



"""simple docstring"""
from math import sqrt





def  lowerCamelCase_						(UpperCamelCase__		:		int							):
    assert isinstance(UpperCamelCase__     ,       UpperCamelCase__							) and (
        number >= 0
    ), "'number' must been an int and positive"

    _UpperCAmelCase							:  Tuple											=    True

    # 0 and 1 are none primes.
    if number <= 1:
        _UpperCAmelCase							:  int											=    False

    for divisor in range(2     ,       int(round(sqrt(UpperCamelCase__							)							)							) + 1							):
        # if 'number' divisible by 'divisor' then sets 'status'
        # of false and break up the loop.
        if number % divisor == 0:
            _UpperCAmelCase							:  str											=    False
            break

    # precondition
    assert isinstance(UpperCamelCase__     ,       UpperCamelCase__							), "'status' must been from type bool"

    return status





def  lowerCamelCase_						(UpperCamelCase__		:		Dict							):
    assert isinstance(UpperCamelCase__     ,       UpperCamelCase__							) and (n > 2), "'N' must been an int and > 2"

    # beginList: contains all natural numbers from 2 up to N
    _UpperCAmelCase							:  int											=    list(range(2     ,       n + 1							)							)

    _UpperCAmelCase							:  List[str]											=    []  # this list will be returns.

    # actual sieve of erathostenes
    for i in range(len(UpperCamelCase__							)							):
        for j in range(i + 1     ,       len(UpperCamelCase__							)							):
            if (begin_list[i] != 0) and (begin_list[j] % begin_list[i] == 0):
                _UpperCAmelCase							:  Any											=    0

    # filters actual prime numbers.
    _UpperCAmelCase							:  Optional[Any]											=    [x for x in begin_list if x != 0]

    # precondition
    assert isinstance(UpperCamelCase__     ,       UpperCamelCase__							), "'ans' must been from type list"

    return ans





def  lowerCamelCase_						(UpperCamelCase__		:		Tuple							):
    assert isinstance(UpperCamelCase__     ,       UpperCamelCase__							) and (n > 2), "'N' must been an int and > 2"

    _UpperCAmelCase							:  Optional[Any]											=    []

    # iterates over all numbers between 2 up to N+1
    # if a number is prime then appends to list 'ans'
    for number in range(2     ,       n + 1							):
        if is_prime(UpperCamelCase__							):
            ans.append(UpperCamelCase__							)

    # precondition
    assert isinstance(UpperCamelCase__     ,       UpperCamelCase__							), "'ans' must been from type list"

    return ans





def  lowerCamelCase_						(UpperCamelCase__		:		List[str]							):
    assert isinstance(UpperCamelCase__     ,       UpperCamelCase__							) and number >= 0, "'number' must been an int and >= 0"

    _UpperCAmelCase							:  Dict											=    []  # this list will be returns of the function.

    # potential prime number factors.

    _UpperCAmelCase							:  Union[str, Any]											=    2

    _UpperCAmelCase							:  str											=    number

    if number == 0 or number == 1:
        ans.append(UpperCamelCase__							)

    # if 'number' not prime then builds the prime factorization of 'number'
    elif not is_prime(UpperCamelCase__							):
        while quotient != 1:
            if is_prime(UpperCamelCase__							) and (quotient % factor == 0):
                ans.append(UpperCamelCase__							)
                quotient /= factor
            else:
                factor += 1

    else:
        ans.append(UpperCamelCase__							)

    # precondition
    assert isinstance(UpperCamelCase__     ,       UpperCamelCase__							), "'ans' must been from type list"

    return ans





def  lowerCamelCase_						(UpperCamelCase__		:		List[Any]							):
    assert isinstance(UpperCamelCase__     ,       UpperCamelCase__							) and (
        number >= 0
    ), "'number' bust been an int and >= 0"

    _UpperCAmelCase							:  List[str]											=    0

    # prime factorization of 'number'
    _UpperCAmelCase							:  int											=    prime_factorization(UpperCamelCase__							)

    _UpperCAmelCase							:  Optional[int]											=    max(UpperCamelCase__							)

    # precondition
    assert isinstance(UpperCamelCase__     ,       UpperCamelCase__							), "'ans' must been from type int"

    return ans





def  lowerCamelCase_						(UpperCamelCase__		:		Any							):
    assert isinstance(UpperCamelCase__     ,       UpperCamelCase__							) and (
        number >= 0
    ), "'number' bust been an int and >= 0"

    _UpperCAmelCase							:  List[Any]											=    0

    # prime factorization of 'number'
    _UpperCAmelCase							:  List[Any]											=    prime_factorization(UpperCamelCase__							)

    _UpperCAmelCase							:  Optional[Any]											=    min(UpperCamelCase__							)

    # precondition
    assert isinstance(UpperCamelCase__     ,       UpperCamelCase__							), "'ans' must been from type int"

    return ans





def  lowerCamelCase_						(UpperCamelCase__		:		Union[str, Any]							):
    assert isinstance(UpperCamelCase__     ,       UpperCamelCase__							), "'number' must been an int"
    assert isinstance(number % 2 == 0     ,       UpperCamelCase__							), "compare bust been from type bool"

    return number % 2 == 0





def  lowerCamelCase_						(UpperCamelCase__		:		List[Any]							):
    assert isinstance(UpperCamelCase__     ,       UpperCamelCase__							), "'number' must been an int"
    assert isinstance(number % 2 != 0     ,       UpperCamelCase__							), "compare bust been from type bool"

    return number % 2 != 0





def  lowerCamelCase_						(UpperCamelCase__		:		Any							):
    assert (
        isinstance(UpperCamelCase__     ,       UpperCamelCase__							) and (number > 2) and is_even(UpperCamelCase__							)
    ), "'number' must been an int, even and > 2"

    _UpperCAmelCase							:  Tuple											=    []  # this list will returned

    # creates a list of prime numbers between 2 up to 'number'
    _UpperCAmelCase							:  Tuple											=    get_prime_numbers(UpperCamelCase__							)
    _UpperCAmelCase							:  int											=    len(UpperCamelCase__							)

    # run variable for while-loops.
    _UpperCAmelCase							:  Optional[Any]											=    0
    _UpperCAmelCase							:  Any											=    None

    # exit variable. for break up the loops
    _UpperCAmelCase							:  Dict											=    True

    while i < len_pn and loop:
        _UpperCAmelCase							:  Any											=    i + 1

        while j < len_pn and loop:
            if prime_numbers[i] + prime_numbers[j] == number:
                _UpperCAmelCase							:  Tuple											=    False
                ans.append(prime_numbers[i]							)
                ans.append(prime_numbers[j]							)

            j += 1

        i += 1

    # precondition
    assert (
        isinstance(UpperCamelCase__     ,       UpperCamelCase__							)
        and (len(UpperCamelCase__							) == 2)
        and (ans[0] + ans[1] == number)
        and is_prime(ans[0]							)
        and is_prime(ans[1]							)
    ), "'ans' must contains two primes. And sum of elements must been eq 'number'"

    return ans





def  lowerCamelCase_						(UpperCamelCase__		:		Optional[Any]     ,       UpperCamelCase__		:		Tuple							):
    assert (
        isinstance(UpperCamelCase__     ,       UpperCamelCase__							)
        and isinstance(UpperCamelCase__     ,       UpperCamelCase__							)
        and (numbera >= 0)
        and (numbera >= 0)
    ), "'number1' and 'number2' must been positive integer."

    _UpperCAmelCase							:  Any											=    0

    while numbera != 0:
        _UpperCAmelCase							:  Dict											=    numbera % numbera
        _UpperCAmelCase							:  Optional[int]											=    numbera
        _UpperCAmelCase							:  Optional[int]											=    rest

    # precondition
    assert isinstance(UpperCamelCase__     ,       UpperCamelCase__							) and (
        numbera >= 0
    ), "'number' must been from type int and positive"

    return numbera





def  lowerCamelCase_						(UpperCamelCase__		:		Optional[int]     ,       UpperCamelCase__		:		Dict							):
    assert (
        isinstance(UpperCamelCase__     ,       UpperCamelCase__							)
        and isinstance(UpperCamelCase__     ,       UpperCamelCase__							)
        and (numbera >= 1)
        and (numbera >= 1)
    ), "'number1' and 'number2' must been positive integer."

    _UpperCAmelCase							:  Dict											=    1  # actual answer that will be return.

    # for kgV (x,1)
    if numbera > 1 and numbera > 1:
        # builds the prime factorization of 'number1' and 'number2'
        _UpperCAmelCase							:  Any											=    prime_factorization(UpperCamelCase__							)
        _UpperCAmelCase							:  Any											=    prime_factorization(UpperCamelCase__							)

    elif numbera == 1 or numbera == 1:
        _UpperCAmelCase							:  Dict											=    []
        _UpperCAmelCase							:  int											=    []
        _UpperCAmelCase							:  int											=    max(UpperCamelCase__     ,       UpperCamelCase__							)

    _UpperCAmelCase							:  Union[str, Any]											=    0
    _UpperCAmelCase							:  str											=    0

    _UpperCAmelCase							:  Optional[int]											=    []  # captured numbers int both 'primeFac1' and 'primeFac2'

    # iterates through primeFac1
    for n in prime_fac_a:
        if n not in done:
            if n in prime_fac_a:
                _UpperCAmelCase							:  Tuple											=    prime_fac_a.count(UpperCamelCase__							)
                _UpperCAmelCase							:  Optional[int]											=    prime_fac_a.count(UpperCamelCase__							)

                for _ in range(max(UpperCamelCase__     ,       UpperCamelCase__							)							):
                    ans *= n

            else:
                _UpperCAmelCase							:  Dict											=    prime_fac_a.count(UpperCamelCase__							)

                for _ in range(UpperCamelCase__							):
                    ans *= n

            done.append(UpperCamelCase__							)

    # iterates through primeFac2
    for n in prime_fac_a:
        if n not in done:
            _UpperCAmelCase							:  Any											=    prime_fac_a.count(UpperCamelCase__							)

            for _ in range(UpperCamelCase__							):
                ans *= n

            done.append(UpperCamelCase__							)

    # precondition
    assert isinstance(UpperCamelCase__     ,       UpperCamelCase__							) and (
        ans >= 0
    ), "'ans' must been from type int and positive"

    return ans





def  lowerCamelCase_						(UpperCamelCase__		:		Optional[Any]							):
    assert isinstance(UpperCamelCase__     ,       UpperCamelCase__							) and (n >= 0), "'number' must been a positive int"

    _UpperCAmelCase							:  List[str]											=    0
    _UpperCAmelCase							:  str											=    2  # this variable holds the answer

    while index < n:
        index += 1

        ans += 1  # counts to the next number

        # if ans not prime then
        # runs to the next prime number.
        while not is_prime(UpperCamelCase__							):
            ans += 1

    # precondition
    assert isinstance(UpperCamelCase__     ,       UpperCamelCase__							) and is_prime(
        UpperCamelCase__							), "'ans' must been a prime number and from type int"

    return ans





def  lowerCamelCase_						(UpperCamelCase__		:		Any     ,       UpperCamelCase__		:		str							):
    assert (
        is_prime(UpperCamelCase__							) and is_prime(UpperCamelCase__							) and (p_number_a < p_number_a)
    ), "The arguments must been prime numbers and 'pNumber1' < 'pNumber2'"

    _UpperCAmelCase							:  str											=    p_number_a + 1  # jump to the next number

    _UpperCAmelCase							:  Union[str, Any]											=    []  # this list will be returns.

    # if number is not prime then
    # fetch the next prime number.
    while not is_prime(UpperCamelCase__							):
        number += 1

    while number < p_number_a:
        ans.append(UpperCamelCase__							)

        number += 1

        # fetch the next prime number.
        while not is_prime(UpperCamelCase__							):
            number += 1

    # precondition
    assert (
        isinstance(UpperCamelCase__     ,       UpperCamelCase__							)
        and ans[0] != p_number_a
        and ans[len(UpperCamelCase__							) - 1] != p_number_a
    ), "'ans' must been a list without the arguments"

    # 'ans' contains not 'pNumber1' and 'pNumber2' !
    return ans





def  lowerCamelCase_						(UpperCamelCase__		:		Optional[int]							):
    assert isinstance(UpperCamelCase__     ,       UpperCamelCase__							) and (n >= 1), "'n' must been int and >= 1"

    _UpperCAmelCase							:  List[Any]											=    []  # will be returned.

    for divisor in range(1     ,       n + 1							):
        if n % divisor == 0:
            ans.append(UpperCamelCase__							)

    # precondition
    assert ans[0] == 1 and ans[len(UpperCamelCase__							) - 1] == n, "Error in function getDivisiors(...)"

    return ans





def  lowerCamelCase_						(UpperCamelCase__		:		Union[str, Any]							):
    assert isinstance(UpperCamelCase__     ,       UpperCamelCase__							) and (
        number > 1
    ), "'number' must been an int and >= 1"

    _UpperCAmelCase							:  Any											=    get_divisors(UpperCamelCase__							)

    # precondition
    assert (
        isinstance(UpperCamelCase__     ,       UpperCamelCase__							)
        and (divisors[0] == 1)
        and (divisors[len(UpperCamelCase__							) - 1] == number)
    ), "Error in help-function getDivisiors(...)"

    # summed all divisors up to 'number' (exclusive), hence [:-1]
    return sum(divisors[:-1]							) == number





def  lowerCamelCase_						(UpperCamelCase__		:		Any     ,       UpperCamelCase__		:		Dict							):
    assert (
        isinstance(UpperCamelCase__     ,       UpperCamelCase__							)
        and isinstance(UpperCamelCase__     ,       UpperCamelCase__							)
        and (denominator != 0)
    ), "The arguments must been from type int and 'denominator' != 0"

    # build the greatest common divisor of numerator and denominator.
    _UpperCAmelCase							:  Dict											=    gcd(abs(UpperCamelCase__							)     ,       abs(UpperCamelCase__							)							)

    # precondition
    assert (
        isinstance(UpperCamelCase__     ,       UpperCamelCase__							)
        and (numerator % gcd_of_fraction == 0)
        and (denominator % gcd_of_fraction == 0)
    ), "Error in function gcd(...,...)"

    return (numerator // gcd_of_fraction, denominator // gcd_of_fraction)





def  lowerCamelCase_						(UpperCamelCase__		:		List[str]							):
    assert isinstance(UpperCamelCase__     ,       UpperCamelCase__							) and (n >= 0), "'n' must been a int and >= 0"

    _UpperCAmelCase							:  List[Any]											=    1  # this will be return.

    for factor in range(1     ,       n + 1							):
        ans *= factor

    return ans





def  lowerCamelCase_						(UpperCamelCase__		:		str							):
    assert isinstance(UpperCamelCase__     ,       UpperCamelCase__							) and (n >= 0), "'n' must been an int and >= 0"

    _UpperCAmelCase							:  Optional[int]											=    0
    _UpperCAmelCase							:  List[str]											=    1
    _UpperCAmelCase							:  Dict											=    1  # this will be return

    for _ in range(n - 1							):
        _UpperCAmelCase							:  Any											=    ans
        ans += fiba
        _UpperCAmelCase							:  str											=    tmp

    return ans







"""simple docstring"""
from __future__ import annotations

import math





def  lowerCamelCase_						(UpperCamelCase__		:		int     ,       UpperCamelCase__		:		int     ,       UpperCamelCase__		:		bool     ,       UpperCamelCase__		:		list[int]     ,       UpperCamelCase__		:		float							):

    if depth < 0:
        raise ValueError('''Depth cannot be less than 0'''							)

    if len(UpperCamelCase__							) == 0:
        raise ValueError('''Scores cannot be empty'''							)

    if depth == height:
        return scores[node_index]

    if is_max:
        return max(
            minimax(depth + 1     ,       node_index * 2     ,       UpperCamelCase__     ,       UpperCamelCase__     ,       UpperCamelCase__							)     ,       minimax(depth + 1     ,       node_index * 2 + 1     ,       UpperCamelCase__     ,       UpperCamelCase__     ,       UpperCamelCase__							)     ,       )

    return min(
        minimax(depth + 1     ,       node_index * 2     ,       UpperCamelCase__     ,       UpperCamelCase__     ,       UpperCamelCase__							)     ,       minimax(depth + 1     ,       node_index * 2 + 1     ,       UpperCamelCase__     ,       UpperCamelCase__     ,       UpperCamelCase__							)     ,       )





def  lowerCamelCase_						():
    _UpperCAmelCase							:  Any											=    [90, 23, 6, 33, 21, 65, 123, 3_4423]
    _UpperCAmelCase							:  Any											=    math.log(len(UpperCamelCase__							)     ,       2							)
    print('''Optimal value : '''     ,       end=''''''							)
    print(minimax(0     ,       0     ,       UpperCamelCase__     ,       UpperCamelCase__     ,       UpperCamelCase__							)							)


if __name__ == "__main__":
  import doctest

  doctest.testmod()
  main()


