Improve live and random live handling

- Add random live handlers, schemas and persistence for lot/play/continue/gameover/reward flow
- Update live status and schedule generation, including special rotation handling and CN timezone alignment
- Add config support for unlocking all daily special rotation songs and document the option
- Refresh live play, reward, precise score and session logic plus bundled main.db updates

Signed-off-by: Sean Du <do4suki@gmail.com>

internal/handler/api/live/status.go

@@ -16,10 +16,14 @@ func liveStatus(ctx *gin.Context) (res any, err error) {
 		return nil, err
 	}
 
-	specialIDs, err := listTodaySpecialRotationDifficultyIDs(ss, time.Now())
+	specialSchedules, err := listCurrentAndNextSpecialRotationSchedules(ss, time.Now())
 	if err != nil {
 		return nil, err
 	}
+	specialIDs := make([]int, 0, len(specialSchedules))
+	for _, schedule := range specialSchedules {
+		specialIDs = append(specialIDs, schedule.LiveDifficultyID)
+	}
 	trainingIDs, err := listAvailableTrainingLiveDifficultyIDs(ss)
 	if err != nil {
 		return nil, err