




from __future__ import annotations



def 						lowerCAmelCase		(		_lowerCAmelCase							:   list[int]	):  # This function is recursive




 """simple docstring"""







 UpperCAmelCase__							 =						len(_lowerCAmelCase	)
 # If the array contains only one element, we return it (it's the stop condition of
 # recursion)
 if array_length <= 1:
  return array
  # Else
 UpperCAmelCase__							 =						array[0]
 UpperCAmelCase__							 =						False
 UpperCAmelCase__							 =						1
 UpperCAmelCase__							 =						[]
 while not is_found and i < array_length:
  if array[i] < pivot:
   UpperCAmelCase__							 =						True
   UpperCAmelCase__							 =						[element for element in array[i:] if element >= array[i]]
   UpperCAmelCase__							 =						longest_subsequence(_lowerCAmelCase	)
   if len(_lowerCAmelCase	) > len(_lowerCAmelCase	):
    UpperCAmelCase__							 =						temp_array
  else:
   i += 1

 UpperCAmelCase__							 =						[element for element in array[1:] if element >= pivot]
 UpperCAmelCase__							 =						[pivot, *longest_subsequence(_lowerCAmelCase	)]
 if len(_lowerCAmelCase	) > len(_lowerCAmelCase	):
  return temp_array
 else:
  return longest_subseq


if __name__ == "__main__":
  import doctest

  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() = }''')





