code: def _find_reasonable_pivot_naive(
      col, iszerofunc=_iszero, simpfunc=None
  ):
      """
      Helper that computes the pivot value and location from a
      sequence of contiguous matrix column elements. As a side effect
      of the pivot search, this function may simplify some of the elements
      of the input column. A list of these simplified entries and their
      indices are also returned.
      This function mimics the behavior of _find_reasonable_pivot(),
      but does less work trying to determine if an indeterminate candidate
      pivot simplifies to zero. This more naive approach can be much faster,
      with the trade-off that it may erroneously return a pivot that is zero.
  
      ``col`` is a sequence of contiguous column entries to be searched for
      a suitable pivot.
      ``iszerofunc`` is a callable that returns a Boolean that indicates
      if its input is zero, or None if no such determination can be made.
      ``simpfunc`` is a callable that simplifies its input. It must return
      its input if it does not simplify its input. Passing in
      ``simpfunc=None`` indicates that the pivot search should not attempt
      to simplify any candidate pivots.
  
      Returns a 4-tuple:
      (pivot_offset, pivot_val, assumed_nonzero, newly_determined)
      ``pivot_offset`` is the sequence index of the pivot.
      ``pivot_val`` is the value of the pivot.
      pivot_val and col[pivot_index] are equivalent, but will be different
      when col[pivot_index] was simplified during the pivot search.
      ``assumed_nonzero`` is a boolean indicating if the pivot cannot be
      guaranteed to be zero. If assumed_nonzero is true, then the pivot
      may or may not be non-zero. If assumed_nonzero is false, then
      the pivot is non-zero.
      ``newly_determined`` is a list of index-value pairs of pivot candidates
      that were simplified during the pivot search.
      """
  
      # indeterminates holds the index-value pairs of each pivot candidate
      # that is neither zero or non-zero, as determined by iszerofunc().
      # If iszerofunc() indicates that a candidate pivot is guaranteed
      # non-zero, or that every candidate pivot is zero then the contents
      # of indeterminates are unused.
      # Otherwise, the only viable candidate pivots are symbolic.
      # In this case, indeterminates will have at least one entry,
      # and all but the first entry are ignored when simpfunc is None.
      indeterminates = []
      for i, col_val in enumerate(col):
          col_val_is_zero = iszerofunc(col_val)
          if col_val_is_zero == False:
              # This pivot candidate is non-zero.
              return i, col_val, False, []
          elif col_val_is_zero is None:
              # The candidate pivot's comparison with zero
              # is indeterminate.
              indeterminates.append((i, col_val))
  
      if len(indeterminates) == 0:
          # All candidate pivots are guaranteed to be zero, i.e. there is
          # no pivot.
          return None, None, False, []
  
      if simpfunc is None:
          # Caller did not pass in a simplification function that might
          # determine if an indeterminate pivot candidate is guaranteed
          # to be nonzero, so assume the first indeterminate candidate
          # is non-zero.
          return indeterminates[0][0], indeterminates[0][1], True, []
  
      # newly_determined holds index-value pairs of candidate pivots
      # that were simplified during the search for a non-zero pivot.
      newly_determined = []
      for i, col_val in indeterminates:
          tmp_col_val = simpfunc(col_val)
          if id(col_val) != id(tmp_col_val):
              # simpfunc() simplified this candidate pivot.
              newly_determined.append((i, tmp_col_val))
              if iszerofunc(tmp_col_val) == False:
                  # Candidate pivot simplified to a guaranteed non-zero value.
                  return i, tmp_col_val, False, newly_determined
  
      return (
          indeterminates[0][0],
          indeterminates[0][1],
          True,
          newly_determined,
      )
  
imports: from types import FunctionType
  from sympy.core.cache import cacheit
  from sympy.core.numbers import Float, Integer
  from sympy.core.singleton import S
  from sympy.core.symbol import uniquely_named_symbol
  from sympy.core.mul import Mul
  from sympy.polys import PurePoly, cancel
  from sympy.functions.combinatorial.numbers import nC
  from sympy.polys.matrices.domainmatrix import DomainMatrix
  from sympy.polys.matrices.ddm import DDM
  from .exceptions import NonSquareMatrixError
  from .utilities import _get_intermediate_simp, _get_intermediate_simp_bool, _iszero, _is_zero_after_expand_mul, _dotprodsimp, _simplify
label_desc: Use image node_id 2 for calling a global function with example usage: _find_reasonable_pivot_naive(col, iszerofunc, simpfunc) and returns: indeterminates,  indeterminates,  True,  newly_determined, None,  None,  False,  list, indeterminates,  indeterminates,  True,  list, i,  col_val,  False,  list, i,  tmp_col_val,  False,  newly_determined