


def 						_UpperCAmelCase				(			SCREAMING_SNAKE_CASE__			:					int			,     SCREAMING_SNAKE_CASE__			:					bool = False				):

     if n == 2:
          return True
     if not n % 2 or n < 2:
          return False
     if n > 5 and n % 10 not in (1, 3, 7, 9):  # can quickly check last digit
          return False
     if n > 3_31_70_44_06_46_79_88_73_85_96_19_81 and not allow_probable:
          raise ValueError(
              'Warning: upper bound of deterministic test is exceeded. '
              'Pass allow_probable=True to allow probabilistic test. '
              'A return value of True indicates a probable prime.'				)
     # array bounds provided by analysis
     __UpperCamelCase    							=[
         20_47,
         1_37_36_53,
         25_32_60_01,
         32_15_03_17_51,
         2_15_23_02_89_87_47,
         3_47_47_49_66_03_83,
         3_41_55_00_71_72_83_21,
         1,
         3_82_51_23_05_65_46_41_30_51,
         1,
         1,
         31_86_65_85_78_34_03_11_51_16_74_61,
         3_31_70_44_06_46_79_88_73_85_96_19_81,
     ]

     __UpperCamelCase    							=[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41]
     for idx, _p in enumerate(__a			,     1				):
          if n < _p:
               # then we have our last prime to check
               __UpperCamelCase    							=primes[:idx]
               break
     __UpperCamelCase							,      __UpperCamelCase    							=n - 1, 0
     # break up n -1 into a power of 2 (s) and
     # remaining odd component
     # essentially, solve for d * 2 ** s == n - 1
     while d % 2 == 0:
          d //= 2
          s += 1
     for prime in plist:
          __UpperCamelCase    							=False
          for r in range(__a				):
               __UpperCamelCase    							=pow(__a			,     d * 2**r			,     __a				)
               # see article for analysis explanation for m
               if (r == 0 and m == 1) or ((m + 1) % n == 0):
                    __UpperCamelCase    							=True
                    # this loop will not determine compositeness
                    break
          if pr:
               continue
          # if pr is False, then the above loop never evaluated to true,
          # and the n MUST be composite
          return False
     return True

def 						_UpperCAmelCase				(			):
     assert not miller_rabin(5_61				)
     assert miller_rabin(5_63				)
     # 2047

     assert not miller_rabin(83_82_01				)
     assert miller_rabin(83_82_07				)
     # 1_373_653

     assert not miller_rabin(17_31_60_01				)
     assert miller_rabin(17_31_60_17				)
     # 25_326_001

     assert not miller_rabin(30_78_38_66_41				)
     assert miller_rabin(30_78_38_66_53				)
     # 3_215_031_751

     assert not miller_rabin(1_71_30_45_57_48_01				)
     assert miller_rabin(1_71_30_45_57_48_19				)
     # 2_152_302_898_747

     assert not miller_rabin(2_77_97_99_72_83_07				)
     assert miller_rabin(2_77_97_99_72_83_27				)
     # 3_474_749_660_383

     assert not miller_rabin(1_13_85_00_23_90_94_41				)
     assert miller_rabin(1_13_85_00_23_90_95_27				)
     # 341_550_071_728_321

     assert not miller_rabin(1_27_50_41_01_88_48_80_43_51				)
     assert miller_rabin(1_27_50_41_01_88_48_80_43_91				)
     # 3_825_123_056_546_413_051

     assert not miller_rabin(7_96_66_46_44_58_50_77_87_79_18_67				)
     assert miller_rabin(7_96_66_46_44_58_50_77_87_79_19_51				)
     # 318_665_857_834_031_151_167_461

     assert not miller_rabin(55_28_40_67_74_46_64_78_97_66_03_33				)
     assert miller_rabin(55_28_40_67_74_46_64_78_97_66_03_59				)
     # 3_317_044_064_679_887_385_961_981
     # upper limit for probabilistic test


if __name__ == "__main__":
   test_miller_rabin()












class   A			(SCREAMING_SNAKE_CASE    ):
			'''simple docstring'''




			pass


class   A			(SCREAMING_SNAKE_CASE    ):
			'''simple docstring'''




			pass


class   A			:
			'''simple docstring'''




			def __init__(			self   :    List[Any]   )	->	str:







					"""simple docstring"""






					A__        =   [
					    [],
					    [],
					    [],
					]






			def    a_       (			self   :    Dict							,	__lowerCAmelCase   :    int							,	__lowerCAmelCase   :    int   )	->	None:







					"""simple docstring"""






					try:
							if len(self.queues[priority]   ) >= 1_00:
									raise OverflowError("""Maximum queue size is 100"""   )
							self.queues[priority].append(__lowerCAmelCase   )
					except IndexError:
							raise ValueError("""Valid priorities are 0, 1, and 2"""   )






			def    a_       (			self   :    Optional[Any]   )	->	int:







					"""simple docstring"""






					for queue in self.queues:
							if queue:
									return queue.pop(0   )
					raise UnderFlowError("""All queues are empty"""   )







			def __str__(			self   :    Tuple   )	->	str:







					"""simple docstring"""






					return "\n".join(f'Priority {i}: {q}' for i, q in enumerate(self.queues   )   )


class   A			:
			'''simple docstring'''




			def __init__(			self   :    int   )	->	str:







					"""simple docstring"""






					A__        =   []






			def    a_       (			self   :    int							,	__lowerCAmelCase   :    int   )	->	None:







					"""simple docstring"""






					if len(self.queue   ) == 1_00:
							raise OverFlowError("""Maximum queue size is 100"""   )
					self.queue.append(__lowerCAmelCase   )






			def    a_       (			self   :    List[str]   )	->	int:







					"""simple docstring"""






					if not self.queue:
							raise UnderFlowError("""The queue is empty"""   )
					else:
							A__        =   min(self.queue   )
							self.queue.remove(__lowerCAmelCase   )
							return data







			def __str__(			self   :    List[Any]   )	->	str:







					"""simple docstring"""






					return str(self.queue   )







def    __lowerCamelCase					(							)		->				Optional[Any]:






		"""simple docstring"""

		A__        =   FixedPriorityQueue()
		fpq.enqueue(0    ,     1_0    )
		fpq.enqueue(1    ,     7_0    )
		fpq.enqueue(0    ,     1_0_0    )
		fpq.enqueue(2    ,     1    )
		fpq.enqueue(2    ,     5    )
		fpq.enqueue(1    ,     7    )
		fpq.enqueue(2    ,     4    )
		fpq.enqueue(1    ,     6_4    )
		fpq.enqueue(0    ,     1_2_8    )
		print(__a    )
		print(fpq.dequeue()    )
		print(fpq.dequeue()    )
		print(fpq.dequeue()    )
		print(fpq.dequeue()    )
		print(fpq.dequeue()    )
		print(__a    )
		print(fpq.dequeue()    )
		print(fpq.dequeue()    )
		print(fpq.dequeue()    )
		print(fpq.dequeue()    )
		print(fpq.dequeue()    )







def    __lowerCamelCase					(							)		->				int:






		"""simple docstring"""

		A__        =   ElementPriorityQueue()
		epq.enqueue(1_0    )
		epq.enqueue(7_0    )
		epq.enqueue(1_0_0    )
		epq.enqueue(1    )
		epq.enqueue(5    )
		epq.enqueue(7    )
		epq.enqueue(4    )
		epq.enqueue(6_4    )
		epq.enqueue(1_2_8    )
		print(__a    )
		print(epq.dequeue()    )
		print(epq.dequeue()    )
		print(epq.dequeue()    )
		print(epq.dequeue()    )
		print(epq.dequeue()    )
		print(__a    )
		print(epq.dequeue()    )
		print(epq.dequeue()    )
		print(epq.dequeue()    )
		print(epq.dequeue()    )
		print(epq.dequeue()    )


if __name__ == "__main__":
					fixed_priority_queue()
					element_priority_queue()



