code: def _find_factor(
      dependent_rows, mark, gauss_matrix, index, smooth_relations, N
  ):
      """Finds proper factor of N. Here, transform the dependent rows as a
      combination of independent rows of the gauss_matrix to form the desired
      relation of the form ``X**2 = Y**2 modN``. After obtaining the desired relation
      we obtain a proper factor of N by `gcd(X - Y, N)`.
  
      Parameters
      ==========
  
      dependent_rows : denoted dependent rows in the reduced matrix form
      mark : boolean array to denoted dependent and independent rows
      gauss_matrix : Reduced form of the smooth relations matrix
      index : denoted the index of the dependent_rows
      smooth_relations : Smooth relations vectors matrix
      N : Number to be factored
      """
      idx_in_smooth = dependent_rows[index][1]
      independent_u = [smooth_relations[idx_in_smooth][0]]
      independent_v = [smooth_relations[idx_in_smooth][1]]
      dept_row = dependent_rows[index][0]
  
      for idx, val in enumerate(dept_row):
          if val == 1:
              for row in range(len(gauss_matrix)):
                  if gauss_matrix[row][idx] == 1 and mark[row] == True:
                      independent_u.append(smooth_relations[row][0])
                      independent_v.append(smooth_relations[row][1])
                      break
  
      u = 1
      v = 1
      for i in independent_u:
          u *= i
      for i in independent_v:
          v *= i
      # assert u**2 % N == v % N
      v = isqrt(v)
      return gcd(u - v, N)
  
imports: from sympy.core.random import _randint
  from sympy.external.gmpy import gcd, invert, sqrt
  from sympy.ntheory.residue_ntheory import _sqrt_mod_prime_power
  from sympy.ntheory import isprime
  from math import log, sqrt
label_desc: Use image node_id 9 for calling a global function with example usage: _find_factor(dependent_rows, mark, gauss_matrix, index, smooth_relations, N) and returns: gcd