





"""simple docstring"""






def    __lowerCamelCase     (							SCREAMING_SNAKE_CASE				)		->							int:






       """simple docstring"""




       if not isinstance(SCREAMING_SNAKE_CASE,SCREAMING_SNAKE_CASE				) or not all(
           isinstance(SCREAMING_SNAKE_CASE,SCREAMING_SNAKE_CASE				) and item >= 0 for item in citations				):
              raise ValueError('The citations should be a list of non negative integers.'				)

       citations.sort()
       _UpperCAmelCase			     =  len(SCREAMING_SNAKE_CASE				)

       for i in range(SCREAMING_SNAKE_CASE				):
              if citations[len_citations - 1 - i] <= i:
                     return i

       return len_citations


if __name__ == "__main__":
    import doctest

    doctest.testmod()













"""simple docstring"""






from __future__ import annotations

def    __lowerCamelCase     (							SCREAMING_SNAKE_CASE,SCREAMING_SNAKE_CASE				)		->							list:






       """simple docstring"""




       _UpperCAmelCase			     =  [None] * (len(SCREAMING_SNAKE_CASE				) + len(SCREAMING_SNAKE_CASE				))

       _UpperCAmelCase			     =  0  # pointer to current index for left Half
       _UpperCAmelCase			     =  0  # pointer to current index for the right Half
       _UpperCAmelCase			     =  0  # pointer to current index for the sorted array Half

       while pointera < len(SCREAMING_SNAKE_CASE				) and pointera < len(SCREAMING_SNAKE_CASE				):
              if left_half[pointera] < right_half[pointera]:
                     _UpperCAmelCase			     =  left_half[pointera]
                     pointera += 1
                     index += 1
              else:
                     _UpperCAmelCase			     =  right_half[pointera]
                     pointera += 1
                     index += 1
       while pointera < len(SCREAMING_SNAKE_CASE				):
              _UpperCAmelCase			     =  left_half[pointera]
              pointera += 1
              index += 1

       while pointera < len(SCREAMING_SNAKE_CASE				):
              _UpperCAmelCase			     =  right_half[pointera]
              pointera += 1
              index += 1

       return sorted_array

def    __lowerCamelCase     (							SCREAMING_SNAKE_CASE				)		->							list:






       """simple docstring"""




       if len(SCREAMING_SNAKE_CASE				) <= 1:
              return array
       # the actual formula to calculate the middle element = left + (right - left) // 2
       # this avoids integer overflow in case of large N
       _UpperCAmelCase			     =  0 + (len(SCREAMING_SNAKE_CASE				) - 0) // 2

       # Split the array into halves till the array length becomes equal to One
       # merge the arrays of single length returned by mergeSort function and
       # pass them into the merge arrays function which merges the array
       _UpperCAmelCase			     =  array[:middle]
       _UpperCAmelCase			     =  array[middle:]

       return merge(merge_sort(SCREAMING_SNAKE_CASE				),merge_sort(SCREAMING_SNAKE_CASE				)				)


if __name__ == "__main__":
    import doctest

    doctest.testmod()






