Home Assistant requests in a task, progress bar, Plex page, bitmap cache

- net/ha_api: the HTTP requests run in a FreeRTOS task on core 0; loop()
  only hands over copies through queues. The web UI's test button polls.
  No more 3-4 s loop() stalls once a minute; timeout raised to 5 s.
- Progress bar page element (page format 4, v3 pages are converted)
- Plex user setting (settings v6): {plex_user} in entity ids
- Plex page (examples): watchers, play/pause icon, title, progress
- Bitmaps cached in RAM: a LittleFS read took ~100 ms on every render
  and web preview; long JSON lists sent in 2 KB pieces
- Backup includes the refresh settings and the Plex user

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

docs/home-assistant.md

@@ -37,8 +37,10 @@ its attributes, as a number (multiplier, decimals, auto unit, without minus sign
 duration, a time from a timestamp, or with own texts per state (see the
 [web UI](web-ui.md#pages)).
 
-**Home Assistant values** on the shown page are read every minute (one entity per loop,
-at most about 3 s). An unknown or unreachable value shows `--`. The display only changes
+**Home Assistant values** on the shown page are read every minute, one entity at a time,
+by a task of their own (on the ESP32's other core), so a slow or unreachable Home
+Assistant never holds up the display or the web UI. An unknown or unreachable value shows
+`--`. The display only changes
 when the shown text changes, so round values with the decimals setting.
 
 ## Rules
@@ -47,3 +49,19 @@ The [schedule](web-ui.md#schedule) can bring a page to the front from a Home Ass
 state, e.g. the 3D printer page while `sensor.mainsailos_current_print_state` is
 `printing`. Each rule is read at its own interval; when Home Assistant can't be reached it
 is read once a minute at most, so the display keeps running.
+
+## Plex and Tautulli
+
+The **Plex** page (in [examples/home-pages.json](../examples/home-pages.json)) uses Home
+Assistant's Plex and Tautulli integrations, no extra firmware:
+
+- `sensor.mangavalkserver` (Plex): how many are watching.
+- `sensor.{plex_user}_state` (Tautulli, per user): `playing`, `paused`, ... chooses the icon
+  (bitmaps `plex playing`, `plex paused`, `plex buffering`).
+- `sensor.{plex_user}_full_title` and `sensor.{plex_user}_progress` (Tautulli, per user):
+  the title and the progress bar. Home Assistant creates these disabled: enable them on
+  the Tautulli user's device in Home Assistant.
+
+`{plex_user}` is the Plex user set in the web UI (Home Assistant tab), written the way Home
+Assistant puts it in entity ids (`MangaValk` -> `mangavalk`). A rule (Schedule tab) adds
+the page to the rotation while `sensor.mangavalkserver` is above 0.