Stage 17 round 6 (#7): reset the nudge cooldown once the player acts

The hourly nudge cooldown now clears as soon as the sender has moved or posted a chat
since their last nudge — engagement lifts the 'don't spam' limit. Backend: Nudge checks
game.LastMoveAt + the sender's last non-nudge chat against the last nudge time
(GameReader gains LastMoveAt). UI: nudgeOnCooldown mirrors it — a chat reset is read from
the message list, a move is tracked client-side (lastActedAt on commit/pass/exchange; the
backend stays authoritative across a reload). Integration test covers the reset.

ui/src/game/Game.svelte

@@ -95,14 +95,23 @@
   // timer while the chat is open, so it re-enables without waiting for a new message.
   const nudgeCooldownSecs = 3600;
   let nudgeTick = $state(0);
+  // Unix seconds of the player's own last move, which resets the nudge cooldown (mirrors the
+  // backend, Stage 17). A chat reset is read from `messages`; a move is tracked client-side
+  // (the backend stays authoritative across a reload).
+  let lastActedAt = $state(0);
   const nudgeOnCooldown = $derived.by(() => {
     void nudgeTick;
     const mine = app.session?.userId ?? '';
-    const last = messages.reduce(
-      (mx, m) => (m.kind === 'nudge' && m.senderId === mine ? Math.max(mx, m.createdAtUnix) : mx),
-      0,
-    );
-    return last > 0 && Date.now() / 1000 - last < nudgeCooldownSecs;
+    let lastNudge = 0;
+    let lastChat = 0;
+    for (const m of messages) {
+      if (m.senderId !== mine) continue;
+      if (m.kind === 'nudge') lastNudge = Math.max(lastNudge, m.createdAtUnix);
+      else lastChat = Math.max(lastChat, m.createdAtUnix);
+    }
+    if (lastNudge === 0 || Date.now() / 1000 - lastNudge >= nudgeCooldownSecs) return false;
+    // Engagement since the nudge clears the cooldown: a chat or a move.
+    return lastChat <= lastNudge && lastActedAt <= lastNudge;
   });
 
   async function load() {
@@ -361,6 +370,7 @@
     busy = true;
     try {
       await gateway.submitPlay(id, sub.dir, sub.tiles, variant);
+      lastActedAt = Date.now() / 1000; // a move resets the nudge cooldown
       telegramHaptic('success');
       zoomed = false;
       await load();
@@ -381,6 +391,7 @@
     busy = true;
     try {
       await gateway.pass(id);
+      lastActedAt = Date.now() / 1000; // a move resets the nudge cooldown
       await load();
     } catch (e) {
       handleError(e);
@@ -461,6 +472,7 @@
     busy = true;
     try {
       await gateway.exchange(id, tiles, variant);
+      lastActedAt = Date.now() / 1000; // a move resets the nudge cooldown
       await load();
     } catch (e) {
       handleError(e);