feat(macos): Natively bump casks during homebrew repo updates

[eduardomozart]

Description

This PR appends the official Homebrew/actions/bump-packages action to the end of the __call-update-homebrew-repo.yml workflow.
When a project (e.g., Sunshine) publishes a new release, this workflow will continue to bump the source Formula using release_homebrew. Once that completes, it will now immediately run a brew livecheck across the entire LizardByte/homebrew-homebrew tap and natively bump any outdated Casks (creating concurrent Pull Requests).
By centralizing this logic, we guarantee that Formula and Cask updates are perfectly synchronized during release events, without needing dedicated cron workflows in the downstream tap repository.
This is the companion PR to LizardByte/homebrew-homebrew#419.

Screenshot

Issues Fixed or Closed

This is the companion PR to LizardByte/homebrew-homebrew#419.

Roadmap Issues

Testing

N/A - CI configuration change.

Type of Change

• feat: New feature (non-breaking change which adds functionality)
• fix: Bug fix (non-breaking change which fixes an issue)
• docs: Documentation only changes
• style: Changes that do not affect the meaning of the code (white-space, formatting, missing semicolons, etc.)
• refactor: Code change that neither fixes a bug nor adds a feature
• perf: Code change that improves performance
• test: Adding missing tests or correcting existing tests
• build: Changes that affect the build system or external dependencies
• ci: Changes to CI configuration files and scripts
• chore: Other changes that don't modify src or test files
• revert: Reverts a previous commit
• BREAKING CHANGE: Introduces a breaking change (can be combined with any type above)

Checklist

• Code follows the style guidelines of this project
• Code has been self-reviewed
• Code has been commented, particularly in hard-to-understand areas
• Code docstring/documentation-blocks for new or existing methods/components have been added or updated
• Tests have been added or updated for new or modified functionality, including regression tests for bug fixes

AI Usage

See our AI usage policy.

• AI tools were used while preparing this PR

[sonarqubecloud]

Quality Gate passed

Issues
0 New issues
0 Accepted issues

Measures
0 Security Hotspots
0.0% Coverage on New Code
0.0% Duplication on New Code

See analysis details on SonarQube Cloud