





'''simple docstring'''
from math import factorial

class      snake_case     :




 """simple docstring"""



 def __init__(       self       :							List[str]							,	__A       :							Any							,	__A       :							Tuple	):
        __UpperCamelCase       =  real
        if isinstance(__A							,	__A	):
               __UpperCamelCase       =  [1] * rank
        else:
               __UpperCamelCase       =  rank






 def __repr__(       self       :							Any	):
        return (
            f'''{self.real}+'''
            f'''{'+'.join(str(__A	)+'E'+str(n+1	)for n,dual in enumerate(self.duals	)	)}'''
        )






 def        _lowerCamelCase					(       self       :							str	):
        __UpperCamelCase       =  self.duals.copy()
        while cur[-1] == 0:
               cur.pop(-1	)
        return Dual(self.real							,	__A	)






 def __add__(       self       :							List[Any]							,	__A       :							Any	):
        if not isinstance(__A							,	__A	):
               return Dual(self.real + other							,	self.duals	)
        __UpperCamelCase       =  self.duals.copy()
        __UpperCamelCase       =  other.duals.copy()
        if len(__A	) > len(__A	):
               o_dual.extend([1] * (len(__A	) - len(__A	))	)
        elif len(__A	) < len(__A	):
               s_dual.extend([1] * (len(__A	) - len(__A	))	)
        __UpperCamelCase       =  []
        for i in range(len(__A	)	):
               new_duals.append(s_dual[i] + o_dual[i]	)
        return Dual(self.real + other.real							,	__A	)

 SCREAMING_SNAKE_CASE_	:					Tuple					=__add__






 def __sub__(       self       :							List[str]							,	__A       :							List[Any]	):
        return self + other * -1






 def __mul__(       self       :							Tuple							,	__A       :							List[str]	):
        if not isinstance(__A							,	__A	):
               __UpperCamelCase       =  []
               for i in self.duals:
                      new_duals.append(i * other	)
               return Dual(self.real * other							,	__A	)
        __UpperCamelCase       =  [0] * (len(self.duals	) + len(other.duals	) + 1)
        for i, item in enumerate(self.duals	):
               for j, jtem in enumerate(other.duals	):
                      new_duals[i + j + 1] += item * jtem
        for k in range(len(self.duals	)	):
               new_duals[k] += self.duals[k] * other.real
        for index in range(len(other.duals	)	):
               new_duals[index] += other.duals[index] * self.real
        return Dual(self.real * other.real							,	__A	)

 SCREAMING_SNAKE_CASE_	:					Optional[Any]					=__mul__






 def __truediv__(       self       :							int							,	__A       :							List[str]	):
        if not isinstance(__A							,	__A	):
               __UpperCamelCase       =  []
               for i in self.duals:
                      new_duals.append(i / other	)
               return Dual(self.real / other							,	__A	)
        raise ValueError






 def __floordiv__(       self       :							Any							,	__A       :							List[Any]	):
        if not isinstance(__A							,	__A	):
               __UpperCamelCase       =  []
               for i in self.duals:
                      new_duals.append(i // other	)
               return Dual(self.real // other							,	__A	)
        raise ValueError






 def __pow__(       self       :							str							,	__A       :							Optional[int]	):
        if n < 0 or isinstance(__A							,	__A	):
               raise ValueError('power must be a positive integer'	)
        if n == 0:
               return 1
        if n == 1:
               return self
        __UpperCamelCase       =  self
        for _ in range(n - 1	):
               x *= self
        return x






def 				lowercase__   (   __lowercase						:  str				,  __lowercase						:  Optional[int]				,  __lowercase						:  List[str]     ) ->      Dict:



       """simple docstring"""






       if not callable(__lowercase     ):
              raise ValueError('differentiate() requires a function as input for func'     )
       if not isinstance(__lowercase				,  (float, int)     ):
              raise ValueError('differentiate() requires a float as input for position'     )
       if not isinstance(__lowercase				,  __lowercase     ):
              raise ValueError('differentiate() requires an int as input for order'     )
       __UpperCamelCase       =  Dual(__lowercase				,  1     )
       __UpperCamelCase       =  func(__lowercase     )
       if order == 0:
              return result.real
       return result.duals[order - 1] * factorial(__lowercase     )


if __name__ == "__main__":
    import doctest

    doctest.testmod()
    def 				lowercase__   (   __lowercase						:  Optional[int]     ) ->      List[Any]:



           """simple docstring"""






           return y**2 * y**4


    print(differentiate(f, 9, 2))




'''simple docstring'''







from __future__ import annotations

from typing import TypedDict







class A__       (						UpperCAmelCase__      ):
      __UpperCamelCase       :  str
      __UpperCamelCase       :  int


def 	SCREAMING_SNAKE_CASE_					(  _UpperCAmelCase	:	str  )  ->			list[str]:
       if not isinstance(_UpperCAmelCase    ,_UpperCAmelCase  ):
              raise TypeError("""The parameter s type must be str."""  )

       return [s[i:] + s[:i] for i in range(len(_UpperCAmelCase  )  )]


def 	SCREAMING_SNAKE_CASE_					(  _UpperCAmelCase	:	str  )  ->			BWTTransformDict:
       if not isinstance(_UpperCAmelCase    ,_UpperCAmelCase  ):
              raise TypeError("""The parameter s type must be str."""  )
       if not s:
              raise ValueError("""The parameter s must not be empty."""  )

       _a    : List[Any]     					=all_rotations(_UpperCAmelCase  )
       rotations.sort()  # sort the list of rotations in alphabetically order
       # make a string composed of the last char of each rotation
       _a    : BWTTransformDict     					={
           "bwt_string": "".join([word[-1] for word in rotations]  ),
           "idx_original_string": rotations.index(_UpperCAmelCase  ),
       }
       return response


def 	SCREAMING_SNAKE_CASE_					(  _UpperCAmelCase	:	str    ,_UpperCAmelCase	:	int  )  ->			str:
       if not isinstance(_UpperCAmelCase    ,_UpperCAmelCase  ):
              raise TypeError("""The parameter bwt_string type must be str."""  )
       if not bwt_string:
              raise ValueError("""The parameter bwt_string must not be empty."""  )
       try:
              _a    : List[str]     					=int(_UpperCAmelCase  )
       except ValueError:
              raise TypeError(
                  """The parameter idx_original_string type must be int or passive"""
                  """ of cast to int."""  )
       if idx_original_string < 0:
              raise ValueError("""The parameter idx_original_string must not be lower than 0."""  )
       if idx_original_string >= len(_UpperCAmelCase  ):
              raise ValueError(
                  """The parameter idx_original_string must be lower than""" """ len(bwt_string)."""  )

       _a    : Optional[int]     					=[""""""] * len(_UpperCAmelCase  )
       for _ in range(len(_UpperCAmelCase  )  ):
              for i in range(len(_UpperCAmelCase  )  ):
                     _a    : int     					=bwt_string[i] + ordered_rotations[i]
              ordered_rotations.sort()
       return ordered_rotations[idx_original_string]


if __name__ == "__main__":
   A__:    Any					    =       '''Provide a string that I will generate its BWT transform: '''
   A__:    Union[str, Any]					    =       input(entry_msg).strip()
   A__:    Optional[int]					    =       bwt_transform(s)
   print(
       F"Burrows Wheeler transform for string '{s}' results "
       F"in '{result['bwt_string']}'"
   )
   A__:    Union[str, Any]					    =       reverse_bwt(result['''bwt_string'''], result['''idx_original_string'''])
   print(
       F"Reversing Burrows Wheeler transform for entry '{result['bwt_string']}' "
       F"we get original string '{original_string}'"
   )


