
"""simple docstring"""







import math
import sys




def       A_      (       lowercase   )  ->				int:


     """simple docstring"""







     if number != int(lowercase   ):
          raise ValueError("""the value of input must be a natural number"""   )
     if number < 0:
          raise ValueError("""the value of input must not be a negative number"""   )
     if number == 0:
          return 1
     UpperCAmelCase_    :     List[Any]            =	[-1] * (number + 1)
     UpperCAmelCase_    :     Optional[int]            =	0
     for i in range(1    ,			number + 1   ):
          UpperCAmelCase_    :     Any            =	sys.maxsize
          UpperCAmelCase_    :     str            =	int(math.sqrt(lowercase   )   )
          for j in range(1    ,			root + 1   ):
               UpperCAmelCase_    :     Optional[Any]            =	1 + answers[i - (j**2)]
               UpperCAmelCase_    :     int            =	min(lowercase    ,			lowercase   )
          UpperCAmelCase_    :     str            =	answer
     return answers[number]


if __name__ == "__main__":
       import doctest

       doctest.testmod()



"""simple docstring"""







def       A_      (       lowercase   )  ->				str:


     """simple docstring"""







     UpperCAmelCase_    :     Any            =	0

     # if input_string is "aba" than new_input_string become "a|b|a"
     UpperCAmelCase_    :     Union[str, Any]            =	""""""
     UpperCAmelCase_    :     List[Any]            =	""""""

     # append each character + "|" in new_string for range(0, length-1)
     for i in input_string[: len(lowercase   ) - 1]:
          new_input_string += i + "|"
     # append last character
     new_input_string += input_string[-1]

     # we will store the starting and ending of previous furthest ending palindromic
     # substring
     UpperCAmelCase_       ,UpperCAmelCase_    :     Dict            =	0, 0

     # length[i] shows the length of palindromic substring with center i
     UpperCAmelCase_    :     Dict            =	[1 for i in range(len(lowercase   )   )]

     # for each character in new_string find corresponding palindromic string
     UpperCAmelCase_    :     Optional[Any]            =	0
     for j in range(len(lowercase   )   ):
          UpperCAmelCase_    :     str            =	1 if j > r else min(length[l + r - j] // 2    ,			r - j + 1   )
          while (
              j - k >= 0
              and j + k < len(lowercase   )
              and new_input_string[k + j] == new_input_string[j - k]
          ):
               k += 1

          UpperCAmelCase_    :     Union[str, Any]            =	2 * k - 1

          # does this string is ending after the previously explored end (that is r) ?
          # if yes the update the new r to the last index of this
          if j + k - 1 > r:
               UpperCAmelCase_    :     List[Any]            =	j - k + 1  # noqa: E741
               UpperCAmelCase_    :     Union[str, Any]            =	j + k - 1

          # update max_length and start position
          if max_length < length[j]:
               UpperCAmelCase_    :     Optional[Any]            =	length[j]
               UpperCAmelCase_    :     int            =	j

    # create that string
     UpperCAmelCase_    :     List[str]            =	new_input_string[start - max_length // 2 : start + max_length // 2 + 1]
     for i in s:
          if i != "|":
               output_string += i

     return output_string


if __name__ == "__main__":
       import doctest

       doctest.testmod()

