Allow stale action to write actions

[ludeeus]

Proposed change

Ref actions/stale#1248

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 logos 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:
• 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.

[home-assistant]

It seems that this PR is targeted against an incorrect branch since it has a parent PR on one of our codebases. Documentation that needs to be updated for an upcoming release should target the next branch. Please change the target branch of this PR to next and rebase if needed.

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[Copilot]

Pull Request Overview

This PR updates the GitHub Actions workflow permissions for the stale action to include write access to actions. This change aligns with recent updates to the stale action that now require additional permissions to function properly.

• Adds actions: write permission to the stale workflow