code: def _find_product_shape(factors, default_point_type):
      """Determine an appropriate shape for the product from the factors."""
      factor_shapes = [factor.shape for factor in factors]
  
      if default_point_type == "auto":
          if _all_equal(factor_shapes):
              return len(factors), *factors[0].shape
          default_point_type = "vector"
      if default_point_type == "vector":
          return (
              sum(
                  math.prod(factor_shape)
                  for factor_shape in factor_shapes
              ),
          )
      if not _all_equal(factor_shapes):
          raise ValueError(
              "A default_point_type of 'matrix' or 'other' can only be used if all "
              "manifolds have the same shape."
          )
      if (
          default_point_type == "matrix"
          and not len(factor_shapes[0]) == 1
      ):
          raise ValueError(
              "A default_point_type of 'matrix' can only be used if all "
              "manifolds have vector type."
          )
      return len(factors), *factors[0].shape
  
imports: import math
  import geomstats.backend
  import geomstats.errors
  from geomstats.geometry.complex_manifold import ComplexManifold
  from geomstats.geometry.complex_riemannian_metric import ComplexRiemannianMetric
  from geomstats.geometry.manifold import Manifold
  from geomstats.geometry.riemannian_metric import RiemannianMetric
  from geomstats.vectorization import get_batch_shape
label_desc: Use image node_id 5 for calling a global function with example usage: _find_product_shape(factors, default_point_type) and returns: len,  unknown, sum, len,  unknown