code: def multiproduct(seq=(), start=1):
      """
      Return the product of a sequence of factors with multiplicities,
      times the value of the parameter ``start``. The input may be a
      sequence of (factor, exponent) pairs or a dict of such pairs.
  
          >>> multiproduct({3:7, 2:5}, 4) # = 3**7 * 2**5 * 4
          279936
  
      """
      if not seq:
          return start
      if isinstance(seq, dict):
          seq = iter(seq.items())
      units = start
      multi = []
      for base, exp in seq:
          if not exp:
              continue
          elif exp == 1:
              units *= base
          else:
              if exp % 2:
                  units *= base
              multi.append((base, exp // 2))
      return units * multiproduct(multi) ** 2
  
imports: from sympy.concrete.summations import summation
  from sympy.core.containers import Dict
  from sympy.core.mul import Mul
  from sympy.core.power import Pow
  from sympy.core.singleton import S
  from sympy.core.symbol import Symbol
  from sympy.functions.combinatorial.factorials import factorial
  from sympy.core.numbers import Integer, Rational
  from sympy.external.gmpy import gcd
  from sympy.ntheory import totient, factorint, primefactors, divisors, nextprime, primerange, pollard_rho, perfect_power, multiplicity, multiplicity_in_factorial, divisor_count, primorial, pollard_pm1, divisor_sigma, factorrat, reduced_totient
  from sympy.ntheory.factor_ import smoothness, smoothness_p, proper_divisors, antidivisors, antidivisor_count, core, udivisors, udivisor_sigma, udivisor_count, proper_divisor_count, primenu, primeomega, mersenne_prime_exponent, is_perfect, is_abundant, is_deficient, is_amicable, dra, drm, _perfect_power
  from sympy.testing.pytest import raises, slow
  from sympy.utilities.iterables import capture
label_desc: Use image node_id 2 for calling a global function with example usage: multiproduct(seq, start) and returns: unknown, start