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/inttest/social_test.go

@@ -353,3 +353,33 @@ func TestNudgeRulesAndRateLimit(t *testing.T) {
 		t.Fatalf("nudge after window: %v", err)
 	}
 }
+
+// TestNudgeCooldownResetsOnAction checks the nudge cooldown clears once the player has
+// acted (moved or chatted) since their last nudge, even within the hour (Stage 17).
+func TestNudgeCooldownResetsOnAction(t *testing.T) {
+	ctx := context.Background()
+	svc := newSocialService()
+	gsvc := newGameService()
+	gameID, seats := newGameWithSeats(t, 2) // seat 0 to move
+
+	if _, err := svc.Nudge(ctx, gameID, seats[1]); err != nil { // the waiting player nudges
+		t.Fatalf("nudge: %v", err)
+	}
+	if _, err := svc.Nudge(ctx, gameID, seats[1]); !errors.Is(err, social.ErrNudgeTooSoon) {
+		t.Fatalf("rapid nudge = %v, want ErrNudgeTooSoon", err)
+	}
+	// Seat 1 takes a turn: seat 0 passes (-> seat 1's turn), seat 1 chats then passes.
+	if _, err := gsvc.Pass(ctx, gameID, seats[0]); err != nil {
+		t.Fatalf("seat0 pass: %v", err)
+	}
+	if _, err := svc.PostMessage(ctx, gameID, seats[1], "thinking", ""); err != nil {
+		t.Fatalf("seat1 chat: %v", err)
+	}
+	if _, err := gsvc.Pass(ctx, gameID, seats[1]); err != nil {
+		t.Fatalf("seat1 pass: %v", err)
+	}
+	// Back on the opponent's turn, the cooldown is reset by the action since the nudge.
+	if _, err := svc.Nudge(ctx, gameID, seats[1]); err != nil {
+		t.Fatalf("nudge after acting = %v, want allowed (cooldown reset)", err)
+	}
+}