refactor: extract GameService for Snake NATS/chat handling

Apply the same service pattern from Connect 4 to Snake game.
Handlers now receive the service and call its methods instead of
managing NATS connections directly. Also aligns heartbeat to 10s
and removes ConnectionIndicator patching (matching C4 changes).

features/snakegame/services/game_service.go

@@ -0,0 +1,62 @@
+// Package services provides the game service layer for Snake,
+// handling NATS subscriptions and chat room management.
+package services
+
+import (
+	"fmt"
+
+	"github.com/nats-io/nats.go"
+
+	"github.com/ryanhamamura/games/chat"
+	chatcomponents "github.com/ryanhamamura/games/chat/components"
+	"github.com/ryanhamamura/games/snake"
+)
+
+func snakeChatColor(slot int) string {
+	if slot >= 0 && slot < len(snake.SnakeColors) {
+		return snake.SnakeColors[slot]
+	}
+	return "#666"
+}
+
+// GameService manages NATS subscriptions and chat for Snake games.
+type GameService struct {
+	nc *nats.Conn
+}
+
+// NewGameService creates a new game service.
+func NewGameService(nc *nats.Conn) *GameService {
+	return &GameService{
+		nc: nc,
+	}
+}
+
+// SubscribeGameUpdates returns a NATS subscription and channel for game state updates.
+func (s *GameService) SubscribeGameUpdates(gameID string) (*nats.Subscription, <-chan *nats.Msg, error) {
+	ch := make(chan *nats.Msg, 64)
+	sub, err := s.nc.ChanSubscribe(snake.GameSubject(gameID), ch)
+	if err != nil {
+		return nil, nil, fmt.Errorf("subscribing to game updates: %w", err)
+	}
+	return sub, ch, nil
+}
+
+// ChatConfig returns the chat configuration for a game.
+func (s *GameService) ChatConfig(gameID string) chatcomponents.Config {
+	return chatcomponents.Config{
+		CSSPrefix:          "snake",
+		PostURL:            fmt.Sprintf("/snake/%s/chat", gameID),
+		Color:              snakeChatColor,
+		StopKeyPropagation: true,
+	}
+}
+
+// ChatRoom returns a chat room for a game (ephemeral, not persisted).
+func (s *GameService) ChatRoom(gameID string) *chat.Room {
+	return chat.NewRoom(s.nc, snake.ChatSubject(gameID))
+}
+
+// PublishGameUpdate sends a notification that the game state has changed.
+func (s *GameService) PublishGameUpdate(gameID string) error {
+	return s.nc.Publish(snake.GameSubject(gameID), nil)
+}