Update litterrobot to cloud push

[natekspencer]

Proposed change

Update the IoT class for litterrobot integration to cloud push

Type of change

• Spelling, grammar or other readability improvements (current branch).
• Adjusted missing or incorrect information in the current documentation (current branch).
• Added documentation for a new integration I'm adding to Home Assistant (next branch).
  • I've opened up a PR to add logo's and icons in Brands repository.
• Added documentation for a new feature I'm adding to Home Assistant (next branch).
• Removed stale or deprecated documentation.

Additional information

• Link to parent pull request in the codebase: Change litterrobot integration to cloud_push core#77741
• Link to parent pull request in the Brands repository:
• This PR fixes or closes issue: fixes #

Checklist

• This PR uses the correct branch, based on one of the following:
  • I made a change to the existing documentation and used the current branch.
  • I made a change that is related to an upcoming version of Home Assistant and used the next branch.
• The documentation follows the Home Assistant documentation standards.

[klaasnicolaas]

✅ Approved. Can be merged as soon as the parent PR gets merged 👍