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.

backend/internal/game/service.go

@@ -226,6 +226,13 @@ func (svc *Service) RobotSchedule(ctx context.Context, gameID uuid.UUID) (seed i
 	return svc.store.RobotSchedule(ctx, gameID)
 }
 
+// LastMoveAt returns the time of an account's most recent move in a game (and whether it
+// has moved). The social service uses it to reset the nudge cooldown once a player has
+// taken a turn (Stage 17).
+func (svc *Service) LastMoveAt(ctx context.Context, gameID, accountID uuid.UUID) (time.Time, bool, error) {
+	return svc.store.LastMoveAt(ctx, gameID, accountID)
+}
+
 // transition validates the actor and turn, applies op under the per-game lock and
 // commits the result.
 func (svc *Service) transition(ctx context.Context, gameID, accountID uuid.UUID, op engineOp) (MoveResult, error) {

backend/internal/game/store.go

@@ -651,6 +651,25 @@ func (s *Store) GameSeed(ctx context.Context, id uuid.UUID) (int64, error) {
 	return row.Seed, nil
 }
 
+// LastMoveAt returns the time of the account's most recent move in the game and true, or
+// the zero time and false when it has not moved. The social service uses it to reset the
+// nudge cooldown once the player has taken a turn (Stage 17).
+func (s *Store) LastMoveAt(ctx context.Context, gameID, accountID uuid.UUID) (time.Time, bool, error) {
+	var at sql.NullTime
+	err := s.db.QueryRowContext(ctx,
+		`SELECT MAX(m.created_at) FROM backend.game_moves m
+		 JOIN backend.game_players p ON p.game_id = m.game_id AND p.seat = m.seat
+		 WHERE m.game_id = $1 AND p.account_id = $2`,
+		gameID, accountID).Scan(&at)
+	if err != nil {
+		return time.Time{}, false, fmt.Errorf("game: last move at %s: %w", gameID, err)
+	}
+	if !at.Valid {
+		return time.Time{}, false, nil
+	}
+	return at.Time, true, nil
+}
+
 // RobotSchedule returns a game's bag seed and current turn-start time. The admin console
 // combines them with the robot strategy to show a robot seat's play-to-win intent and its
 // next-move ETA. Both are server-only state, never part of the public game view.