

from typing import Any

import numpy as np







class 		__magic_name__    :

   '''simple docstring'''




   def __init__(       self,					lowercase_,					lowercase_ )							->					None:






     """simple docstring"""






     if np.any(tableau[:, -1],					where=tableau[:, -1] < 0 ):
       raise ValueError('''RHS must be > 0''' )

     a__						=tableau
     a__,   a__						=tableau.shape

     # Number of decision variables x1, x2, x3...
     a__						=n_vars

     # Number of artificial variables to be minimised
     a__						=len(np.where(tableau[self.n_vars : -1] == -1 )[0] )

     # 2 if there are >= or == constraints (nonstandard), 1 otherwise (std)
     a__						=(self.n_art_vars > 0) + 1

     # Number of slack variables added to make inequalities into equalities
     a__						=self.n_rows - self.n_stages

     # Objectives for each stage
     a__						=['''max''']

     # In two stage simplex, first minimise then maximise
     if self.n_art_vars:
       self.objectives.append('''min''' )

     a__						=['''''']

     # Index of current pivot row and column
     a__						=None
     a__						=None

     # Does objective row only contain (non)-negative values?
     a__						=False



   @staticmethod
   def 				_UpperCAmelCase				(       *lowercase_ )							->					list[str]:






     """simple docstring"""






     if len(lowercase_ ) != 3:
       raise ValueError('''Must provide n_vars, n_slack, and n_art_vars''' )

     if not all(x >= 0 and isinstance(lowercase_,					lowercase_ ) for x in args ):
       raise ValueError('''All arguments must be non-negative integers''' )

     # decision | slack | artificial
     a__						=['''x''', '''s''', '''a''']
     a__						=[]
     for i in range(3 ):
       for j in range(args[i] ):
         titles.append(string_starts[i] + str(j + 1 ) )
     titles.append('''RHS''' )
     return titles



   def 				_UpperCAmelCase				(       self,					lowercase_ )							->					tuple[Any, Any]:






     """simple docstring"""






     a__						=self.objectives[-1]

     # Find entries of highest magnitude in objective rows
     a__						=(objective == '''min''') - (objective == '''max''')
     a__						=np.argmax(sign * tableau[0, : self.n_vars] )

     # Choice is only valid if below 0 for maximise, and above for minimise
     if sign * self.tableau[0, col_idx] <= 0:
       a__						=True
       return 0, 0

     # Pivot row is chosen as having the lowest quotient when elements of
     # the pivot column divide the right-hand side

     # Slice excluding the objective rows
     a__						=slice(self.n_stages,					self.n_rows )

     # RHS
     a__						=tableau[s, -1]

     # Elements of pivot column within slice
     a__						=tableau[s, col_idx]

     # Array filled with nans
     a__						=np.full(self.n_rows - self.n_stages,					np.nan )

     # If element in pivot column is greater than zeron_stages, return
     # quotient or nan otherwise
     a__						=np.divide(lowercase_,					lowercase_,					out=lowercase_,					where=divisor > 0 )

     # Arg of minimum quotient excluding the nan values. n_stages is added
     # to compensate for earlier exclusion of objective columns
     a__						=np.nanargmin(lowercase_ ) + self.n_stages
     return row_idx, col_idx



   def 				_UpperCAmelCase				(       self,					lowercase_,					lowercase_,					lowercase_ )							->					np.ndarray:






     """simple docstring"""






     a__						=tableau[row_idx].copy()

     a__						=piv_row[col_idx]

     # Entry becomes 1
     piv_row *= 1 / piv_val

     # Variable in pivot column becomes basic, ie the only non-zero entry
     for idx, coeff in enumerate(tableau[:, col_idx] ):
       tableau[idx] += -coeff * piv_row
     a__						=piv_row
     return tableau



   def 				_UpperCAmelCase				(       self,					lowercase_ )							->					np.ndarray:






     """simple docstring"""






     self.objectives.pop()

     if not self.objectives:
       return tableau

     # Slice containing ids for artificial columns
     a__						=slice(-self.n_art_vars - 1,					-1 )

     # Delete the artificial variable columns
     a__						=np.delete(lowercase_,					lowercase_,					axis=1 )

     # Delete the objective row of the first stage
     a__						=np.delete(lowercase_,					0,					axis=0 )

     a__						=1
     self.n_rows -= 1
     a__						=0
     a__						=False
     return tableau



   def 				_UpperCAmelCase				(       self )							->					dict[Any, Any]:






     """simple docstring"""






     for _ in range(100 ):
       # Completion of each stage removes an objective. If both stages
       # are complete, then no objectives are left
       if not self.objectives:
         a__						=self.generate_col_titles(
             self.n_vars,					self.n_slack,					self.n_art_vars )

         # Find the values of each variable at optimal solution
         return self.interpret_tableau(self.tableau,					self.col_titles )

       a__,   a__						=self.find_pivot(self.tableau )

       # If there are no more negative values in objective row
       if self.stop_iter:
         # Delete artificial variable columns and rows. Update attributes
         a__						=self.change_stage(self.tableau )
       else:
         a__						=self.pivot(self.tableau,					lowercase_,					lowercase_ )
     return {}



   def 				_UpperCAmelCase				(       self,					lowercase_,					lowercase_ )							->					dict[str, float]:






     """simple docstring"""






     a__						={'''P''': abs(tableau[0, -1] )}

     for i in range(self.n_vars ):
       # Gives ids of nonzero entries in the ith column
       a__						=np.nonzero(tableau[:, i] )
       a__						=len(nonzero[0] )

       # First entry in the nonzero ids
       a__						=nonzero[0][0]
       a__						=tableau[nonzero_rowidx, i]

       # If there is only one nonzero value in column, which is one
       if n_nonzero == nonzero_val == 1:
         a__						=tableau[nonzero_rowidx, -1]
         a__						=rhs_val

        # Check for basic variables
     for title in col_titles:
       # Don't add RHS or slack variables to output dict
       if title[0] not in "R-s-a":
         output_dict.setdefault(lowercase_,					0 )
     return output_dict


if __name__ == "__main__":
       import doctest

       doctest.testmod()



import math
from decimal import Decimal, getcontext




def     UpperCAmelCase__       (    _A   :				int = 4_00_00_00							):





  '''simple docstring'''







  try:
    a__						=int(_A							)
  except (TypeError, ValueError):
    raise TypeError('''Parameter n must be int or castable to int.'''							)
  if n <= 0:
    raise ValueError('''Parameter n must be greater than or equal to one.'''							)
  a__						=1_00
  a__						=(Decimal(5							) ** Decimal(0.5							) + 1) / Decimal(2							)

  a__						=(math.floor(math.log(n * (phi + 2)			,   _A							) - 1							) // 3) * 3 + 2
  a__						=Decimal(round(phi ** Decimal(index + 1							)							)							) / (phi + 2)
  a__						=num // 2
  return int(_A							)


if __name__ == "__main__":
       print(f"""{solution() = }""")
