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/social/social.go

@@ -28,6 +28,9 @@ type GameReader interface {
 	// SharedGame reports whether two accounts are seated together in any game
 	// (active or finished); it gates the "befriend an opponent" request path.
 	SharedGame(ctx context.Context, a, b uuid.UUID) (bool, error)
+	// LastMoveAt is the time of an account's most recent move in a game (and whether it
+	// has moved); the nudge cooldown resets once the player has taken a turn.
+	LastMoveAt(ctx context.Context, gameID, accountID uuid.UUID) (time.Time, bool, error)
 }
 
 // Sentinel errors returned by the service.