code: def add_output_dim(
      X: Tensor, original_batch_shape: torch.Size
  ) -> Tuple[Tensor, int]:
      r"""Insert the output dimension at the correct location.
  
      The trailing batch dimensions of X must match the original batch dimensions
      of the training inputs, but can also include extra batch dimensions.
  
      Args:
          X: A `(new_batch_shape) x (original_batch_shape) x n x d` tensor of
              features.
          original_batch_shape: the batch shape of the model's training inputs.
  
      Returns:
          2-element tuple containing
  
          - A `(new_batch_shape) x (original_batch_shape) x m x n x d` tensor of
              features.
          - The index corresponding to the output dimension.
      """
      X_batch_shape = X.shape[:-2]
      if len(X_batch_shape) > 0 and len(original_batch_shape) > 0:
          # check that X_batch_shape supports broadcasting or augments
          # original_batch_shape with extra batch dims
          try:
              torch.broadcast_shapes(
                  X_batch_shape, original_batch_shape
              )
          except RuntimeError:
              raise RuntimeError(
                  "The trailing batch dimensions of X must match the trailing "
                  f"batch dimensions of the training inputs. Got {X.shape=} "
                  f"and {original_batch_shape=}."
              )
      # insert `m` dimension
      X = X.unsqueeze(-3)
      output_dim_idx = max(
          len(original_batch_shape), len(X_batch_shape)
      )
      return X, output_dim_idx
  
imports: from __future__ import annotations
  import warnings
  from contextlib import contextmanager, ExitStack
  from typing import Iterator, List, Optional, Tuple
  import torch
  from botorch import settings
  from botorch.exceptions import InputDataError, InputDataWarning
  from botorch.settings import _Flag
  from gpytorch import settings
  from gpytorch.module import Module
  from torch import Tensor
label_desc: Use image node_id 3 for calling a global function with example usage: add_output_dim(X, original_batch_shape) and returns: X,  output_dim_idx