feat(plex): every show card plays the episode it would put on

A show card's picture opened the show page everywhere but Home, whose shelves carried the
resolved `next`. Now every Plex picture plays: a show asks `GET /api/plex/show/{id}/next`
(the in-progress episode, else the first unwatched, else S1·E1 — the rule Home and the show
page's Resume follow) on the click, fresh each time. A started show shows Continue + Restart;
its title, an ⓘ and the menu's "Open show" row open its page; one with no episodes opens there.

The question is asked per show, never carried on every card: three probes a show would cost a
grid page up to 180 queries. The hover preview asks it once it opens and names the episode
("Next: S6·E16 — Title"). Related shows on a show page play the same way.

docs/DESIGN.md

@@ -400,7 +400,9 @@ pick a density step; it never changes the anatomy or loses an action. [Nielsen #
   `mediaTitleClass`, `mediaPictureBox`).
 - **Play only where the click plays.** The hover layer dims the picture and shows `MediaPlayGlyph`, the
   one round media Play (the button family's `media` variant), only on a card whose click starts
-  playback; a show without a next episode opens instead and shows none. The hover row (watched, save or
+  playback. Every Plex picture plays, a show too: the episode in progress (Continue + Restart), else the
+  first unwatched, else the first, which the click asks the server for (`/show/{id}/next`); its title
+  and an ⓘ open the show's page, and one with no episodes opens there. The hover row (watched, save or
   add, "…") docks at the picture's bottom-right; the length badge steps aside while it shows.
 - **Density.** A grid card: a two-line title and the full meta (Plex: year · ★ score; a video: the
   channel, then views · age). On a shelf (any horizontal strip) the title is one line, cut with an
@@ -413,7 +415,8 @@ pick a density step; it never changes the anatomy or loses an action. [Nielsen #
   had become the strip's.
 - **The preview (the owner's pick C).** Resting the mouse on a Plex card for ~0.9 s opens a panel beside
   it: the whole title, year · length · ★, and the synopsis in short lines (`ui/MediaPreview` +
-  `useMediaPreview`). Mouse only (a keyboard walk or a tap never opens it; "…" → Media info has the same),
+  `useMediaPreview`); a show's adds the episode it would play ("Next: S6·E16 — Title"), asked as the
+  panel opens. Mouse only (a keyboard walk or a tap never opens it; "…" → Media info has the same),
   passive (it never takes a click), and any press on the card closes it. Long text never goes in a
   native `title` tooltip.
 - **Home's hero is a card** (the owner's pick A): the same hover row and Play on its picture, the same
@@ -718,3 +721,4 @@ of a full-width panel and nothing on the right.
 | 2026-10-08 | Plex subtitles and the control bar | The player draws the cues itself; while the bar shows they rise above it (200 ms), then settle back | The browser's own cues, moved by `line` | On an iPad the bar covered the text; a native cue cannot be animated or measured, and our layer behaves the same on every engine |
 | 2026-10-08 | Hover on a button over a picture | The `media` variant's hover is a 2px accent ring over a darker body (Plex + YT cards, rail arrows; option B) | An accent fill (reads as "selected"); a white wash (weak on a bright poster) | The 60 → 75% black step was invisible, and over a card a rail arrow could not be told from the card's own Play under it |
 | 2026-10-08 | Plex "Add to collection" | A movie or show card's menu row opens a panel beside the card: its library's manual collections tick in place, a name + Enter creates one with the title in it (admin; option A) | An Add submenu with a "New…" dialog; the per-title editor modal from the menu | A collection took a trip to the Collections page, a name, a title search, then a second search per title; the panel is the video card's "Add to playlist" |
+| 2026-10-09 | A Plex show card plays | Its picture plays the show's next episode everywhere (in progress → Continue + Restart, else the first unwatched, else S1·E1), resolved on the click; the preview names it once up; title and ⓘ open the show (option A + hover ask) | The picture opens the show page (as before); every card carrying `next` | A show card on Discover had no Play while Home's had one; carrying `next` costs a grid page up to 180 queries for the few cards anyone plays |