code: def _trial_division_stage(
      N,
      M,
      factor_base,
      sieve_array,
      sieve_poly,
      partial_relations,
      ERROR_TERM,
  ):
      """Trial division stage. Here we trial divide the values generetated
      by sieve_poly in the sieve interval and if it is a smooth number then
      it is stored in `smooth_relations`. Moreover, if we find two partial relations
      with same large prime then they are combined to form a smooth relation.
      First we iterate over sieve array and look for values which are greater
      than accumulated_val, as these values have a high chance of being smooth
      number. Then using these values we find smooth relations.
      In general, let ``t**2 = u*p modN`` and ``r**2 = v*p modN`` be two partial relations
      with the same large prime p. Then they can be combined ``(t*r/p)**2 = u*v modN``
      to form a smooth relation.
  
      Parameters
      ==========
  
      N : Number to be factored
      M : sieve interval
      factor_base : factor_base primes
      sieve_array : stores log_p values
      sieve_poly : polynomial from which we find smooth relations
      partial_relations : stores partial relations with one large prime
      ERROR_TERM : error term for accumulated_val
      """
      sqrt_n = isqrt(N)
      accumulated_val = log(M * sqrt_n) * 2**10 - ERROR_TERM
      smooth_relations = []
      proper_factor = set()
      partial_relation_upper_bound = 128 * factor_base[-1].prime
      for idx, val in enumerate(sieve_array):
          if val < accumulated_val:
              continue
          x = idx - M
          v = sieve_poly.eval(x)
          vec, is_smooth = _check_smoothness(v, factor_base)
          if is_smooth is None:  # Neither smooth nor partial
              continue
          u = sieve_poly.a * x + sieve_poly.b
          # Update the partial relation
          # If 2 partial relation with same large prime is found then generate smooth relation
          if is_smooth is False:  # partial relation found
              large_prime = vec
              # Consider the large_primes under 128*F
              if large_prime > partial_relation_upper_bound:
                  continue
              if large_prime not in partial_relations:
                  partial_relations[large_prime] = (u, v)
                  continue
              else:
                  u_prev, v_prev = partial_relations[large_prime]
                  partial_relations.pop(large_prime)
                  try:
                      large_prime_inv = invert(large_prime, N)
                  except ZeroDivisionError:  # if large_prime divides N
                      proper_factor.add(large_prime)
                      continue
                  u = u * u_prev * large_prime_inv
                  v = v * v_prev // (large_prime * large_prime)
                  vec, is_smooth = _check_smoothness(v, factor_base)
          # assert u*u % N == v % N
          smooth_relations.append((u, v, vec))
      return smooth_relations, proper_factor
  
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 6 for calling a global function with example usage: _trial_division_stage(N, M, factor_base, sieve_array, sieve_poly, partial_relations, ERROR_TERM) and returns: smooth_relations,  proper_factor