code: 
  
  """simple docstring"""
  from itertools import product
  
  
  
  
  def 				a__	(	lowerCAmelCase = 15						)  ->   int:
        UpperCAmelCase__ :       float						=						0.0
        UpperCAmelCase__ :       float
        UpperCAmelCase__ :       int
        UpperCAmelCase__ :       int
        UpperCAmelCase__ :       int
        UpperCAmelCase__ :       int
        UpperCAmelCase__ :       tuple[int, ...]
  
        for series in product(range(2						) ,  repeat=lowerCAmelCase						):
              UpperCAmelCase__ :       int						=						series.count(1						)
              UpperCAmelCase__ :       List[str]						=						num_turns - num_blue
              if num_red >= num_blue:
                    continue
              UpperCAmelCase__ :       List[Any]						=						1.0
              for ind, col in enumerate(lowerCAmelCase ,  2						):
                    if col == 0:
                          prob *= (ind - 1) / ind
                    else:
                          prob *= 1 / ind
  
              total_prob += prob
  
        return int(1 / total_prob						)
  
  
  if __name__ == "__main__":
      print(f'''{solution() = }''')
  
  
style_context: 
  
  """simple docstring"""
  import requests
  
  try:
      from django.contrib.auth import authenticate, login
      from django.shortcuts import redirect, render
  except ImportError:
      _A						=     _A						=     _A						=     _A						=     print
  
  
  
  
  def 				a__	(	lowerCAmelCase						)  ->   str:
        # Enter your recaptcha secret key here
        UpperCAmelCase__ :       int						=						"""secretKey"""
        UpperCAmelCase__ :       Any						=						"""https://www.google.com/recaptcha/api/siteverify"""
  
        # when method is not POST, direct user to login page
        if request.method != "POST":
              return render(lowerCAmelCase ,  """login.html"""						)
  
        # from the frontend, get username, password, and client_key
        UpperCAmelCase__ :       List[Any]						=						request.POST.get("""username"""						)
        UpperCAmelCase__ :       Optional[Any]						=						request.POST.get("""password"""						)
        UpperCAmelCase__ :       List[str]						=						request.POST.get("""g-recaptcha-response"""						)
  
        # post recaptcha response to Google's recaptcha api
        UpperCAmelCase__ :       Any						=						requests.post(lowerCAmelCase ,  data={"""secret""": secret_key, """response""": client_key}						)
        # if the recaptcha api verified our keys
        if response.json().get("""success""" ,  lowerCAmelCase						):
              # authenticate the user
              UpperCAmelCase__ :       List[str]						=						authenticate(lowerCAmelCase ,  username=lowerCAmelCase ,  password=lowerCAmelCase						)
              if user_in_database:
                    login(lowerCAmelCase ,  lowerCAmelCase						)
                    return redirect("""/your-webpage"""						)
        return render(lowerCAmelCase ,  """login.html"""						)
  
  