






"""simple docstring"""



def 			_snake_case    (					lowerCamelCase__     :   int = 10**12  )    ->     int:
							lowerCamelCase_	:							Optional[int]									=1
							lowerCamelCase_	:							Union[str, Any]									=0

							lowerCamelCase_	:							List[Any]									=1
							lowerCamelCase_	:							Union[str, Any]									=1

							while numerator <= 2 * min_total - 1:
														prev_numerator += 2 * numerator
														numerator += 2 * prev_numerator

														prev_denominator += 2 * denominator
														denominator += 2 * prev_denominator

							return (denominator + 1) // 2


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











"""simple docstring"""



def 			_snake_case    (					lowerCamelCase__     :   list[list[int]]    ,						lowerCamelCase__     :   int    ,						lowerCamelCase__     :   int    ,						lowerCamelCase__     :   list[int]  )    ->     bool:

							# 1. Validate that path exists between current and next vertices
							if graph[path[curr_ind - 1]][next_ver] == 0:
														return False

							# 2. Validate that next vertex is not already in path
							return not any(vertex == next_ver for vertex in path  )

def 			_snake_case    (					lowerCamelCase__     :   list[list[int]]    ,						lowerCamelCase__     :   list[int]    ,						lowerCamelCase__     :   int  )    ->     bool:

							# Base Case
							if curr_ind == len(lowerCamelCase__  ):
														# return whether path exists between current and starting vertices
														return graph[path[curr_ind - 1]][path[0]] == 1

							# Recursive Step
							for next_ver in range(0    ,						len(lowerCamelCase__  )  ):
														if valid_connection(lowerCamelCase__    ,						lowerCamelCase__    ,						lowerCamelCase__    ,						lowerCamelCase__  ):
																					# Insert current vertex  into path as next transition
																					lowerCamelCase_	:							Tuple									=next_ver
																					# Validate created path
																					if util_hamilton_cycle(lowerCamelCase__    ,						lowerCamelCase__    ,						curr_ind + 1  ):
																												return True
																					# Backtrack
																					lowerCamelCase_	:							int									=-1
							return False

def 			_snake_case    (					lowerCamelCase__     :   list[list[int]]    ,						lowerCamelCase__     :   int = 0  )    ->     list[int]:
							lowerCamelCase_	:							Optional[Any]									=[-1] * (len(lowerCamelCase__  ) + 1)
							# initialize start and end of path with starting index
							lowerCamelCase_	:							Optional[int]									=start_index
							# evaluate and if we find answer return path either return empty array
							return path if util_hamilton_cycle(lowerCamelCase__    ,						lowerCamelCase__    ,						1  ) else []



