


def 						lowerCAmelCase		(		_lowerCAmelCase							:   Any	):




 """simple docstring"""







 UpperCAmelCase__							 =						[]
 if len(_lowerCAmelCase	) == 1:
  return [nums.copy()]
 for _ in range(len(_lowerCAmelCase	)	):
  UpperCAmelCase__							 =						nums.pop(0	)
  UpperCAmelCase__							 =						permute(_lowerCAmelCase	)
  for perm in permutations:
   perm.append(_lowerCAmelCase	)
  result.extend(_lowerCAmelCase	)
  nums.append(_lowerCAmelCase	)
 return result



def 						lowerCAmelCase		(		_lowerCAmelCase							:   List[Any]	):




 """simple docstring"""







 def backtrack(_lowerCAmelCase							:   Optional[int]	):
  if start == len(_lowerCAmelCase	) - 1:
   output.append(nums[:]	)
  else:
   for i in range(_lowerCAmelCase						,							len(_lowerCAmelCase	)	):
    UpperCAmelCase__ , UpperCAmelCase__							 =						nums[i], nums[start]
    backtrack(start + 1	)
    UpperCAmelCase__ , UpperCAmelCase__							 =						nums[i], nums[start]  # backtrack

 UpperCAmelCase__							 =						[]
 backtrack(0	)
 return output


if __name__ == "__main__":
  import doctest

  # use res to print the data in permute2 function
  _lowerCAmelCase						:   Optional[Any]		=   permutea([1, 2, 3])
  print(res)
  doctest.testmod()











import math



def 						lowerCAmelCase		(		_lowerCAmelCase							:   int	):




 """simple docstring"""







 UpperCAmelCase__							 =						math.loga(math.sqrt(4 * positive_integer + 1	) / 2 + 1 / 2	)

 return exponent == int(_lowerCAmelCase	)



def 						lowerCAmelCase		(		_lowerCAmelCase							:   float = 1 / 1_2345	):




 """simple docstring"""







 UpperCAmelCase__							 =						0
 UpperCAmelCase__							 =						0

 UpperCAmelCase__							 =						3
 while True:
  UpperCAmelCase__							 =						(integer**2 - 1) / 4
  # if candidate is an integer, then there is a partition for k
  if partition_candidate == int(_lowerCAmelCase	):
   UpperCAmelCase__							 =						int(_lowerCAmelCase	)
   total_partitions += 1
   if check_partition_perfect(_lowerCAmelCase	):
    perfect_partitions += 1
  if perfect_partitions > 0:
   if perfect_partitions / total_partitions < max_proportion:
    return int(_lowerCAmelCase	)
  integer += 1


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





