t _R>
bool
operator!=(
    const subtract_with_carry_engine<_UI, _W, _S, _R>& __x,
    const subtract_with_carry_engine<_UI, _W, _S, _R>& __y);

template <class _CharT, class _Traits,
          class _UI, size_t _W, size_t _S, size_t _R>
basic_ostream<_CharT, _Traits>&
operator<<(basic_ostream<_CharT, _Traits>& __os,
           const subtract_with_carry_engine<_UI, _W, _S, _R>& __x);

template <class _CharT, class _Traits,
          class _UI, size_t _W, size_t _S, size_t _R>
basic_istream<_CharT, _Traits>&
operator>>(basic_istream<_CharT, _Traits>& __is,
           subtract_with_carry_engine<_UI, _W, _S, _R>& __x);

template<class _UIntType, size_t __w, size_t __s, size_t __r>
class subtract_with_carry_engine
{
public:
    // types
    typedef _UIntType result_type;

private:
    result_type __x_[__r];
    result_type  __c_;
    size_t      __i_;

    static const result_type _Dt = numeric_limits<result_type>::digits;
    static_assert(  0 <  __w, "subtract_with_carry_engine invalid parameters");
    static_assert(__w <= _Dt, "subtract_with_carry_engine invalid parameters");
    static_assert(  0 <  __s, "subtract_with_carry_engine invalid parameters");
    static_assert(__s <  __r, "subtract_with_carry_engine invalid parameters");
public:
    static const result_type _Min = 0;
    static const result_type _Max = __w == _Dt ? result_type(~0) :
                                   (result_type(1) << __w) - result_type(1);
    static_assert(_Min < _Max, "subtract_with_carry_engine invalid parameters");

    // engine characteristics
    static const/*expr*/ size_t word_size = __w;
    static const/*expr*/ size_t short_lag = __s;
    static const/*expr*/ size_t long_lag = __r;
    static const/*expr*/ result_type min() { return _Min; }
    static const/*expr*/ result_type max() { return _Max; }
    static const/*expr*/ result_type default_seed = 19780503u;

    // constructors and seeding functions
    explicit subtract_with_carry_engine(result_type __sd = default_seed)
        {seed(__sd);}
    template<class _Sseq> explicit subtract_with_carry_engine(_Sseq& __q)
        {seed(__q);}
    void seed(result_type __sd = default_seed)
        {seed(__sd, integral_constant<unsigned, 1 + (__w - 1) / 32>());}
    template<class _Sseq>
        typename enable_if
        <
            !is_convertible<_Sseq, result_type>::value,
            void
        >::type
        seed(_Sseq& __q)
            {__seed(__q, integral_constant<unsigned, 1 + (__w - 1) / 32>());}

    // generating functions
    result_type operator()();
    void discard(unsigned long long __z) {for (; __z; --__z) operator()();}

    template<class _UI, size_t _W, size_t _S, size_t _R>
    friend
    bool
    operator==(
        const subtract_with_carry_engine<_UI, _W, _S, _R>& __x,
        const subtract_with_carry_engine<_UI, _W, _S, _R>& __y);
    
    template<class _UI, size_t _W, size_t _S, size_t _R>
    friend
    bool
    operator!=(
        const subtract_with_carry_engine<_UI, _W, _S, _R>& __x,
        const subtract_with_carry_engine<_UI, _W, _S, _R>& __y);
    
    template <class _CharT, class _Traits,
              class _UI, size_t _W, size_t _S, size_t _R>
    friend
    basic_ostream<_CharT, _Traits>&
    operator<<(basic_ostream<_CharT, _Traits>& __os,
               const subtract_with_carry_engine<_UI, _W, _S, _R>& __x);
    
    template <class _CharT, class _Traits,
              class _UI, size_t _W, size_t _S, size_t _R>
    friend
    basic_istream<_CharT, _Traits>&
    operator>>(basic_istream<_CharT, _Traits>& __is,
               subtract_with_carry_engine<_UI, _W, _S, _R>& __x);

private:

    void seed(result_type __sd, integral_constant<unsigned, 1>);
    void seed(result_type __sd, integral_constant<unsigned, 2>);
    template<class _Sseq>
        void __seed(_Sseq& __q, integral_constant<unsigned, 1>);
    template<class _Sseq>
        void __seed(_Sseq& __q, integral_constant<unsigned, 2>);
};

template<class _UIntType, size_t __w, size_t __s, size_t __r>
void
subtract_with_carry_engine<_UIntType, __w, __s, __r>::seed(result_type __sd,
        integral_constant<unsigned, 1>)
{
    linear_congruential_engine<result_type, 40014u, 0u, 2147483563u>
        __e(__sd == 0u ? default_seed : __sd);
    for (size_t __i = 0; __i < __r; ++__i)
        __x_[__i] = static_cast<result_type>(__e() & _Max);
    __c_ = __x_[__r-1] == 0;
    __i_ = 0;
}

template<class _UIntType, size_t __w, size_t __s, size_t __r>
void
subtract_with_carry_engine<_UIntType, __w, __s, __r>::seed(result_type __sd,
        integral_constant<unsigned, 2>)
{
    linear_congruential_engine<result_type, 40014u, 0u, 2147483563u>
        __e(__sd == 0u ? default_seed : __sd);
    for (size_t __i = 0; __i < __r; ++__i)
        __x_[__i] = static_cast<result_type>(
                                    (__e() + ((uint64_t)__e() << 32)) & _Max);
    __c_ = __x_[__r-1] == 0;
    __i_ = 0;
}

template<class _UIntType, size_t __w, size_t __s, size_t __r>
template<class _Sseq>
void
subtract_with_carry_engine<_UIntType, __w, __s, __r>::__seed(_Sseq& __q,
        integral_constant<unsigned, 1>)
{
    const unsigned __k = 1;
    uint32_t __ar[__r * __k];
    __q.generate(__ar, __ar + __r * __k);
    for (size_t __i = 0; __i < __r; ++__i)
        __x_[__i] = static_cast<result_type>(__ar[__i] & _Max);
    __c_ = __x_[__r-1] == 0;
    __i_ = 0;
}

template<class _UIntType, size_t __w, size_t __s, size_t __r>
template<class _Sseq>
void
subtract_with_carry_engine<_UIntType, __w, __s, __r>::__seed(_Sseq& __q,
        integral_constant<unsigned, 2>)
{
    const unsigned __k = 2;
    uint32_t __ar[__r * __k];
    __q.generate(__ar, __ar + __r * __k);
    for (size_t __i = 0; __i < __r; ++__i)
        __x_[__i] = static_cast<result_type>(
                  (__ar[2 * __i] + ((uint64_t)__ar[2 * __i + 1] << 32)) & _Max);
    __c_ = __x_[__r-1] == 0;
    __i_ = 0;
}

template<class _UIntType, size_t __w, size_t __s, size_t __r>
_UIntType
subtract_with_carry_engine<_UIntType, __w, __s, __r>::operator()()
{
    const result_type& __xs = __x_[(__i_ + (__r - __s)) % __r];
    result_type& __xr = __x_[__i_];
    result_type __new_c = __c_ == 0 ? __xs < __xr : __xs != 0 ? __xs <= __xr : 1;
    __xr = (__xs - __xr - __c_) & _Max;
    __c_ = __new_c;
    __i_ = (__i_ + 1) % __r;
    return __xr;
}

template<class _UI, size_t _W, size_t _S, size_t _R>
bool
operator==(
    const subtract_with_carry_engine<_UI, _W, _S, _R>& __x,
    const subtract_with_carry_engine<_UI, _W, _S, _R>& __y)
{
    if (__x.__c_ != __y.__c_)
        return false;
    if (__x.__i_ == __y.__i_)
        return _STD::equal(__x.__x_, __x.__x_ + _R, __y.__x_);
    if (__x.__i_ == 0 || __y.__i_ == 0)
    {
        size_t __j = _STD::min(_R - __x.__i_, _R - __y.__i_);
        if (!_STD::equal(__x.__x_ + __x.__i_, __x.__x_ + __x.__i_ + __j,
                         __y.__x_ + __y.__i_))
            return false;
        if (__x.__i_ == 0)
            return _STD::equal(__x.__x_ + __j, __x.__x_ + _R, __y.__x_);
        return _STD::equal(__x.__x_, __x.__x_ + (_R - __j), __y.__x_ + __j);
    }
    if (__x.__i_ < __y.__i_)
    {
        size_t __j = _R - __y.__i_;
        if (!_STD::equal(__x.__x_ + __x.__i_, __x.__x_ + (__x.__i_ + __j),
                         __y.__x_ + __y.__i_))
            return false;
        if (!_STD::equal(__x.__x_ + (__x.__i_ + __j), __x.__x_ + _R,
                         __y.__x_))
            return false;
        return _STD::equal(__x.__x_, __x.__x_ + __x.__i_,
                           __y.__x_ + (_R - (__x.__i_ + __j)));
    }
    size_t __j = _R - __x.__i_;
    if (!_STD::equal(__y.__x_ + __y.__i_, __y.__x_ + (__y.__i_ + __j),
                     __x.__x_ + __x.__i_))
        return false;
    if (!_STD::equal(__y.__x_ + (__y.__i_ + __j), __y.__x_ + _R,
                     __x.__x_))
        return false;
    return _STD::equal(__y.__x_, __y.__x_ + __y.__i_,
                       __x.__x_ + (_R - (__y.__i_ + __j)));
}

template<class _UI, size_t _W, size_t _S, size_t _R>
inline
bool
operator!=(
    const subtract_with_carry_engine<_UI, _W, _S, _R>& __x,
    const subtract_with_carry_engine<_UI, _W, _S, _R>& __y)
{
    return !(__x == __y);
}

template <class _CharT, class _Traits,
          class _UI, size_t _W, size_t _S, size_t _R>
basic_ostream<_CharT, _Traits>&
operator<<(basic_ostream<_CharT, _Traits>& __os,
           const subtract_with_carry_engine<_UI, _W, _S, _R>& __x)
{
    __save_flags<_CharT, _Traits> _(__os);
    __os.flags(ios_base::dec | ios_base::left);
    _CharT __sp = __os.widen(' ');
    __os.fill(__sp);
    __os << __x.__x_[__x.__i_];
    for (size_t __j = __x.__i_ + 1; __j < _R; ++__j)
        __os << __sp << __x.__x_[__j];
    for (size_t __j = 0; __j < __x.__i_; ++__j)
        __os << __sp << __x.__x_[__j];
    __os << __sp << __x.__c_;
    return __os;
}

template <class _CharT, class _Traits,
          class _UI, size_t _W, size_t _S, size_t _R>
basic_istream<_CharT, _Traits>&
operator>>(basic_istream<_CharT, _Traits>& __is,
           subtract_with_carry_engine<_UI, _W, _S, _R>& __x)
{
    __save_flags<_CharT, _Traits> _(__is);
    __is.flags(ios_base::dec | ios_base::skipws);
    _UI __t[_R+1];
    for (size_t __i = 0; __i < _R+1; ++__i)
        __is >> __t[__i];
    if (!__is.fail())
    {
        for (size_t __i = 0; __i < _R; ++__i)
            __x.__x_[__i] = __t[__i];
        __x.__c_ = __t[_R];
        __x.__i_ = 0;
    }
    return __is;
}

typedef subtract_with_carry_engine<uint_fast32_t, 24, 10, 24>     ranlux24_base;
typedef subtract_with_carry_engine<uint_fast64_t, 48,  5, 12>     ranlux48_base;

// discard_block_engine

template<class _Engine, size_t __p, size_t __r>
class discard_block_engine
{
    _Engine __e_;
    int     __n_;

    static_assert(  0 <  __r, "discard_block_engine invalid parameters");
    static_assert(__r <= __p, "discard_block_engine invalid parameters");
public:
    // types
    typedef typename _Engine::result_type result_type;

    // engine characteristics
    static const/*expr*/ size_t block_size = __p;
    static const/*expr*/ size_t used_block = __r;

    // Temporary work around for lack of constexpr
    static const result_type _Min = _Engine::_Min;
    static const result_type _Max = _Engine::_Max;

    static const/*expr*/ result_type min() { return _Engine::min(); }
    static const/*expr*/ result_type max() { return _Engine::max(); }

    // constructors and seeding functions
    discard_block_engine() : __n_(0) {}
//     explicit discard_block_engine(const _Engine& __e);
//     explicit discard_block_engine(_Engine&& __e);
    explicit discard_block_engine(result_type __sd) : __e_(__sd), __n_(0) {}
    template<class _Sseq> explicit discard_block_engine(_Sseq& __q)
        : __e_(__q), __n_(0) {}
    void seed() {__e_.seed(); __n_ = 0;}
    void seed(result_type __sd) {__e_.seed(__sd); __n_ = 0;}
    template<class _Sseq> void seed(_Sseq& __q) {__e_.seed(__q); __n_ = 0;}

    // generating functions
    result_type operator()();
    void discard(unsigned long long __z) {for (; __z; --__z) operator()();}

    // property functions
    const _Engine& base() const {return __e_;}

    template<class _Eng, size_t _P, size_t _R>
    friend
    bool
    operator==(
        const discard_block_engine<_Eng, _P, _R>& __x,
        const discard_block_engine<_Eng, _P, _R>& __y);
    
    template<class _Eng, size_t _P, size_t _R>
    friend
    bool
    operator!=(
        const discard_block_engine<_Eng, _P, _R>& __x,
        const discard_block_engine<_Eng, _P, _R>& __y);
    
    template <class _CharT, class _Traits,
              class _Eng, size_t _P, size_t _R>
    friend
    basic_ostream<_CharT, _Traits>&
    operator<<(basic_ostream<_CharT, _Traits>& __os,
               const discard_block_engine<_Eng, _P, _R>& __x);
    
    template <class _CharT, class _Traits,
              class _Eng, size_t _P, size_t _R>
    friend
    basic_istream<_CharT, _Traits>&
    operator>>(basic_istream<_CharT, _Traits>& __is,
               discard_block_engine<_Eng, _P, _R>& __x);
};

template<class _Engine, size_t __p, size_t __r>
typename discard_block_engine<_Engine, __p, __r>::result_type
discard_block_engine<_Engine, __p, __r>::operator()()
{
    if (__n_ >= __r)
    {
        __e_.discard(__p - __r);
        __n_ = 0;
    }
    ++__n_;
    return __e_();
}

template<class _Eng, size_t _P, size_t _R>
inline
bool
operator==(const discard_block_engine<_Eng, _P, _R>& __x,
           const discard_block_engine<_Eng, _P, _R>& __y)
{
    return __x.__n_ == __y.__n_ && __x.__e_ == __y.__e_;
}

template<class _Eng, size_t _P, size_t _R>
inline
bool
operator!=(const discard_block_engine<_Eng, _P, _R>& __x,
           const discard_block_engine<_Eng, _P, _R>& __y)
{
    return !(__x == __y);
}

template <class _CharT, class _Traits,
          class _Eng, size_t _P, size_t _R>
basic_ostream<_CharT, _Traits>&
operator<<(basic_ostream<_CharT, _Traits>& __os,
           const discard_block_engine<_Eng, _P, _R>& __x)
{
    __save_flags<_CharT, _Traits> _(__os);
    __os.flags(ios_base::dec | ios_base::left);
    _CharT __sp = __os.widen(' ');
    __os.fill(__sp);
    return __os << __x.__e_ << __sp << __x.__n_;
}

template <class _CharT, class _Traits,
          class _Eng, size_t _P, size_t _R>
basic_istream<_CharT, _Traits>&
operator>>(basic_istream<_CharT, _Traits>& __is,
           discard_block_engine<_Eng, _P, _R>& __x)
{
    __save_flags<_CharT, _Traits> _(__is);
    __is.flags(ios_base::dec | ios_base::skipws);
    _Eng __e;
    int __n;
    __is >> __e >> __n;
    if (!__is.fail())
    {
        __x.__e_ = __e;
        __x.__n_ = __n;
    }
    return __is;
}

typedef discard_block_engine<ranlux24_base, 223, 23> ranlux24;
typedef discard_block_engine<ranlux48_base, 389, 11> ranlux48;

// independent_bits_engine

template <unsigned long long _X, size_t _R>
struct __log2_imp
{
    static const size_t value = _X & ((unsigned long long)(1) << _R) ? _R
                                           : __log2_imp<_X, _R - 1>::value;
};

template <unsigned long long _X>
struct __log2_imp<_X, 0>
{
    static const size_t value = 0;
};

template <size_t _R>
struct __log2_imp<0, _R>
{
    static const size_t value = _R + 1;
};

template <class _UI, _UI _X>
struct __log2
{
    static const size_t value = __log2_imp<_X,
                                         sizeof(_UI) * __CHAR_BIT__ - 1>::value;
};

template<class _Engine, size_t __w, class _UIntType>
class independent_bits_engine
{
    template <class _UI, _UI _R0, size_t _W, size_t _M>
    class __get_n
    {
        static const size_t _Dt = numeric_limits<_UI>::digits;
        static const size_t _N = _W / _M + (_W % _M != 0);
        static const size_t _W0 = _W / _N;
        static const _UI _Y0 = _W0 >= _Dt ? 0 : (_R0 >> _W0) << _W0;
    public:
        static const size_t value = _R0 - _Y0 > _Y0 / _N ? _N + 1 : _N;
    };
public:
    // types
    typedef _UIntType result_type;

private:
    _Engine __e_;

    static const result_type _Dt = numeric_limits<result_type>::digits;
    static_assert(  0 <  __w, "independent_bits_engine invalid parameters");
    static_assert(__w <= _Dt, "independent_bits_engine invalid parameters");

    typedef typename _Engine::result_type _Engine_result_type;
    typedef typename conditional
        <
            sizeof(_Engine_result_type) <= sizeof(result_type),
                result_type,
                _Engine_result_type
        >::type _Working_result_type;
    // Temporary work around for lack of constexpr
    static const _Working_result_type _R = _Engine::_Max - _Engine::_Min
                                                         + _Working_result_type(1);
    static const size_t __m = __log2<_Working_result_type, _R>::value;
    static const size_t __n = __get_n<_Working_result_type, _R, __w, __m>::value;
    static const size_t __w0 = __w / __n;
    static const size_t __n0 = __n - __w % __n;
    static const size_t _WDt = numeric_limits<_Working_result_type>::digits;
    static const size_t _EDt = numeric_limits<_Engine_result_type>::digits;
    static const _Working_result_type __y0 = __w0 >= _WDt ? 0 :
                                                   (_R >> __w0) << __w0;
    static const _Working_result_type __y1 = __w0 >= _WDt - 1 ? 0 :
                                                   (_R >> (__w0+1)) << (__w0+1);
    static const _Engine_result_type __mask0 = __w0 > 0 ?
                                _Engine_result_type(~0) >> (_EDt - __w0) :
                                _Engine_result_type(0);
    static const _Engine_result_type __mask1 = __w0 < _EDt - 1 ?
                                _Engine_result_type(~0) >> (_EDt - (__w0 + 1)) :
                                _Engine_result_type(~0);
public:
    static const result_type _Min = 0;
    static const result_type _Max = __w == _Dt ? result_type(~0) :
                                   (result_type(1) << __w) - result_type(1);
    static_assert(_Min < _Max, "independent_bits_engine invalid parameters");

    // engine characteristics
    static const/*expr*/ result_type min() { return _Min; }
    static const/*expr*/ result_type max() { return _Max; }

    // constructors and seeding functions
    independent_bits_engine() {}
//    explicit independent_bits_engine(const _Engine& __e);
//    explicit independent_bits_engine(_Engine&& __e);
    explicit independent_bits_engine(result_type __sd) : __e_(__sd) {}
    template<class _Sseq> explicit independent_bits_engine(_Sseq& __q)
         : __e_(__q) {}
    void seed() {__e_.seed();}
    void seed(result_type __sd) {__e_.seed(__sd);}
    template<class _Sseq> void seed(_Sseq& __q) {__e_.seed(__q);}

    // generating functions
    result_type operator()() {return __eval(integral_constant<bool, _R != 0>());}
    void discard(unsigned long long __z) {for (; __z; --__z) operator()();}

    // property functions
    const _Engine& base() const {return __e_;}

    template<class _Eng, size_t _W, class _UI>
    friend
    bool
    operator==(
        const independent_bits_engine<_Eng, _W, _UI>& __x,
        const independent_bits_engine<_Eng, _W, _UI>& __y);
    
    template<class _Eng, size_t _W, class _UI>
    friend
    bool
    operator!=(
        const independent_bits_engine<_Eng, _W, _UI>& __x,
        const independent_bits_engine<_Eng, _W, _UI>& __y);
    
    template <class _CharT, class _Traits,
              class _Eng, size_t _W, class _UI>
    friend
    basic_ostream<_CharT, _Traits>&
    operator<<(basic_ostream<_CharT, _Traits>& __os,
               const independent_bits_engine<_Eng, _W, _UI>& __x);
    
    template <class _CharT, class _Traits,
              class _Eng, size_t _W, class _UI>
    friend
    basic_istream<_CharT, _Traits>&
    operator>>(basic_istream<_CharT, _Traits>& __is,
               independent_bits_engine<_Eng, _W, _UI>& __x);

private:
    result_type __eval(false_type);
    result_type __eval(true_type);

    template <size_t __count>
        static
        typename enable_if
        <
            __count < _Dt,
            result_type
        >::type
        __lshift(result_type __x) {return __x << __count;}

    template <size_t __count>
        static
        typename enable_if
        <
            (__count >= _Dt),
            result_type
        >::type
        __lshift(result_type __x) {return result_type(0);}
};

template<class _Engine, size_t __w, class _UIntType>
inline
_UIntType
independent_bits_engine<_Engine, __w, _UIntType>::__eval(false_type)
{
    return static_cast<result_type>(__e_() & __mask0);
}

template<class _Engine, size_t __w, class _UIntType>
_UIntType
independent_bits_engine<_Engine, __w, _UIntType>::__eval(true_type)
{
    result_type _S = 0;
    for (size_t __k = 0; __k < __n0; ++__k)
    {
        _Engine_result_type __u;
        do
        {
            __u = __e_() - _Engine::min();
        } while (__u >= __y0);
        _S = static_cast<result_type>(__lshift<__w0>(_S) + (__u & __mask0));
    }
    for (size_t __k = __n0; __k < __n; ++__k)
    {
        _Engine_result_type __u;
        do
        {
            __u = __e_() - _Engine::min();
        } while (__u >= __y1);
        _S = static_cast<result_type>(__lshift<__w0+1>(_S) + (__u & __mask1));
    }
    return _S;
}

template<class _Eng, size_t _W, class _UI>
inline
bool
operator==(
    const independent_bits_engine<_Eng, _W, _UI>& __x,
    const independent_bits_engine<_Eng, _W, _UI>& __y)
{
    return __x.base() == __y.base();
}

template<class _Eng, size_t _W, class _UI>
inline
bool
operator!=(
    const independent_bits_engine<_Eng, _W, _UI>& __x,
    const independent_bits_engine<_Eng, _W, _UI>& __y)
{
    return !(__x == __y);
}

template <class _CharT, class _Traits,
          class _Eng, size_t _W, class _UI>
basic_ostream<_CharT, _Traits>&
operator<<(basic_ostream<_CharT, _Traits>& __os,
           const independent_bits_engine<_Eng, _W, _UI>& __x)
{
    return __os << __x.base();
}

template <class _CharT, class _Traits,
          class _Eng, size_t _W, class _UI>
basic_istream<_CharT, _Traits>&
operator>>(basic_istream<_CharT, _Traits>& __is,
           independent_bits_engine<_Eng, _W, _UI>& __x)
{
    _Eng __e;
    __is >> __e;
    if (!__is.fail())
        __x.__e_ = __e;
    return __is;
}

// shuffle_order_engine

template <uint64_t _Xp, uint64_t _Yp>
struct __ugcd
{
    static const uint64_t value = __ugcd<_Yp, _Xp % _Yp>::value;
};

template <uint64_t _Xp>
struct __ugcd<_Xp, 0>
{
    static const uint64_t value = _Xp;
};

template <uint64_t _N, uint64_t _D>
class __uratio
{
    static_assert(_D != 0, "__uratio divide by 0");
    static const uint64_t __gcd = __ugcd<_N, _D>::value;
public:
    static const uint64_t num = _N / __gcd;
    static const uint64_t den = _D / __gcd;

    typedef __uratio<num, den> type;
};

template<class _Engine, size_t __k>
class shuffle_order_engine
{
    static_assert(0 < __k, "shuffle_order_engine invalid parameters");
public:
    // types
    typedef typename _Engine::result_type result_type;

private:
    _Engine __e_;
    result_type _V_[__k];
    result_type _Y_;

public:
    // engine characteristics
    static const/*expr*/ size_t table_size = __k;
    
    static const result_type _Min = _Engine::_Min;
    static const result_type _Max = _Engine::_Max;
    static_assert(_Min < _Max, "shuffle_order_engine invalid parameters");
    static const/*expr*/ result_type min() { return _Min; }
    static const/*expr*/ result_type max() { return _Max; }

    static const unsigned long long _R = _Max - _Min + 1ull;

    // constructors and seeding functions
    shuffle_order_engine() {__init();}
//    explicit shuffle_order_engine(const _Engine& __e);
//    explicit shuffle_order_engine(_Engine&& e);
    explicit shuffle_order_engine(result_type __sd) : __e_(__sd) {__init();}
    template<class _Sseq> explicit shuffle_order_engine(_Sseq& __q)
         : __e_(__q) {__init();}
    void seed() {__e_.seed(); __init();}
    void seed(result_type __sd) {__e_.seed(__sd); __init();}
    template<class _Sseq> void seed(_Sseq& __q) {__e_.seed(__q); __init();}

    // generating functions
    result_type operator()() {return __eval(integral_constant<bool, _R != 0>());}
    void discard(unsigned long long __z) {for (; __z; --__z) operator()();}

    // property functions
    const _Engine& base() const {return __e_;}

private:
    template<class _Eng, size_t _K>
    friend
    bool
    operator==(
        const shuffle_order_engine<_Eng, _K>& __x,
        const shuffle_order_engine<_Eng, _K>& __y);
    
    template<class _Eng, size_t _K>
    friend
    bool
    operator!=(
        const shuffle_order_engine<_Eng, _K>& __x,
        const shuffle_order_engine<_Eng, _K>& __y);
    
    template <class _CharT, class _Traits,
              class _Eng, size_t _K>
    friend
    basic_ostream<_CharT, _Traits>&
    operator<<(basic_ostream<_CharT, _Traits>& __os,
               const shuffle_order_engine<_Eng, _K>& __x);
    
    template <class _CharT, class _Traits,
              class _Eng, size_t _K>
    friend
    basic_istream<_CharT, _Traits>&
    operator>>(basic_istream<_CharT, _Traits>& __is,
               shuffle_order_engine<_Eng, _K>& __x);

    void __init()
    {
        for (size_t __i = 0; __i < __k; ++__i)
            _V_[__i] = __e_();
        _Y_ = __e_();
    }

    result_type __eval(false_type) {return __eval2(integral_constant<bool, __k & 1>());}
    result_type __eval(true_type) {return __eval(__uratio<__k, _R>());}

    result_type __eval2(false_type) {return __eval(__uratio<__k/2, 0x8000000000000000ull>());}
    result_type __eval2(true_type) {return __evalf<__k, 0>();}

    template <uint64_t _N, uint64_t _D>
        typename enable_if
        <
            (__uratio<_N, _D>::num > 0xFFFFFFFFFFFFFFFFull / (_Max - _Min)),
            result_type
        >::type
        __eval(__uratio<_N, _D>)
            {return __evalf<__uratio<_N, _D>::num, __uratio<_N, _D>::den>();}

    template <uint64_t _N, uint64_t _D>
        typename enable_if
        <
            __uratio<_N, _D>::num <= 0xFFFFFFFFFFFFFFFFull / (_Max - _Min),
            result_type
        >::type
        __eval(__uratio<_N, _D>)
        {
            const size_t __j = static_cast<size_t>(__uratio<_N, _D>::num * (_Y_ - _Min)
                                                   / __uratio<_N, _D>::den);
            _Y_ = _V_[__j];
            _V_[__j] = __e_();
            return _Y_;
        }

    template <uint64_t __n, uint64_t __d>
        result_type __evalf()
        {
            const double _F = __d == 0 ?
                __n / (2. * 0x8000000000000000ull) :
                __n / (double)__d;
            const size_t __j = static_cast<size_t>(_F * (_Y_ - _Min));
            _Y_ = _V_[__j];
            _V_[__j] = __e_();
            return _Y_;
        }
};

template<class _Eng, size_t _K>
bool
operator==(
    const shuffle_order_engine<_Eng, _K>& __x,
    const shuffle_order_engine<_Eng, _K>& __y)
{
    return __x._Y_ == __y._Y_ && _STD::equal(__x._V_, __x._V_ + _K, __y._V_) &&
           __x.__e_ == __y.__e_;
}

template<class _Eng, size_t _K>
inline
bool
operator!=(
    const shuffle_order_engine<_Eng, _K>& __x,
    const shuffle_order_engine<_Eng, _K>& __y)
{
    return !(__x == __y);
}

template <class _CharT, class _Traits,
          class _Eng, size_t _K>
basic_ostream<_CharT, _Traits>&
operator<<(basic_ostream<_CharT, _Traits>& __os,
           const shuffle_order_engine<_Eng, _K>& __x)
{
    __save_flags<_CharT, _Traits> _(__os);
    __os.flags(ios_base::dec | ios_base::left);
    _CharT __sp = __os.widen(' ');
    __os.fill(__sp);
    __os << __x.__e_ << __sp << __x._V_[0];
    for (size_t __i = 1; __i < _K; ++__i)
        __os << __sp << __x._V_[__i];
    return __os << __sp << __x._Y_;
}

template <class _CharT, class _Traits,
          class _Eng, size_t _K>
basic_istream<_CharT, _Traits>&
operator>>(basic_istream<_CharT, _Traits>& __is,
           shuffle_order_engine<_Eng, _K>& __x)
{
    typedef typename shuffle_order_engine<_Eng, _K>::result_type result_type;
    __save_flags<_CharT, _Traits> _(__is);
    __is.flags(ios_base::dec | ios_base::skipws);
    _Eng __e;
    result_type _V[_K+1];
    __is >> __e;
    for (size_t __i = 0; __i < _K+1; ++__i)
        __is >> _V[__i];
    if (!__is.fail())
    {
        __x.__e_ = __e;
        for (size_t __i = 0; __i < _K; ++__i)
            __x._V_[__i] = _V[__i];
        __x._Y_ = _V[_K];
    }
    return __is;
}

typedef shuffle_order_engine<minstd_rand0, 256>                         knuth_b;

// random_device

class random_device
{
    int __f_;
public:
    // types
    typedef unsigned result_type;

    // generator characteristics
    static const result_type _Min = 0;
    static const result_type _Max = 0xFFFFFFFFu;

    static const/*expr*/ result_type min() { return _Min;}
    static const/*expr*/ result_type max() { return _Max;}

    // constructors
    explicit random_device(const string& __token = "/dev/urandom");
    ~random_device();

    // generating functions
    result_type operator()();

    // property functions
    double entropy() const;

private:
    // no copy functions
    random_device(const random_device&); // = delete;
    random_device& operator=(const random_device&); // = delete;
};

// seed_seq

class seed_seq
{
public:
    // types
    typedef uint32_t result_type;

private:
    vector<result_type> __v_;

    template<class _InputIterator>
        void init(_InputIterator __first, _InputIterator __last);
public:
    // constructors
    seed_seq() {}
    template<class _Tp>
        seed_seq(initializer_list<_Tp> __il) {init(__il.begin(), __il.end());}
            
    template<class _InputIterator>
        seed_seq(_InputIterator __first, _InputIterator __last)
             {init(__first, __last);}

    // generating functions
    template<class _RandomAccessIterator>
        void generate(_RandomAccessIterator __first, _RandomAccessIterator __last);

    // property functions
    size_t size() const {return __v_.size();}
    template<class _OutputIterator>
        void param(_OutputIterator __dest) const
            {_STD::copy(__v_.begin(), __v_.end(), __dest);}

private:
    // no copy functions
    seed_seq(const seed_seq&); // = delete;
    void operator=(const seed_seq&); // = delete;

    static result_type _T(result_type __x) {return __x ^ (__x >> 27);}
};

template<class _InputIterator>
void
seed_seq::init(_InputIterator __first, _InputIterator __last)
{
    for (_InputIterator __s = __first; __s != __last; ++__s)
        __v_.push_back(*__s & 0xFFFFFFFF);
}

template<class _RandomAccessIterator>
void
seed_seq::generate(_RandomAccessIterator __first, _RandomAccessIterator __last)
{
    if (__first != __last)
    {
        _STD::fill(__first, __last, 0x8b8b8b8b);
        const size_t __n = static_cast<size_t>(__last - __first);
        const size_t __s = __v_.size();
        const size_t __t = (__n >= 623) ? 11
                         : (__n >= 68) ? 7
                         : (__n >= 39) ? 5
                         : (__n >= 7)  ? 3
                         : (__n - 1) / 2;
        const size_t __p = (__n - __t) / 2;
        const size_t __q = __p + __t;
        const size_t __m = _STD::max(__s + 1, __n);
        // __k = 0;
        {
            result_type __r = 1664525 * _T(__first[0] ^ __first[__p]
                                                      ^  __first[__n - 1]);
            __first[__p] += __r;
            __r += __s;
            __first[__q] += __r;
            __first[0] = __r;
        }
        for (size_t __k = 1; __k <= __s; ++__k)
        {
            const size_t __kmodn = __k % __n;
            const size_t __kpmodn = (__k + __p) % __n;
            result_type __r = 1664525 * _T(__first[__kmodn] ^ __first[__kpmodn]
                                           ^ __first[(__k - 1) % __n]);
            __first[__kpmodn] += __r;
            __r +=  __kmodn + __v_[__k-1];
            __first[(__k + __q) % __n] += __r;
            __first[__kmodn] = __r;
        }
        for (size_t __k = __s + 1; __k < __m; ++__k)
        {
            const size_t __kmodn = __k % __n;
            const size_t __kpmodn = (__k + __p) % __n;
            result_type __r = 1664525 * _T(__first[__kmodn] ^ __first[__kpmodn]
                                           ^ __first[(__k - 1) % __n]);
            __first[__kpmodn] += __r;
            __r +=  __kmodn;
            __first[(__k + __q) % __n] += __r;
            __first[__kmodn] = __r;
        }
        for (size_t __k = __m; __k < __m + __n; ++__k)
        {
            const size_t __kmodn = __k % __n;
            const size_t __kpmodn = (__k + __p) % __n;
            result_type __r = 1566083941 * _T(__first[__kmodn] +
                                              __first[__kpmodn] +
                                              __first[(__k - 1) % __n]);
            __first[__kpmodn] ^= __r;
            __r -= __kmodn;
            __first[(__k + __q) % __n] ^= __r;
            __first[__kmodn] = __r;
        }
    }
}

template<class _RealType, size_t __bits, class _URNG>
_RealType
generate_canonical(_URNG& __g)
{
    const size_t _Dt = numeric_limits<_RealType>::digits;
    const size_t __b = _Dt < __bits ? _Dt : __bits;
    const size_t __logR = __log2<uint64_t, _URNG::_Max - _URNG::_Min + uint64_t(1)>::value;
    const size_t __k = __b / __logR + (__b % __logR != 0) + (__b == 0);
    const _RealType _R = _URNG::_Max - _URNG::_Min + _RealType(1);
    _RealType __base = _R;
    _RealType _S = __g() - _URNG::_Min;
    for (size_t __i = 1; __i < __k; ++__i, __base *= _R)
        _S += (__g() - _URNG::_Min) * __base;
    return _S / __base;
}

// __independent_bits_engine

template<class _Engine, class _UIntType>
class __independent_bits_engine
{
public:
    // types
    typedef _UIntType result_type;

private:
    typedef typename _Engine::result_type _Engine_result_type;
    typedef typename conditional
        <
            sizeof(_Engine_result_type) <= sizeof(result_type),
                result_type,
                _Engine_result_type
        >::type _Working_result_type;

    _Engine& __e_;
    size_t __w_;
    size_t __w0_;
    size_t __n_;
    size_t __n0_;
    _Working_result_type __y0_;
    _Working_result_type __y1_;
    _Engine_result_type __mask0_;
    _Engine_result_type __mask1_;

    static const _Working_result_type _R = _Engine::_Max - _Engine::_Min
                                                         + _Working_result_type(1);
    static const size_t __m = __log2<_Working_result_type, _R>::value;
    static const size_t _WDt = numeric_limits<_Working_result_type>::digits;
    static const size_t _EDt = numeric_limits<_Engine_result_type>::digits;

public:
    // constructors and seeding functions
    __independent_bits_engine(_Engine& __e, size_t __w);

    // generating functions
    result_type operator()() {return __eval(integral_constant<bool, _R != 0>());}

private:
    result_type __eval(false_type);
    result_type __eval(true_type);
};

template<class _Engine, class _UIntType>
__independent_bits_engine<_Engine, _UIntType>
    ::__independent_bits_engine(_Engine& __e, size_t __w)
        : __e_(__e),
          __w_(__w)
{
    __n_ = __w_ / __m + (__w_ % __m != 0);
    __w0_ = __w_ / __n_;
    if (_R == 0)
        __y0_ = _R;
    else if (__w0_ < _WDt)
        __y0_ = (_R >> __w0_) << __w0_;
    else
        __y0_ = 0;
    if (_R - __y0_ > __y0_ / __n_)
    {
        ++__n_;
        __w0_ = __w_ / __n_;
        if (__w0_ < _WDt)
            __y0_ = (_R >> __w0_) << __w0_;
        else
            __y0_ = 0;
    }
    __n0_ = __n_ - __w_ % __n_;
    if (__w0_ < _WDt - 1)
        __y1_ = (_R >> (__w0_ + 1)) << (__w0_ + 1);
    else
        __y1_ = 0;
    __mask0_ = __w0_ > 0 ? _Engine_result_type(~0) >> (_EDt - __w0_) :
                          _Engine_result_type(0);
    __mask1_ = __w0_ < _EDt - 1 ?
                               _Engine_result_type(~0) >> (_EDt - (__w0_ + 1)) :
                               _Engine_result_type(~0);
}

template<class _Engine, class _UIntType>
inline
_UIntType
__independent_bits_engine<_Engine, _UIntType>::__eval(false_type)
{
    return static_cast<result_type>(__e_() & __mask0_);
}

template<class _Engine, class _UIntType>
_UIntType
__independent_bits_engine<_Engine, _UIntType>::__eval(true_type)
{
    result_type _S = 0;
    for (size_t __k = 0; __k < __n0_; ++__k)
    {
        _Engine_result_type __u;
        do
        {
            __u = __e_() - _Engine::min();
        } while (__u >= __y0_);
        if (__w0_ < _EDt)
            _S <<= __w0_;
        else
            _S = 0;
        _S += __u & __mask0_;
    }
    for (size_t __k = __n0_; __k < __n_; ++__k)
    {
        _Engine_result_type __u;
        do
        {
            __u = __e_() - _Engine::min();
        } while (__u >= __y1_);
        if (__w0_ < _EDt - 1)
            _S <<= __w0_ + 1;
        else
            _S = 0;
        _S += __u & __mask1_;
    }
    return _S;
}

template<class _IntType = int>
class uniform_int_distribution
{
public:
    // types
    typedef _IntType result_type;

    class param_type
    {
        result_type __a_;
        result_type __b_;
    public:
        typedef uniform_int_distribution distribution_type;

        explicit param_type(result_type __a = 0,
                            result_type __b = numeric_limits<result_type>::max())
            : __a_(__a), __b_(__b) {}

        result_type a() const {return __a_;}
        result_type b() const {return __b_;}

        friend bool operator==(const param_type& __x, const param_type& __y)
            {return __x.__a_ == __y.__a_ && __x.__b_ == __y.__b_;}
        friend bool operator!=(const param_type& __x, const param_type& __y)
            {return !(__x == __y);}
    };

private:
    param_type __p_;

public:
    // constructors and reset functions
    explicit uniform_int_distribution(result_type __a = 0,
                                      result_type __b = numeric_limits<result_type>::max())
        : __p_(param_type(__a, __b)) {}
    explicit uniform_int_distribution(const param_type& __p) : __p_(__p) {}
    void reset() {}

    // generating functions
    template<class _URNG> result_type operator()(_URNG& __g)
        {return (*this)(__g, __p_);}
    template<class _URNG> result_type operator()(_URNG& __g, const param_type& __p);

    // property functions
    result_type a() const {return __p_.a();}
    result_type b() const {return __p_.b();}

    param_type param() const {return __p_;}
    void param(const param_type& __p) {__p_ = __p;}

    result_type min() const {return a();}
    result_type max() const {return b();}

    friend bool operator==(const uniform_int_distribution& __x,
                           const uniform_int_distribution& __y)
        {return __x.__p_ == __y.__p_;}
    friend bool operator!=(const uniform_int_distribution& __x,
                           const uniform_int_distribution& __y)
            {return !(__x == __y);}
};

template<class _IntType>
template<class _URNG>
typename uniform_int_distribution<_IntType>::result_type
uniform_int_distribution<_IntType>::operator()(_URNG& __g, const param_type& __p)
{
    typedef typename conditional<sizeof(result_type) <= sizeof(uint32_t),
                                            uint32_t, uint64_t>::type _UIntType;
    const _UIntType _R = __p.b() - __p.a() + _UIntType(1);
    if (_R == 1)
        return __p.a();
    const size_t _Dt = numeric_limits<_UIntType>::digits;
    typedef __independent_bits_engine<_URNG, _UIntType> _Eng;
    if (_R == 0)
        return static_cast<result_type>(_Eng(__g, _Dt)());
    size_t __w = _Dt - __clz(_R) - 1;
    if ((_R & (_UIntType(~0) >> (_Dt - __w))) != 0)
        ++__w;
    _Eng __e(__g, __w);
    _UIntType __u;
    do
    {
        __u = __e();
    } while (__u >= _R);
    return static_cast<result_type>(__u + __p.a());
}

template <class _CharT, class _Traits, class _IT>
basic_ostream<_CharT, _Traits>&
operator<<(basic_ostream<_CharT, _Traits>& __os,
           const uniform_int_distribution<_IT>& __x)
{
    __save_flags<_CharT, _Traits> _(__os);
    __os.flags(ios_base::dec | ios_base::left);
    _CharT __sp = __os.widen(' ');
    __os.fill(__sp);
    return __os << __x.a() << __sp << __x.b();
}

template <class _CharT, class _Traits, class _IT>
basic_istream<_CharT, _Traits>&
operator>>(basic_istream<_CharT, _Traits>& __is,
           uniform_int_distribution<_IT>& __x)
{
    typedef uniform_int_distribution<_IT> _Eng;
    typedef typename _Eng::result_type result_type;
    typedef typename _Eng::param_type param_type;
    __save_flags<_CharT, _Traits> _(__is);
    __is.flags(ios_base::dec | ios_base::skipws);
    result_type __a;
    result_type __b;
    __is >> __a >> __b;
    if (!__is.fail())
        __x.param(param_type(__a, __b));
    return __is;
}

template<class _RealType = double>
class uniform_real_distribution
{
public:
    // types
    typedef _RealType result_type;

    class param_type
    {
        result_type __a_;
        result_type __b_;
    public:
        typedef uniform_real_distribution distribution_type;

        explicit param_type(result_type __a = 0,
                            result_type __b = 1)
            : __a_(__a), __b_(__b) {}

        result_type a() const {return __a_;}
        result_type b() const {return __b_;}

        friend bool operator==(const param_type& __x, const param_type& __y)
            {return __x.__a_ == __y.__a_ && __x.__b_ == __y.__b_;}
        friend bool operator!=(const param_type& __x, const param_type& __y)
            {return !(__x == __y);}
    };

private:
    param_type __p_;

public:
    // constructors and reset functions
    explicit uniform_real_distribution(result_type __a = 0, result_type __b = 1)
        : __p_(param_type(__a, __b)) {}
    explicit uniform_real_distribution(const param_type& __p) : __p_(__p) {}
    void reset() {}

    // generating functions
    template<class _URNG> result_type operator()(_URNG& __g)
        {return (*this)(__g, __p_);}
    template<class _URNG> result_type operator()(_URNG& __g, const param_type& __p);

    // property functions
    result_type a() const {return __p_.a();}
    result_type b() const {return __p_.b();}

    param_type param() const {return __p_;}
    void param(const param_type& __p) {__p_ = __p;}

    result_type min() const {return a();}
    result_type max() const {return b();}

    friend bool operator==(const uniform_real_distribution& __x,
                           const uniform_real_distribution& __y)
        {return __x.__p_ == __y.__p_;}
    friend bool operator!=(const uniform_real_distribution& __x,
                           const uniform_real_distribution& __y)
        {return !(__x == __y);}
};

template<class _RealType>
template<class _URNG>
inline
typename uniform_real_distribution<_RealType>::result_type
uniform_real_distribution<_RealType>::operator()(_URNG& __g, const param_type& __p)
{
    return (__p.b() - __p.a())
        * _STD::generate_canonical<_RealType, numeric_limits<_RealType>::digits>(__g)
        + __p.a();
}

template <class _CharT, class _Traits, class _RT>
basic_ostream<_CharT, _Traits>&
operator<<(basic_ostream<_CharT, _Traits>& __os,
           const uniform_real_distribution<_RT>& __x)
{
    __save_flags<_CharT, _Traits> _(__os);
    __os.flags(ios_base::dec | ios_base::left);
    _CharT __sp = __os.widen(' ');
    __os.fill(__sp);
    return __os << __x.a() << __sp << __x.b();
}

template <class _CharT, class _Traits, class _RT>
basic_istream<_CharT, _Traits>&
operator>>(basic_istream<_CharT, _Traits>& __is,
           uniform_real_distribution<_RT>& __x)
{
    typedef uniform_real_distribution<_RT> _Eng;
    typedef typename _Eng::result_type result_type;
    typedef typename _Eng::param_type param_type;
    __save_flags<_CharT, _Traits> _(__is);
    __is.flags(ios_base::dec | ios_base::skipws);
    result_type __a;
    result_type __b;
    __is >> __a >> __b;
    if (!__is.fail())
        __x.param(param_type(__a, __b));
    return __is;
}

class bernoulli_distribution
{
public:
    // types
    typedef bool result_type;

    class param_type
    {
        double __p_;
    public:
        typedef bernoulli_distribution distribution_type;

        explicit param_type(double __p = 0.5) : __p_(__p) {}

        double p() const {return __p_;}

        friend bool operator==(const param_type& __x, const param_type& __y)
            {return __x.__p_ == __y.__p_;}
        friend bool operator!=(const param_type& __x, const param_type& __y)
            {return !(__x == __y);}
    };

private:
    param_type __p_;

public:
    // constructors and reset functions
    explicit bernoulli_distribution(double __p = 0.5)
        : __p_(param_type(__p)) {}
    explicit bernoulli_distribution(const param_type& __p) : __p_(__p) {}
    void reset() {}

    // generating functions
    template<class _URNG> result_type operator()(_URNG& __g)
        {return (*this)(__g, __p_);}
    template<class _URNG> result_type operator()(_URNG& __g, const param_type& __p)
        {
            return _STD::generate_canonical<double, numeric_limits<double>::digits>(__g)
                   < __p.p();
        }

    // property functions
    double p() const {return __p_.p();}

    param_type param() const {return __p_;}
    void param(const param_type& __p) {__p_ = __p;}

    result_type min() const {return false;}
    result_type max() const {return true;}

    friend bool operator==(const bernoulli_distribution& __x,
                          const bernoulli_distribution& __y)
        {return __x.__p_ == __y.__p_;}
    friend bool operator!=(const bernoulli_distribution& __x,
                          const bernoulli_distribution& __y)
        {return !(__x == __y);}
};

template <class _CharT, class _Traits>
basic_ostream<_CharT, _Traits>&
operator<<(basic_ostream<_CharT, _Traits>& __os, const bernoulli_distribution& __x)
{
    __save_flags<_CharT, _Traits> _(__os);
    __os.flags(ios_base::dec | ios_base::left);
    _CharT __sp = __os.widen(' ');
    __os.fill(__sp);
    return __os << __x.p();
}

template <class _CharT, class _Traits>
basic_istream<_CharT, _Traits>&
operator>>(basic_istream<_CharT, _Traits>& __is, bernoulli_distribution& __x)
{
    typedef bernoulli_distribution _Eng;
    typedef typename _Eng::param_type param_type;
    __save_flags<_CharT, _Traits> _(__is);
    __is.flags(ios_base::dec | ios_base::skipws);
    double __p;
    __is >> __p;
    if (!__is.fail())
        __x.param(param_type(__p));
    return __is;
}

_LIBCPP_END_NAMESPACE_STD

#endif  // _LIBCPP_RANDOM