code: 
  
  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()
  
style_context: 
  
  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() = }""")
  