





"""simple docstring"""
def   __lowerCamelCase     (							a_  :	list[list[int]]							)    ->   str:
      if len(a_							) == 0 or not all(len(a_							) == 2 for pair in coordinates							):
            raise ValueError('''The program cannot work out a fitting polynomial.'''							)

      if len({tuple(a_							) for pair in coordinates}							) != len(a_							):
            raise ValueError('''The program cannot work out a fitting polynomial.'''							)

      __SCREAMING_SNAKE_CASE     :List[Any]	       =						{x for x, _ in coordinates}
      if len(a_							) == 1:
            return f'''x={coordinates[0][0]}'''

      if len(a_							) != len(a_							):
            raise ValueError('''The program cannot work out a fitting polynomial.'''							)

      __SCREAMING_SNAKE_CASE     :Tuple	       =						len(a_							)

      __SCREAMING_SNAKE_CASE     :Any	       =						0
      __SCREAMING_SNAKE_CASE     :list[list[float]]	       =						[]
      # put the x and x to the power values in a matrix
      while count_of_line < x:
            __SCREAMING_SNAKE_CASE     :List[Any]	       =						0
            __SCREAMING_SNAKE_CASE     :int	       =						coordinates[count_of_line][0]
            __SCREAMING_SNAKE_CASE     :list[float]	       =						[]
            while count_in_line < x:
                  count_line.append(a ** (x - (count_in_line + 1))							)
                  count_in_line += 1
            matrix.append(a_							)
            count_of_line += 1

      __SCREAMING_SNAKE_CASE     :str	       =						0
      # put the y values into a vector
      __SCREAMING_SNAKE_CASE     :list[float]	       =						[]
      while count_of_line < x:
            vector.append(coordinates[count_of_line][1]							)
            count_of_line += 1

      __SCREAMING_SNAKE_CASE     :List[Any]	       =						0

      while count < x:
            __SCREAMING_SNAKE_CASE     :Optional[int]	       =						0
            while zahlen < x:
                  if count == zahlen:
                        zahlen += 1
                  if zahlen == x:
                        break
                  __SCREAMING_SNAKE_CASE     :Dict	       =						matrix[zahlen][count] / matrix[count][count]
                  for counting_columns, item in enumerate(matrix[count]							):
                        # manipulating all the values in the matrix
                        matrix[zahlen][counting_columns] -= item * bruch
                  # manipulating the values in the vector
                  vector[zahlen] -= vector[count] * bruch
                  zahlen += 1
            count += 1

      __SCREAMING_SNAKE_CASE     :int	       =						0
      # make solutions
      __SCREAMING_SNAKE_CASE     :list[str]	       =						[]
      while count < x:
            solution.append(str(vector[count] / matrix[count][count]							)							)
            count += 1

      __SCREAMING_SNAKE_CASE     :str	       =						0
      __SCREAMING_SNAKE_CASE     :List[str]	       =						'''f(x)='''

      while count < x:
            __SCREAMING_SNAKE_CASE     :list[str]	       =						solution[count].split('''E'''							)
            if len(a_							) > 1:
                  __SCREAMING_SNAKE_CASE     :Union[str, Any]	       =						f'''{remove_e[0]}*10^{remove_e[1]}'''
            solved += f'''x^{x - (count + 1)}*{solution[count]}'''
            if count + 1 != x:
                  solved += "+"
            count += 1

      return solved


if __name__ == "__main__":
  print(points_to_polynomial([]))
  print(points_to_polynomial([[]]))
  print(points_to_polynomial([[1, 0], [2, 0], [3, 0]]))
  print(points_to_polynomial([[1, 1], [2, 1], [3, 1]]))
  print(points_to_polynomial([[1, 3], [2, 3], [3, 3]]))
  print(points_to_polynomial([[1, 1], [2, 2], [3, 3]]))
  print(points_to_polynomial([[1, 1], [2, 4], [3, 9]]))
  print(points_to_polynomial([[1, 3], [2, 6], [3, 1_1]]))
  print(points_to_polynomial([[1, -3], [2, -6], [3, -1_1]]))
  print(points_to_polynomial([[1, 5], [2, 2], [3, 9]]))






"""simple docstring"""
# Lint as: python3

import hashlib
from typing import Union



def   __lowerCamelCase     (							a_  :	Union[str, int, bytes]							)    ->   bytes:
      if isinstance(a_				,     a_							):
            # Data already in bytes, returns as it as
            return hash_data
      elif isinstance(a_				,     a_							):
            # We keep the data as it as for it ot be later encoded to UTF-8
            # However replace `\\` with `/` for Windows compatibility
            __SCREAMING_SNAKE_CASE     :str	       =						hash_data.replace('''\\'''				,     '''/'''							)
      elif isinstance(a_				,     a_							):
            __SCREAMING_SNAKE_CASE     :Union[str, Any]	       =						str(a_							)
      else:
            # If data is not of the required type, raise error
            raise InvalidKeyError(a_							)

      return hash_data.encode('''utf-8'''							)






class       _SCREAMING_SNAKE_CASE(     A						):




       def __init__(		self	,SCREAMING_SNAKE_CASE__							)    ->   Optional[int]:





             """simple docstring"""






             __SCREAMING_SNAKE_CASE     :Tuple	       =						'''\nFAILURE TO GENERATE DATASET: Invalid key type detected'''
             __SCREAMING_SNAKE_CASE     :int	       =						f'''\nFound Key {hash_data} of type {type(SCREAMING_SNAKE_CASE__							)}'''
             __SCREAMING_SNAKE_CASE     :Optional[int]	       =						'''\nKeys should be either str, int or bytes type'''
             super().__init__(f'''{self.prefix}{self.err_msg}{self.suffix}'''							)






class       _SCREAMING_SNAKE_CASE(     A						):




       def __init__(		self	,SCREAMING_SNAKE_CASE__	,SCREAMING_SNAKE_CASE__	,SCREAMING_SNAKE_CASE__=""							)    ->   List[str]:





             """simple docstring"""






             __SCREAMING_SNAKE_CASE     :str	       =						key
             __SCREAMING_SNAKE_CASE     :str	       =						duplicate_key_indices
             __SCREAMING_SNAKE_CASE     :Union[str, Any]	       =						fix_msg
             __SCREAMING_SNAKE_CASE     :int	       =						'''Found multiple examples generated with the same key'''
             if len(SCREAMING_SNAKE_CASE__							) <= 20:
                   __SCREAMING_SNAKE_CASE     :Dict	       =						f'''\nThe examples at index {", ".join(SCREAMING_SNAKE_CASE__							)} have the key {key}'''
             else:
                   __SCREAMING_SNAKE_CASE     :List[str]	       =						f'''\nThe examples at index {", ".join(duplicate_key_indices[:20]							)}... ({len(SCREAMING_SNAKE_CASE__							) - 20} more) have the key {key}'''
             __SCREAMING_SNAKE_CASE     :Union[str, Any]	       =						'''\n''' + fix_msg if fix_msg else ''''''
             super().__init__(f'''{self.prefix}{self.err_msg}{self.suffix}'''							)






class       _SCREAMING_SNAKE_CASE:




       def __init__(		self	,SCREAMING_SNAKE_CASE__							)    ->   Union[str, Any]:





             """simple docstring"""






             __SCREAMING_SNAKE_CASE     :Dict	       =						hashlib.mda(_as_bytes(SCREAMING_SNAKE_CASE__							)							)




       def 					_UpperCamelCase  (		self	,SCREAMING_SNAKE_CASE__							)    ->   int:





             """simple docstring"""






             __SCREAMING_SNAKE_CASE     :Tuple	       =						self._split_mda.copy()
             __SCREAMING_SNAKE_CASE     :Optional[Any]	       =						_as_bytes(SCREAMING_SNAKE_CASE__							)
             mda.update(SCREAMING_SNAKE_CASE__							)
             # Convert to integer with hexadecimal conversion
             return int(mda.hexdigest()	,16							)