code: def _last_modified_time(filename: str | os.PathLike[str]) -> int:
      """Return the last modified time of ``filename``.
  
      The time is returned as integer microseconds.
      The lowest common denominator of modern file-systems seems to be
      microsecond-level precision.
  
      We prefer to err on the side of re-rendering a file,
      so we round up to the nearest microsecond.
      """
      # upside-down floor division to get the ceiling
      return -(os.stat(filename).st_mtime_ns // -1_000)
  
imports: from __future__ import annotations
  import functools
  import os
  import pickle
  import time
  from collections import defaultdict
  from copy import copy
  from os import path
  from typing import TYPE_CHECKING, Any, Callable, NoReturn
  from sphinx import addnodes
  from sphinx.environment.adapters import toctree
  from sphinx.errors import BuildEnvironmentError, DocumentError, ExtensionError, SphinxError
  from sphinx.locale import __
  from sphinx.transforms import SphinxTransformer
  from sphinx.util import DownloadFiles, FilenameUniqDict, logging
  from sphinx.util.docutils import LoggingReporter
  from sphinx.util.i18n import CatalogRepository, docname_to_domain
  from sphinx.util.nodes import is_translatable
  from sphinx.util.osutil import canon_path, os_path
label_desc: Use image node_id 1 for calling a global function with example usage: _last_modified_time(filename) and returns: unknown