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# std.random

Facilities for `random` number generation.
Disclaimer: The random number generators and API provided in this module are not designed to be cryptographically secure, and are therefore unsuitable for cryptographic or security-related purposes such as generating authentication tokens or network sequence numbers. For such needs, please use a reputable cryptographic library instead.
The new-style generator objects hold their own state so they are immune of threading issues. The generators feature a number of well-known and well-documented methods of generating `random` numbers. An overall fast and reliable means to generate `random` numbers is the Mt19937 generator, which derives its name from "Mersenne Twister with a period of 2 to the power of 19937". In memory-constrained situations, linear congruential generators such as MinstdRand0 and MinstdRand might be useful. The standard library provides an alias Random for whichever generator it considers the most fit for the target environment.
In addition to `random` number generators, this module features distributions, which skew a generator's output statistical distribution in various ways. So far the uniform distribution for integers and real numbers have been implemented.

Source std/random.d

License:
Boost License 1.0.
Authors:
Andrei Alexandrescu Masahiro Nakagawa (Xorshift `random` generator) Joseph Rushton Wakeling (Algorithm D for `random` sampling) Ilya Yaroshenko (Mersenne Twister implementation, adapted from mir-random)

Credits The entire `random` number library architecture is derived from the excellent C++0X `random` number facility proposed by Jens Maurer and contributed to by researchers at the Fermi laboratory (excluding Xorshift).

Examples:

    // seed a random generator with a constant
    auto rnd = Random(42);

    // Generate a uniformly-distributed integer in the range [0, 14]
    // If no random generator is passed, the global `rndGen` would be used
    auto i = uniform(0, 15, rnd);
    assert(i >= 0 && i < 15);

    // Generate a uniformly-distributed real in the range [0, 100)
    auto r = uniform(0.0L, 100.0L, rnd);
    assert(r >= 0 && r < 100);

    // Generate a 32-bit random number
    auto u = uniform!uint(rnd);
    static assert(is(typeof(u) == uint));

enum bool `isUniformRNG`(Rng, ElementType);
enum bool `isUniformRNG`(Rng);
  Test if Rng is a random-number generator. The overload taking a ElementType also makes sure that the Rng generates values of that type.
  A random-number generator has at least the following features:
  - it's an InputRange
  - it has a 'bool isUniformRandom' field readable in CTFE
enum bool `isSeedable`(Rng, SeedType);
enum bool `isSeedable`(Rng);
  Test if Rng is seedable. The overload taking a SeedType also makes sure that the Rng can be seeded with SeedType.
  A seedable random-number generator has the following additional features:
  - it has a 'seed(ElementType)' function
struct `LinearCongruentialEngine`(UIntType, UIntType a, UIntType c, UIntType m) if (isUnsigned!UIntType);
  Linear Congruential generator.
  enum bool `isUniformRandom`;
    Mark this as a Rng
  enum bool `hasFixedRange`;
    Does this generator have a fixed range? (`true`).
  enum UIntType `min`;
    Lowest generated value (1 if c == 0, 0 otherwise).
  enum UIntType `max`;
    Highest generated value (modulus - 1).
  enum UIntType `multiplier`;
  enum UIntType `increment`;
  enum UIntType `modulus`;
    The parameters of this distribution. The random number is x = (x * multipler + `increment`) % `modulus`.
  pure nothrow @nogc @safe this(UIntType `x0`);
    Constructs a LinearCongruentialEngine generator seeded with `x0`.
  pure nothrow @nogc @safe void `seed`(UIntType `x0` = 1);
    (Re)seeds the generator.
  pure nothrow @nogc @safe void `popFront`();
    Advances the random sequence.
  const pure nothrow @nogc @property @safe UIntType `front`();
    Returns the current number in the random sequence.
  pure nothrow @nogc @property @safe typeof(this) `save`();
  enum bool `empty`;
    Always `false` (random generators are infinite ranges).
  const pure nothrow @nogc @safe bool `opEquals`(ref const LinearCongruentialEngine `rhs`);
    Compares against `rhs` for equality.
alias `MinstdRand0` = LinearCongruentialEngine!(uint, 16807u, 0u, 2147483647u).LinearCongruentialEngine;
alias `MinstdRand` = LinearCongruentialEngine!(uint, 48271u, 0u, 2147483647u).LinearCongruentialEngine;
  Define LinearCongruentialEngine generators with well-chosen parameters. `MinstdRand0` implements Park and Miller's "minimal standard" generator that uses 16807 for the multiplier. `MinstdRand` implements a variant that has slightly better spectral behavior by using the multiplier 48271. Both generators are rather simplistic.
  Examples:

      // seed with a constant
      auto rnd0 = MinstdRand0(1);
      auto n = rnd0.front; // same for each run
      // Seed with an unpredictable value
      rnd0.seed(unpredictableSeed);
      n = rnd0.front; // different across runs

struct `MersenneTwisterEngine`(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) if (isUnsigned!UIntType);
  The Mersenne Twister generator.
  enum bool `isUniformRandom`;
    Mark this as a Rng
  enum size_t `wordSize`;
  enum size_t `stateSize`;
  enum size_t `shiftSize`;
  enum size_t `maskBits`;
  enum UIntType `xorMask`;
  enum size_t `temperingU`;
  enum UIntType `temperingD`;
  enum size_t `temperingS`;
  enum UIntType `temperingB`;
  enum size_t `temperingT`;
  enum UIntType `temperingC`;
  enum size_t `temperingL`;
  enum UIntType `initializationMultiplier`;
    Parameters for the generator.
  enum UIntType `min`;
    Smallest generated value (0).
  enum UIntType `max`;
    Largest generated value.
  enum UIntType `defaultSeed`;
    The default seed value.
  pure nothrow @nogc @safe this(UIntType `value`);
    Constructs a MersenneTwisterEngine object.
  pure nothrow @nogc @safe void `seed`()(UIntType `value` = defaultSeed);
    Seeds a MersenneTwisterEngine object.

    Note This `seed` function gives 2^w starting points (the lowest w bits of the `value` provided will be used). To allow the RNG to be started in any one of its internal states use the `seed` overload taking an InputRange.

  void `seed`(T)(T `range`)
  if (isInputRange!T && is(Unqual!(ElementType!T) == UIntType));
    Seeds a MersenneTwisterEngine object using an InputRange.
    Throws:
    Exception if the InputRange didn't provide enough elements to `seed` the generator. The number of elements required is the 'n' template parameter of the MersenneTwisterEngine struct.
  pure nothrow @nogc @safe void `popFront`();
    Advances the generator.
  const pure nothrow @nogc @property @safe UIntType `front`();
    Returns the current random value.
  pure nothrow @nogc @property @safe typeof(this) `save`();
  enum bool `empty`;
    Always `false`.
alias `Mt19937` = MersenneTwisterEngine!(uint, 32u, 624u, 397u, 31u, 2567483615u, 11u, 4294967295u, 7u, 2636928640u, 15u, 4022730752u, 18u, 1812433253u).MersenneTwisterEngine;
  A MersenneTwisterEngine instantiated with the parameters of the original engine MT19937, generating uniformly-distributed 32-bit numbers with a period of 2 to the power of 19937. Recommended for random number generation unless memory is severely restricted, in which case a LinearCongruentialEngine would be the generator of choice.
  Examples:

      // seed with a constant
      Mt19937 gen;
      auto n = gen.front; // same for each run
      // Seed with an unpredictable value
      gen.seed(unpredictableSeed);
      n = gen.front; // different across runs

alias `Mt19937_64` = MersenneTwisterEngine!(ulong, 64u, 312u, 156u, 31u, 13043109905998158313LU, 29u, 6148914691236517205LU, 17u, 8202884508482404352LU, 37u, 18444473444759240704LU, 43u, 6364136223846793005LU).MersenneTwisterEngine;
  A MersenneTwisterEngine instantiated with the parameters of the original engine MT19937-64, generating uniformly-distributed 64-bit numbers with a period of 2 to the power of 19937.
  Examples:

      // Seed with a constant
      auto gen = Mt19937_64(12345);
      auto n = gen.front; // same for each run
      // Seed with an unpredictable value
      gen.seed(unpredictableSeed);
      n = gen.front; // different across runs

struct `XorshiftEngine`(UIntType, UIntType bits, UIntType a, UIntType b, UIntType c) if (isUnsigned!UIntType);
  Xorshift generator using 32bit algorithm.
  Implemented according to Xorshift RNGs. Supporting bits are below, bits means second parameter of `XorshiftEngine`.

  bits | period
  --- | ---
  32 | 2^32 - 1
  64 | 2^64 - 1
  96 | 2^96 - 1
  128 | 2^128 - 1
  160 | 2^160 - 1
  192 | 2^192 - 2^32

  enum bool `isUniformRandom`;
    Mark this as a Rng
  enum auto `empty`;
    Always `false` (random generators are infinite ranges).
  enum UIntType `min`;
    Smallest generated value.
  enum UIntType `max`;
    Largest generated value.
  pure nothrow @nogc @safe this(UIntType `x0`);
    Constructs a XorshiftEngine generator seeded with `x0`.
  pure nothrow @nogc @safe void `seed`(UIntType `x0`);
    (Re)seeds the generator.
  const pure nothrow @nogc @property @safe UIntType `front`();
    Returns the current number in the random sequence.
  pure nothrow @nogc @safe void `popFront`();
    Advances the random sequence.
  pure nothrow @nogc @property @safe typeof(this) `save`();
    Captures a range state.
  const pure nothrow @nogc @safe bool `opEquals`(ref const XorshiftEngine `rhs`);
    Compares against `rhs` for equality.
alias `Xorshift32` = XorshiftEngine!(uint, 32u, 13u, 17u, 15u).XorshiftEngine;
alias `Xorshift64` = XorshiftEngine!(uint, 64u, 10u, 13u, 10u).XorshiftEngine;
alias `Xorshift96` = XorshiftEngine!(uint, 96u, 10u, 5u, 26u).XorshiftEngine;
alias `Xorshift128` = XorshiftEngine!(uint, 128u, 11u, 8u, 19u).XorshiftEngine;
alias `Xorshift160` = XorshiftEngine!(uint, 160u, 2u, 1u, 4u).XorshiftEngine;
alias `Xorshift192` = XorshiftEngine!(uint, 192u, 2u, 1u, 4u).XorshiftEngine;
alias `Xorshift` = XorshiftEngine!(uint, 128u, 11u, 8u, 19u).XorshiftEngine;
  Define XorshiftEngine generators with well-chosen parameters. See each bits examples of "`Xorshift` RNGs". `Xorshift` is a `Xorshift128`'s alias because 128bits implementation is mostly used.
  Examples:

      // Seed with a constant
      auto rnd = Xorshift(1);
      auto num = rnd.front;  // same for each run

      // Seed with an unpredictable value
      rnd.seed(unpredictableSeed);
      num = rnd.front; // different across rnd

nothrow @nogc @property @trusted uint `unpredictableSeed`();
  A "good" seed for initializing random number engines. Initializing with `unpredictableSeed` makes engines generate different random number sequences every run.
  Returns:
  A single unsigned integer seed value, different on each successive call
  Examples:

      auto rnd = Random(unpredictableSeed);
      auto n = rnd.front;
      static assert(is(typeof(n) == uint));

alias `Random` = MersenneTwisterEngine!(uint, 32u, 624u, 397u, 31u, 2567483615u, 11u, 4294967295u, 7u, 2636928640u, 15u, 4022730752u, 18u, 1812433253u).MersenneTwisterEngine;
  The "default", "favorite", "suggested" random number generator type on the current platform. It is an alias for one of the previously-defined generators. You may want to use it if (1) you need to generate some nice random numbers, and (2) you don't care for the minutiae of the method being used.
@property ref @safe Random `rndGen`();
  Global random number generator used by various functions in this module whenever no generator is specified. It is allocated per-thread and initialized to an unpredictable value for each thread.
  Returns:
  A singleton instance of the default random number generator
auto `uniform`(string boundaries = "[)", T1, T2)(T1 `a`, T2 `b`)
if (!is(CommonType!(T1, T2) == void));
auto `uniform`(string boundaries = "[)", T1, T2, UniformRandomNumberGenerator)(T1 `a`, T2 `b`, ref UniformRandomNumberGenerator `urng`)
if (isFloatingPoint!(CommonType!(T1, T2)) && isUniformRNG!UniformRandomNumberGenerator);
  Generates `a` number between `a` and `b`. The boundaries parameter controls the shape of the interval (open vs. closed on either side). Valid values for boundaries are "[]", "(]", "[)", and "()". The default interval is closed to the left and open to the right. The version that does not take `urng` uses the default generator rndGen.
  Parameters:

  T1 `a` | lower bound of the uniform distribution
  T2 `b` | upper bound of the uniform distribution
  UniformRandomNumberGenerator `urng` | (optional) random number generator to use; if not specified, defaults to rndGen

  Returns:
  A single random variate drawn from the uniform distribution between `a` and `b`, whose type is the common type of these parameters
  Examples:

      auto gen = Random(unpredictableSeed);
      // Generate an integer in [0, 1023]
      auto a = uniform(0, 1024, gen);
      // Generate a float in [0, 1)
      auto b = uniform(0.0f, 1.0f, gen);

auto `uniform`(T, UniformRandomNumberGenerator)(ref UniformRandomNumberGenerator `urng`)
if (!is(T == enum) && (isIntegral!T || isSomeChar!T) && isUniformRNG!UniformRandomNumberGenerator);
auto `uniform`(T)()
if (!is(T == enum) && (isIntegral!T || isSomeChar!T));
  Generates a uniformly-distributed number in the range [T.min, T.max] for any integral or character type T. If no random number generator is passed, uses the default rndGen.
  Parameters:

  UniformRandomNumberGenerator `urng` | (optional) random number generator to use; if not specified, defaults to rndGen

  Returns:
  Random variate drawn from the uniform distribution across all possible values of the integral or character type T.
auto `uniform`(E, UniformRandomNumberGenerator)(ref UniformRandomNumberGenerator `urng`)
if (is(E == enum) && isUniformRNG!UniformRandomNumberGenerator);
auto `uniform`(E)()
if (is(E == enum));
  Returns a uniformly selected member of enum E. If no random number generator is passed, uses the default rndGen.
  Parameters:

  UniformRandomNumberGenerator `urng` | (optional) random number generator to use; if not specified, defaults to rndGen

  Returns:
  Random variate drawn with equal probability from any of the possible values of the enum E.
  Examples:

      enum Fruit { apple, mango, pear }
      auto randFruit = uniform!Fruit();

T `uniform01`(T = double)()
if (isFloatingPoint!T);
T `uniform01`(T = double, UniformRNG)(ref UniformRNG `rng`)
if (isFloatingPoint!T && isUniformRNG!UniformRNG);
  Generates a uniformly-distributed floating point number of type T in the range [0, 1). If no random number generator is specified, the default RNG rndGen will be used as the source of randomness.
  `uniform01` offers a faster generation of random variates than the equivalent uniform!"[)"(0.0, 1.0) and so may be preferred for some applications.
  Parameters:

  UniformRNG `rng` | (optional) random number generator to use; if not specified, defaults to rndGen

  Returns:
  Floating-point random variate of type T drawn from the uniform distribution across the half-open interval [0, 1).
F[] `uniformDistribution`(F = double)(size_t `n`, F[] `useThis` = null)
if (isFloatingPoint!F);
  Generates a uniform probability distribution of size `n`, i.e., an array of size `n` of positive numbers of type F that sum to 1. If `useThis` is provided, it is used as storage.
ref auto `choice`(Range, RandomGen = Random)(auto ref Range `range`, ref RandomGen `urng` = rndGen)
if (isRandomAccessRange!Range && hasLength!Range && isUniformRNG!RandomGen);
  Returns a random, uniformly chosen, element e from the supplied Range `range`. If no random number generator is passed, the default rndGen is used.
  Parameters:

  Range `range` | a random access `range` that has the length property defined
  RandomGen `urng` | (optional) random number generator to use; if not specified, defaults to rndGen

  Returns:
  A single random element drawn from the `range`. If it can, it will return a ref to the `range` element, otherwise it will return a copy.
  Examples:

      import std.algorithm.searching : canFind;

      auto array = [1, 2, 3, 4, 5];
      auto elem = choice(array);

      assert(canFind(array, elem),
             "Choice did not return a valid element from the given Range");

      auto urng = Random(unpredictableSeed);
      elem = choice(array, urng);

      assert(canFind(array, elem),
             "Choice did not return a valid element from the given Range");

void `randomShuffle`(Range, RandomGen)(Range `r`, ref RandomGen `gen`)
if (isRandomAccessRange!Range && isUniformRNG!RandomGen);
void `randomShuffle`(Range)(Range `r`)
if (isRandomAccessRange!Range);
  Shuffles elements of `r` using `gen` as a shuffler. `r` must be a random-access range with length. If no RNG is specified, rndGen will be used.
  Parameters:

  Range `r` | random-access range whose elements are to be shuffled
  RandomGen `gen` | (optional) random number generator to use; if not specified, defaults to rndGen

void `partialShuffle`(Range, RandomGen)(Range `r`, in size_t `n`, ref RandomGen `gen`)
if (isRandomAccessRange!Range && isUniformRNG!RandomGen);
void `partialShuffle`(Range)(Range `r`, in size_t `n`)
if (isRandomAccessRange!Range);
  Partially shuffles the elements of `r` such that upon returning `r`[0 .. `n`] is a random subset of `r` and is randomly ordered. `r`[`n` .. `r`.length] will contain the elements not in `r`[0 .. `n`]. These will be in an undefined order, but will not be random in the sense that their order after `partialShuffle` returns will not be independent of their order before `partialShuffle` was called.
  `r` must be a random-access range with length. `n` must be less than or equal to `r`.length. If no RNG is specified, rndGen will be used.
  Parameters:

  Range `r` | random-access range whose elements are to be shuffled
  size_t `n` | number of elements of `r` to shuffle (counting from the beginning); must be less than `r`.length
  RandomGen `gen` | (optional) random number generator to use; if not specified, defaults to rndGen

size_t `dice`(Rng, Num)(ref Rng `rnd`, Num[] `proportions`...)
if (isNumeric!Num && isForwardRange!Rng);
size_t `dice`(R, Range)(ref R `rnd`, Range `proportions`)
if (isForwardRange!Range && isNumeric!(ElementType!Range) && !isArray!Range);
size_t `dice`(Range)(Range `proportions`)
if (isForwardRange!Range && isNumeric!(ElementType!Range) && !isArray!Range);
size_t `dice`(Num)(Num[] `proportions`...)
if (isNumeric!Num);
  Rolls a `dice` with relative probabilities stored in `proportions`. Returns the index in `proportions` that was chosen.
  Parameters:

  Rng `rnd` | (optional) random number generator to use; if not specified, defaults to rndGen
  Num[] `proportions` | forward range or list of individual values whose elements correspond to the probabilities with which to choose the corresponding index value

  Returns:
  Random variate drawn from the index values [0, ... `proportions`.length - 1], with the probability of getting an individual index value i being proportional to `proportions`[i].
  Examples:

      auto x = dice(0.5, 0.5);   // x is 0 or 1 in equal proportions
      auto y = dice(50, 50);     // y is 0 or 1 in equal proportions
      auto z = dice(70, 20, 10); // z is 0 70% of the time, 1 20% of the time,
                                 // and 2 10% of the time

struct `RandomCover`(Range, UniformRNG = void) if (isRandomAccessRange!Range && (isUniformRNG!UniformRNG || is(UniformRNG == void)));
auto `randomCover`(Range, UniformRNG)(Range `r`, auto ref UniformRNG `rng`)
if (isRandomAccessRange!Range && isUniformRNG!UniformRNG);
auto `randomCover`(Range)(Range `r`)
if (isRandomAccessRange!Range);
  Covers a given range `r` in a random manner, i.e. goes through each element of `r` once and only once, just in a random order. `r` must be a random-access range with length.
  If no random number generator is passed to `randomCover`, the thread-global RNG rndGen will be used internally.
  Parameters:

  Range `r` | random-access range to cover
  UniformRNG `rng` | (optional) random number generator to use; if not specified, defaults to rndGen

  Returns:
  Range whose elements consist of the elements of `r`, in random order. Will be a forward range if both `r` and `rng` are forward ranges, an input range otherwise.

  Example

      int[] a = [ 0, 1, 2, 3, 4, 5, 6, 7, 8 ];
      foreach (e; randomCover(a))
      {
          writeln(e);
      }

  WARNING: If an alternative RNG is desired, it is essential for this to be a new RNG seeded in an unpredictable manner. Passing it a RNG used elsewhere in the program will result in unintended correlations, due to the current implementation of RNGs as value types.

  Example

      int[] a = [ 0, 1, 2, 3, 4, 5, 6, 7, 8 ];
      foreach (e; randomCover(a, Random(unpredictableSeed)))  // correct!
      {
          writeln(e);
      }

      foreach (e; randomCover(a, rndGen))  // DANGEROUS!! rndGen gets copied by value
      {
          writeln(e);
      }

      foreach (e; randomCover(a, rndGen))  // ... so this second random cover
      {                                    // will output the same sequence as
          writeln(e);                      // the previous one.
      }

struct `RandomSample`(Range, UniformRNG = void) if (isInputRange!Range && (isUniformRNG!UniformRNG || is(UniformRNG == void)));
auto `randomSample`(Range)(Range `r`, size_t `n`, size_t `total`)
if (isInputRange!Range);
auto `randomSample`(Range)(Range `r`, size_t `n`)
if (isInputRange!Range && hasLength!Range);
auto `randomSample`(Range, UniformRNG)(Range `r`, size_t `n`, size_t `total`, auto ref UniformRNG `rng`)
if (isInputRange!Range && isUniformRNG!UniformRNG);
auto `randomSample`(Range, UniformRNG)(Range `r`, size_t `n`, auto ref UniformRNG `rng`)
if (isInputRange!Range && hasLength!Range && isUniformRNG!UniformRNG);
  Selects a random subsample out of `r`, containing exactly `n` elements. The order of elements is the same as in the original range. The `total` length of `r` must be known. If `total` is passed in, the `total` number of sample is considered to be `total`. Otherwise, `RandomSample` uses `r`.length.
  Parameters:

  Range `r` | range to sample from
  size_t `n` | number of elements to include in the sample; must be less than or equal to the `total` number of elements in `r` and/or the parameter `total` (if provided)
  size_t `total` | (semi-optional) number of elements of `r` from which to select the sample (counting from the beginning); must be less than or equal to the `total` number of elements in `r` itself. May be omitted if `r` has the .length property and the sample is to be drawn from all elements of `r`.
  UniformRNG `rng` | (optional) random number generator to use; if not specified, defaults to rndGen

  Returns:
  Range whose elements consist of a randomly selected subset of the elements of `r`, in the same order as these elements appear in `r` itself. Will be a forward range if both `r` and `rng` are forward ranges, an input range otherwise.
  `RandomSample` implements Jeffrey Scott Vitter's Algorithm D (see Vitter 1984, 1987), which selects a sample of size `n` in O(`n`) steps and requiring O(`n`) random variates, regardless of the size of the data being sampled. The exception to this is if traversing k elements on the input range is itself an O(k) operation (e.g. when sampling lines from an input file), in which case the sampling calculation will inevitably be of O(`total`).
  `RandomSample` will throw an exception if `total` is verifiably less than the `total` number of elements available in the input, or if `n` > `total`.
  If no random number generator is passed to `randomSample`, the thread-global RNG rndGen will be used internally.

  Example

      int[] a = [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ];
      // Print 5 random elements picked off from a
      foreach (e; randomSample(a, 5))
      {
          writeln(e);
      }

  WARNING: If an alternative RNG is desired, it is essential for this to be a new RNG seeded in an unpredictable manner. Passing it a RNG used elsewhere in the program will result in unintended correlations, due to the current implementation of RNGs as value types.

  Example

      int[] a = [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ];
      foreach (e; randomSample(a, 5, Random(unpredictableSeed)))  // correct!
      {
          writeln(e);
      }

      foreach (e; randomSample(a, 5, rndGen))  // DANGEROUS!! rndGen gets
      {                                        // copied by value
          writeln(e);
      }

      foreach (e; randomSample(a, 5, rndGen))  // ... so this second random
      {                                        // sample will select the same
          writeln(e);                          // values as the previous one.
      }

  const @property bool `empty`();
  @property ref auto `front`();
  void `popFront`();
  @property typeof(this) `save`();
  @property size_t `length`();
    Range primitives.
  @property size_t `index`();
    Returns the `index` of the visited record.

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