Fixes#1364
<!-- CURSOR_SUMMARY -->
> [!NOTE]
> Refreshes the Supabase prompt’s login state example to use the latest
onAuthStateChange event handling and unsubscribe pattern.
>
> - **Prompts**:
> - **Supabase system prompt (`src/prompts/supabase_prompt.ts`)**:
> - Revise login state management example to use
`supabase.auth.onAuthStateChange` with explicit events:
`INITIAL_SESSION`, `SIGNED_IN`, `SIGNED_OUT`, `PASSWORD_RECOVERY`,
`TOKEN_REFRESHED`, `USER_UPDATED`.
> - Update unsubscribe usage to `data.subscription.unsubscribe()` and
remove outdated async/getSession snippet.
>
> <sup>Written by [Cursor
Bugbot](https://cursor.com/dashboard?tab=bugbot) for commit
239d20769a02d102d168126f973a6cb4bf6d9e48. This will update automatically
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[here](https://cursor.com/dashboard?tab=bugbot).</sup>
<!-- /CURSOR_SUMMARY -->
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