






import math

def   SCREAMING_SNAKE_CASE_				(				__magic_name__   :    Union[str, Any]   )						->     Any:






	"""simple docstring"""



	if n <= 0:
		raise ValueError("""Only positive integers have prime factors"""   )
	UpperCamelCase	:Tuple		=      []
	while n % 2 == 0:
		pf.append(2   )
		UpperCamelCase	:List[Any]		=      int(n / 2   )
	for i in range(3      ,     int(math.sqrt(UpperCAmelCase_   )   ) + 1      ,     2   ):
		while n % i == 0:
			pf.append(UpperCAmelCase_   )
			UpperCamelCase	:Optional[int]		=      int(n / i   )
	if n > 2:
		pf.append(UpperCAmelCase_   )
	return pf

def   SCREAMING_SNAKE_CASE_				(				__magic_name__   :    Optional[Any]   )						->     List[Any]:






	"""simple docstring"""



	if n <= 0:
		raise ValueError("""Only positive numbers are accepted"""   )
	UpperCamelCase	:Optional[Any]		=      1
	UpperCamelCase	:List[str]		=      1
	while n % 2 == 0:
		temp += 1
		UpperCamelCase	:Tuple		=      int(n / 2   )
	div *= temp
	for i in range(3      ,     int(math.sqrt(UpperCAmelCase_   )   ) + 1      ,     2   ):
		UpperCamelCase	:Union[str, Any]		=      1
		while n % i == 0:
			temp += 1
			UpperCamelCase	:Union[str, Any]		=      int(n / i   )
		div *= temp
	if n > 1:
		div *= 2
	return div

def   SCREAMING_SNAKE_CASE_				(				__magic_name__   :    Union[str, Any]   )						->     Tuple:






	"""simple docstring"""



	if n <= 0:
		raise ValueError("""Only positive numbers are accepted"""   )
	UpperCamelCase	:Union[str, Any]		=      1
	UpperCamelCase	:Tuple		=      1
	while n % 2 == 0:
		temp += 1
		UpperCamelCase	:Optional[Any]		=      int(n / 2   )
	if temp > 1:
		s *= (2**temp - 1) / (2 - 1)
	for i in range(3      ,     int(math.sqrt(UpperCAmelCase_   )   ) + 1      ,     2   ):
		UpperCamelCase	:Dict		=      1
		while n % i == 0:
			temp += 1
			UpperCamelCase	:Optional[Any]		=      int(n / i   )
		if temp > 1:
			s *= (i**temp - 1) / (i - 1)
	return int(UpperCAmelCase_   )

def   SCREAMING_SNAKE_CASE_				(				__magic_name__   :    Any   )						->     Dict:






	"""simple docstring"""



	UpperCamelCase	:Tuple		=      n
	for x in set(prime_factors(UpperCAmelCase_   )   ):
		s *= (x - 1) / x
	return int(UpperCAmelCase_   )


if __name__ == "__main__":
	import doctest

	doctest.testmod()






from itertools import count
from math import asin, pi, sqrt
def 							__lowerCAmelCase			(UpperCAmelCase_		):
    return (
        (1 - 2 * point) * sqrt(point - point**2		) + 2 * point + asin(sqrt(1 - point		)		)
    ) / 4
def 							__lowerCAmelCase			(UpperCAmelCase_		):
    a__  			=							(circles_number + 1 - sqrt(2 * circles_number		)) / (
        2 * (circles_number**2 + 1)
    )
    a__  			=							circles_number * intersection_y

    a__  			=							intersection_x * intersection_y / 2
    a__  			=							circle_bottom_arc_integral(
        1 / 2		) - circle_bottom_arc_integral(UpperCAmelCase_		)

    return triangle_area + concave_region_area
def 							__lowerCAmelCase			(UpperCAmelCase_ = 1 / 1_000		):
    a__  			=							(1 - pi / 4) / 4

    for n in count(1		):
        if concave_triangle_area(UpperCAmelCase_		) / l_section_area < fraction:
            return n

    return -1


if __name__ == "__main__":
 print(F"{solution() = }")






