
def      _a     (   __UpperCamelCase       :  int    ):
   if len(__lowerCAmelCase    ) == 0 or not all(len(__lowerCAmelCase    ) == 2 for pair in coordinates    ):
      raise ValueError('''The program cannot work out a fitting polynomial.'''    )

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

   lowerCAmelCase__ :    List[str]				      =			{x for x, _ in coordinates}
   if len(__lowerCAmelCase    ) == 1:
      return f'''x={coordinates[0][0]}'''

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

   lowerCAmelCase__ :    Any				      =			len(__lowerCAmelCase    )

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

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

   lowerCAmelCase__ :    Tuple				      =			0

   while count < x:
      lowerCAmelCase__ :    Dict				      =			0
      while zahlen < x:
         if count == zahlen:
            zahlen += 1
         if zahlen == x:
            break
         lowerCAmelCase__ :    Optional[Any]				      =			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

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

   lowerCAmelCase__ :    int				      =			0
   lowerCAmelCase__ :    List[str]				      =			"f(x)="

   while count < x:
      lowerCAmelCase__ :    list[str]				      =			solution[count].split('''E'''    )
      if len(__lowerCAmelCase    ) > 1:
         lowerCAmelCase__ :    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]]))





import abc
import copy
import dataclasses
from dataclasses import dataclass
from typing import ClassVar, Dict, Type, TypeVar

from ..features import Features


A__       :   Tuple   						=					TypeVar("""T""", bound="""TaskTemplate""")





@dataclass(frozen=__UpperCamelCase					)
class 						lowercase       (  abc.ABC					):
					# `task` is not a ClassVar since we want it to be part of the `asdict` output for JSON serialization
					__a					=    42
					__a					=    42
					__a					=    42
					def lowercase_			(		self ,	SCREAMING_SNAKE_CASE__    ):
								"""simple docstring"""







								return copy.deepcopy(self    )
					@property
					def lowercase_			(		self    ):
								"""simple docstring"""







								return Features(**self.input_schema ,	**self.label_schema    )
					@property
					@abc.abstractmethod
					def lowercase_			(		self    ):
								"""simple docstring"""







								raise NotImplementedError
					@classmethod
					def lowercase_			(		cls ,	SCREAMING_SNAKE_CASE__    ):
								"""simple docstring"""







								lowerCAmelCase__ :    List[Any]				      =			{f.name for f in dataclasses.fields(cls    )}
								return cls(**{k: v for k, v in template_dict.items() if k in field_names}    )



