code: def _augment_factors_with_cached_factors(
      q: int,
      N: int,
      omega_f_nat_mean: Tensor,
      cached_omega_f_nat_mean: Tensor,
      omega_f_nat_cov: Tensor,
      cached_omega_f_nat_cov: Tensor,
  ) -> Tuple[Tensor, Tensor]:
      r"""Incorporate the cached Pareto updated factors in the forward call and
      augment them with the previously computed factors.
  
      Args:
          q: The batch size.
          N: The number of training points.
          omega_f_nat_mean: A `batch_shape x num_pareto_samples x M x (q + P) x P x 2`
              -dim Tensor containing the omega natural mean for the objective at `X`.
          cached_omega_f_nat_mean: A `num_pareto_samples x M x (N + P) x P x 2`-dim
              Tensor containing the omega natural mean for the objective at `X`.
          omega_f_nat_cov: A `batch_shape x num_pareto_samples x M x (q + P) x P x 2
              x 2` -dim Tensor containing the omega natural covariance for the
              objective at `X`.
          cached_omega_f_nat_cov: A `num_pareto_samples x M x (N + P) x P x 2 x 2`-dim
              Tensor containing the omega covariance mean for the objective at `X`.
  
      Returns:
          A two-element tuple containing
  
          - omega_f_nat_mean_new: A `batch_shape x num_pareto_samples x M x (q + P)
              x P x 2`-dim Tensor containing the omega natural mean for the objective
              at `X`.
          - omega_f_nat_cov_new: A `batch_shape x num_pareto_samples x M x (q + P) x
              P x 2 x 2`-dim Tensor containing the omega natural covariance for the
              objective at `X`.
      """
      ##############################################################################
      # omega_f_nat_mean
      ##############################################################################
      # retrieve the natural mean contribution of the Pareto block omega(x_p, x_p) for
      # the objective
      exp_cached_omega_f_nat_mean = cached_omega_f_nat_mean[
          ..., N:, :, :
      ].expand(omega_f_nat_mean[..., q:, :, :].shape)
      omega_f_nat_mean[..., q:, :, :] = exp_cached_omega_f_nat_mean
      ##############################################################################
      # omega_f_nat_cov
      ##############################################################################
      # retrieve the natural covariance contribution of the Pareto block
      # omega(x_p, x_p) for the objective
      exp_omega_f_nat_cov = cached_omega_f_nat_cov[
          ..., N:, :, :, :
      ].expand(omega_f_nat_cov[..., q:, :, :, :].shape)
      omega_f_nat_cov[..., q:, :, :, :] = exp_omega_f_nat_cov
  
      return omega_f_nat_mean, omega_f_nat_cov
  
imports: from __future__ import annotations
  from typing import Any, Optional, Tuple
  import torch
  from botorch.acquisition.acquisition import AcquisitionFunction
  from botorch.exceptions import InputDataError
  from botorch.exceptions.errors import UnsupportedError
  from botorch.models.model import Model
  from botorch.models.model_list_gp_regression import ModelListGP
  from botorch.models.utils import check_no_nans
  from botorch.utils.transforms import concatenate_pending_points, t_batch_mode_transform
  from torch import Tensor
  from torch.distributions import Normal
label_desc: Use image node_id 11 for calling a global function with example usage: _augment_factors_with_cached_factors(q, N, omega_f_nat_mean, cached_omega_f_nat_mean, omega_f_nat_cov, cached_omega_f_nat_cov) and returns: omega_f_nat_mean,  omega_f_nat_cov