Persist Apps SDK widget UI state across renders with window.openai.setWidgetState and restore it from window.openai.widgetState

Documented steps

1. On mount, initialize component state with a fallback chain: `const initial = window.openai?.widgetState ?? window.openai?.toolOutput`.
2. After each meaningful UI change (selection, filter, favorite, draft), call `window.openai.setWidgetState(nextState)` with the full next-state snapshot, not a partial patch.
3. Do not await it as a persistence barrier — the host persists widget state asynchronously and there is nothing meaningful to await.
4. For widgets that produce content the model should see later, use the documented shape `{ modelContent, privateContent, imageIds }`: `modelContent` is exposed to the model on follow-up turns, `privateContent` stays UI-only.
5. Use `window.openai.uploadFile(file, { library })`, `selectFiles()`, and `getFileDownloadUrl({ fileId })` for file-backed widget content instead of base64-embedding files in state.
6. Treat `widgetState` as scoped to the widget instance for a specific message/conversation — not a durable cross-conversation store.
7. For data that must survive across separate conversations or devices, persist it in your own backend keyed by the authenticated user, and hydrate on the next tool call.
8. Official docs: https://developers.openai.com/apps-sdk/build/state-management and https://developers.openai.com/apps-sdk/reference