feat(tv-ui): add slideshow pattern mode and improve admin readability

lib/providers.dart

@@ -13,6 +13,27 @@ import 'data/services/sync_service.dart';
 // ──────────────────────────────────────────────
 final mockTimeOffsetProvider = StateProvider<Duration>((ref) => Duration.zero);
 
+enum BackgroundRotateAction { next, previous, random }
+
+class BackgroundRotateCommand {
+  final int nonce;
+  final BackgroundRotateAction action;
+
+  const BackgroundRotateCommand({
+    required this.nonce,
+    required this.action,
+  });
+
+  const BackgroundRotateCommand.initial()
+      : nonce = 0,
+        action = BackgroundRotateAction.random;
+}
+
+final backgroundRotateCommandProvider =
+    StateProvider<BackgroundRotateCommand>(
+  (ref) => const BackgroundRotateCommand.initial(),
+);
+
 // ──────────────────────────────────────────────
 // CLOCK PROVIDER — fires every second
 // ──────────────────────────────────────────────
@@ -263,6 +284,22 @@ final rotationIndexProvider =
   return RotationNotifier(ref);
 });
 
+bool _isMainPhaseForSettings(
+  int phaseIndex,
+  AppSettings settings, {
+  required bool hasContent,
+}) {
+  if (!hasContent) return true;
+  if (settings.slideshowPatternMode == SlideshowPatternMode.burst) {
+    final slidesBetweenMain = settings.slideshowSlidesPerMain.clamp(1, 20);
+    final cycleLength = slidesBetweenMain + 1; // main + N slides
+    return phaseIndex % cycleLength == 0;
+  }
+
+  // Default alternating pattern.
+  return phaseIndex % 2 == 0;
+}
+
 class RotationNotifier extends StateNotifier<int> {
   final Ref _ref;
   Timer? _timer;
@@ -301,7 +338,11 @@ class RotationNotifier extends StateNotifier<int> {
         return;
       }
 
-      final isMainScreen = state % 2 == 0;
+      final isMainScreen = _isMainPhaseForSettings(
+        state,
+        settings,
+        hasContent: hasContent,
+      );
       final duration = isMainScreen
           ? _resolveMainPhaseDuration(settings)
           : settings.slideDurationSec.clamp(1, 600);
@@ -346,8 +387,5 @@ final isMainScreenProvider = Provider<bool>((ref) {
   final validSlides =
       settings.slideshowImages.where((i) => i.trim().isNotEmpty).toList();
   final hasContent = validSlides.isNotEmpty;
-  if (!hasContent) return true; // always stay on main screen
-
-  // Even = main, Odd = slideshow
-  return index % 2 == 0;
+  return _isMainPhaseForSettings(index, settings, hasContent: hasContent);
 });