Allow `openai/codex-action` to authenticate through the existing Codex GitHub integration instead of requiring a separate OpenAI API key

Feature request

Allow openai/codex-action to authenticate through the existing Codex GitHub
integration instead of requiring a separate OpenAI API key.

Problem

openai/codex-action currently requires users to configure an OpenAI API key as
a GitHub secret. This makes sense for API-based usage, but it creates extra
friction for users who already have Codex connected to GitHub through the product
integration.

For simple workflows, this means users need:

• a separate API key
• separate API billing
• GitHub secret configuration
• key rotation/management
• additional security considerations

This is surprising when Codex is already available in the repository through
GitHub integration or @codex workflows.

Expected behavior

Provide an authentication mode that can use the existing Codex GitHub integration,
when installed and authorized for the repository.

For example:

uses: openai/codex-action@v1
with:
  auth-mode: codex-github-integration

or another managed-auth mechanism that does not require users to provide an
OpenAI API key manually.

Why this matters
The current API-key-only model makes codex-action less accessible for personal
and small repositories. Users who already pay for Codex through ChatGPT plans may
not want separate API billing just to run simple repository automation.

A managed integration-based auth mode would reduce setup friction, avoid
unnecessary secret management, and make Codex automation easier to adopt.

Start by reviewing openai/codex-action's existing API-key authentication path and how its GitHub Action workflow is configured. Compare that with the repository's existing Codex GitHub integration and @codex workflows, then define the authorization and installation behavior needed for a managed auth mode without manual API keys.