from math import factorial

class 					_lowercase    :






    '''simple docstring'''




    def __init__(    self       ,						lowerCamelCase__       ,						lowerCamelCase__ ):
       lowerCAmelCase_:							int       =	real
       if isinstance(A_       ,						A_ ):
          lowerCAmelCase_:							Dict       =	[1] * rank
       else:
          lowerCAmelCase_:							Optional[int]       =	rank




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




    def      _a    (    self ):
       lowerCAmelCase_:							List[Any]       =	self.duals.copy()
       while cur[-1] == 0:
          cur.pop(-1 )
       return Dual(self.real       ,						A_ )




    def __add__(    self       ,						lowerCamelCase__ ):
       if not isinstance(A_       ,						A_ ):
          return Dual(self.real + other       ,						self.duals )
       lowerCAmelCase_:							List[str]       =	self.duals.copy()
       lowerCAmelCase_:							Union[str, Any]       =	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_ )) )
       lowerCAmelCase_:							Union[str, Any]       =	[]
       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:				Any          =    __add__




    def __sub__(    self       ,						lowerCamelCase__ ):
       return self + other * -1




    def __mul__(    self       ,						lowerCamelCase__ ):
       if not isinstance(A_       ,						A_ ):
          lowerCAmelCase_:							Dict       =	[]
          for i in self.duals:
             new_duals.append(i * other )
          return Dual(self.real * other       ,						A_ )
       lowerCAmelCase_:							Tuple       =	[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:				str          =    __mul__




    def __truediv__(    self       ,						lowerCamelCase__ ):
       if not isinstance(A_       ,						A_ ):
          lowerCAmelCase_:							Any       =	[]
          for i in self.duals:
             new_duals.append(i / other )
          return Dual(self.real / other       ,						A_ )
       raise ValueError




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




    def __pow__(    self       ,						lowerCamelCase__ ):
       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
       lowerCAmelCase_:							Dict       =	self
       for _ in range(n - 1 ):
          x *= self
       return x





def 		snake_case__	(		lowercase							,				lowercase							,				lowercase					):
   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"					)
   lowerCAmelCase_:							Dict       =	Dual(lowercase__							,				1					)
   lowerCAmelCase_:							List[Any]       =	func(lowercase__					)
   if order == 0:
      return result.real
   return result.duals[order - 1] * factorial(lowercase__					)


if __name__ == "__main__":
 import doctest

 doctest.testmod()
 def 		snake_case__	(		lowercase					):
    return y**2 * y**4







 print(differentiate(f, 9, 2))



"""simple docstring"""




from collections import defaultdict





def       a_ (	lowercase__ :int     ):
			__lowerCamelCase          =      1
			__lowerCamelCase          =      True
			for v in tree[start]:
						if v not in visited:
									ret += dfs(lowercase__     )
			if ret % 2 == 0:
						cuts.append(lowercase__     )
			return ret





def       a_ (	):
			dfs(1     )


if __name__ == "__main__":
					__magic_name__  , __magic_name__						: Tuple    =				1_0, 9
					__magic_name__						: Tuple    =				defaultdict(list)
					__magic_name__						: dict[int, bool]    =				{}
					__magic_name__						: list[int]    =				[]
					__magic_name__						: List[str]    =				0
					__magic_name__						: Tuple    =				[(2, 1), (3, 1), (4, 3), (5, 2), (6, 1), (7, 2), (8, 6), (9, 8), (1_0, 8)]
					for u, v in edges:
										tree[u].append(v)
										tree[v].append(u)
					even_tree()
					print(len(cuts) - 1)




