




from functools import reduce

UpperCAmelCase_		:					Optional[int]          =  (
    "73167176531330624919225119674426574742355349194934"
    "96983520312774506326239578318016984801869478851843"
    "85861560789112949495459501737958331952853208805511"
    "12540698747158523863050715693290963295227443043557"
    "66896648950445244523161731856403098711121722383113"
    "62229893423380308135336276614282806444486645238749"
    "30358907296290491560440772390713810515859307960866"
    "70172427121883998797908792274921901699720888093776"
    "65727333001053367881220235421809751254540594752243"
    "52584907711670556013604839586446706324415722155397"
    "53697817977846174064955149290862569321978468622482"
    "83972241375657056057490261407972968652414535100474"
    "82166370484403199890008895243450658541227588666881"
    "16427171479924442928230863465674813919123162824586"
    "17866458359124566529476545682848912883142607690042"
    "24219022671055626321111109370544217506941658960408"
    "07198403850962455444362981230987879927244284909188"
    "84580156166097919133875499200524063689912560717606"
    "05886116467109405077541002256983155200055935729725"
    "71636269561882670428252483600823257530420752963450"
)




def 		lowerCAmelCase_ (						lowerCamelCase = N ):
				return max(
				    # mypy cannot properly interpret reduce
				    int(reduce(lambda lowerCamelCase						,  lowerCamelCase : str(int(lowerCamelCase ) * int(lowerCamelCase ) )						,  n[i : i + 13] ) )
				    for i in range(len(lowerCamelCase ) - 12 ) )


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










from collections.abc import Callable
from math import pi, sqrt
from random import uniform
from statistics import mean




def 		lowerCAmelCase_ (						lowerCamelCase ):

				# A local function to see if a dot lands in the circle.
				def is_in_circle(lowerCamelCase						,  lowerCamelCase ) -> bool:
								__magic_name__						:						Dict										=sqrt((x**2) + (y**2) )
								# Our circle has a radius of 1, so a distance
								# greater than 1 would land outside the circle.
								return distance_from_centre <= 1

				# The proportion of guesses that landed in the circle
				__magic_name__						:						Union[str, Any]										=mean(
				    int(is_in_circle(uniform(-1.0						,  1.0 )						,  uniform(-1.0						,  1.0 ) ) )
				    for _ in range(lowerCamelCase ) )
				# The ratio of the area for circle to square is pi/4.
				__magic_name__						:						List[Any]										=proportion * 4
				print(F"The estimated value of pi is {pi_estimate}" )
				print(F"The numpy value of pi is {pi}" )
				print(F"The total error is {abs(pi - pi_estimate )}" )




def 		lowerCAmelCase_ (						lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase = 0.0						,  lowerCamelCase = 1.0						,  ):
				return mean(
				    function_to_integrate(uniform(lowerCamelCase						,  lowerCamelCase ) ) for _ in range(lowerCamelCase ) ) * (max_value - min_value)




def 		lowerCAmelCase_ (						lowerCamelCase						,  lowerCamelCase = 0.0						,  lowerCamelCase = 1.0 ):

				def identity_function(lowerCamelCase ) -> float:
								return x

				__magic_name__						:						Optional[int]										=area_under_curve_estimator(
				    lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase )
				__magic_name__						:						str										=(max_value * max_value - min_value * min_value) / 2

				print("""******************""" )
				print(F"Estimating area under y=x where x varies from {min_value} to {max_value}" )
				print(F"Estimated value is {estimated_value}" )
				print(F"Expected value is {expected_value}" )
				print(F"Total error is {abs(estimated_value - expected_value )}" )
				print("""******************""" )




def 		lowerCAmelCase_ (						lowerCamelCase ):

				def function_to_integrate(lowerCamelCase ) -> float:
								return sqrt(4.0 - x * x )

				__magic_name__						:						Dict										=area_under_curve_estimator(
				    lowerCamelCase						,  lowerCamelCase						,  0.0						,  2.0 )

				print("""******************""" )
				print("""Estimating pi using area_under_curve_estimator""" )
				print(F"Estimated value is {estimated_value}" )
				print(F"Expected value is {pi}" )
				print(F"Total error is {abs(estimated_value - pi )}" )
				print("""******************""" )


if __name__ == "__main__":
							import doctest

							doctest.testmod()




