fix: resolve Gemini 400 error for multi-account paths

[rennzhang]

Description

Fixes Gemini API 400 errors when users are logged into multiple Google accounts. The /u/N/ path prefix (e.g., /u/0/, /u/1/) was not being included in API requests, causing authentication failures.

Changes:

• Extract /u/N path prefix from the current URL via extractPathPrefix()
• Include the prefix in the batchexecute endpoint, source-path query param, and referrer header
• Backward compatible: single-account users (no /u/N/ prefix) are unaffected

Type of Change

• Bug fix (non-breaking change which fixes an issue)
• New feature (non-breaking change which adds functionality)
• Breaking change (fix or feature that would cause existing functionality to not work as expected)
• Documentation update

Checklist

• My code follows the style guidelines of this project
• I have performed a self-review of my code
• I have commented my code, particularly in hard-to-understand areas
• I have made corresponding changes to the documentation
• My changes generate no new warnings
• Any dependent changes have been merged and published