Refactor logic out of YAML/Bash and into a TypeScript package

Description:

The action.yml currently contains significant amounts of complex logic embedded directly within bash run steps. This approach makes the action difficult to test, maintain, and debug. Refactoring this logic into a dedicated TypeScript package will improve code quality, enable proper unit testing, and enhance reusability.

Specific areas for refactoring:

1. Input Validation Logic: The Validate Inputs step contains extensive conditional logic for validating authentication methods and input combinations. This should be moved into a TypeScript module.
2. Gemini CLI Installation Logic: The Install Gemini CLI step handles different installation methods based on the gemini_cli_version. This logic, including npm and git operations, should be encapsulated in a TypeScript function.
3. Gemini CLI Execution and Output Handling: The Run Gemini CLI step manages the execution of the Gemini CLI, captures its stdout/stderr, and handles error conditions. The output parsing and error reporting could be made more robust and testable in a TypeScript module.
4. Telemetry Upload Logic: The Upload Telemetry to Google Cloud step involves Docker commands, file manipulation, and a polling mechanism. This complex sequence should be refactored into a dedicated TypeScript module for better testability and maintainability.

Acceptance Criteria:

• All identified complex bash logic is replaced with calls to functions within a new TypeScript package.
• The action.yml remains focused on orchestrating steps rather than implementing detailed logic.
• A clear project structure for the TypeScript package is established (e.g., src/, dist/, package.json).