automatic backup / logging of chat

## Problem
Amazon Q authentication expired, and the chat was lost with all context

## Expected behavior
Chats to be stored in log/archive for recovery (much like Chat-GPT) or re-using as future context

No file, test, or entry point is named. Start by locating the Amazon Q chat handling in the extension and determine how chat context is currently held; done means chats are recoverable after authentication expiry and can be reused as future context.

Written by the indexing model from the issue text.