Part of https://linear.app/a8c/issue/WOOAI-360

Proposed Changes

• Save new checkpoint ID in the cookies for every user message sent to the AI assistant

Why are these changes being made?

See 3459-ghe-Automattic/ciab-admin for details.

Testing Instructions

TBD

Pre-merge Checklist

• Has the general commit checklist been followed? (PCYsg-hS-p2)
• Have you written new tests for your changes?
• Have you tested the feature in Simple (P9HQHe-k8-p2), Atomic (P9HQHe-jW-p2), and self-hosted Jetpack sites (PCYsg-g6b-p2)?
• Have you checked for TypeScript, React or other console errors?
• Have you tested accessibility for your changes? Ensure the feature remains usable with various user agents (e.g., browsers), interfaces (e.g., keyboard navigation), and assistive technologies (e.g., screen readers) (PCYsg-S3g-p2).
• Have you used memoizing on expensive computations? More info in Memoizing with create-selector and Using memoizing selectors and Our Approach to Data
• Have we added the "[Status] String Freeze" label as soon as any new strings were ready for translation (p4TIVU-5Jq-p2)?
  • For UI changes, have we tested the change in various languages (for example, ES, PT, FR, or DE)? The length of text and words vary significantly between languages.
• For changes affecting Jetpack: Have we added the "[Status] Needs Privacy Updates" label if this pull request changes what data or activity we track or use (p4TIVU-aUh-p2)?