



'''simple docstring'''




lowerCamelCase							:int       = tuple[float, float, float]
lowerCamelCase							:Union[str, Any]       = tuple[float, float, float]



def      a      (			lowerCamelCase__						,				lowerCamelCase__   ):
			'''simple docstring'''





			A_     :						Optional[Any]     =					end_pointa[0] - end_pointa[0]
			A_     :						Union[str, Any]     =					end_pointa[1] - end_pointa[1]
			A_     :						Tuple     =					end_pointa[2] - end_pointa[2]
			return (x, y, z)



def      a      (			lowerCamelCase__						,				lowerCamelCase__   ):
			'''simple docstring'''





			A_     :						Optional[Any]     =					ab[1] * ac[2] - ab[2] * ac[1]  # *i
			A_     :						str     =					(ab[0] * ac[2] - ab[2] * ac[0]) * -1  # *j
			A_     :						int     =					ab[0] * ac[1] - ab[1] * ac[0]  # *k
			return (x, y, z)



def      a      (			lowerCamelCase__						,				lowerCamelCase__   ):
			'''simple docstring'''





			return tuple(round(lowerCamelCase__						,				lowerCamelCase__   ) for x in vector   ) == (0, 0, 0)



def      a      (			lowerCamelCase__						,				lowerCamelCase__						,				lowerCamelCase__						,				lowerCamelCase__ = 10   ):
			'''simple docstring'''





			A_     :						Optional[int]     =					create_vector(lowerCamelCase__						,				lowerCamelCase__   )
			A_     :						List[str]     =					create_vector(lowerCamelCase__						,				lowerCamelCase__   )
			return is_zero_vector(get_ad_vectors_cross(lowerCamelCase__						,				lowerCamelCase__   )						,				lowerCamelCase__   )




'''simple docstring'''




from __future__ import annotations

from functools import lru_cache
from math import ceil

lowerCamelCase							:List[str]       = 1_0_0

lowerCamelCase							:Dict       = set(range(3, NUM_PRIMES, 2))
primes.add(2)
lowerCamelCase							:int

for prime in range(3, ceil(NUM_PRIMES**0.5), 2):
						if prime not in primes:
												continue
						primes.difference_update(set(range(prime * prime, NUM_PRIMES, prime)))



@lru_cache(maxsize=1_00   )
def      a      (			lowerCamelCase__   ):
			'''simple docstring'''





			if number_to_partition < 0:
						return set()
			elif number_to_partition == 0:
						return {1}

			A_     :						set[int]     =					set()
			A_     :						int
			A_     :						int

			for prime in primes:
						if prime > number_to_partition:
									continue
						for sub in partition(number_to_partition - prime   ):
									ret.add(sub * prime   )

			return ret



def      a      (			lowerCamelCase__ = 50_00   ):
			'''simple docstring'''





			for number_to_partition in range(1						,				lowerCamelCase__   ):
						if len(partition(lowerCamelCase__   )   ) > number_unique_partitions:
									return number_to_partition
			return None


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