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) {