feat(home): implement count for home auth dispatch requests and enable usage statistics

- Added `count` attribute to `homeAuthCount` requests to improve home message batching.
- Enabled usage statistics for home mode by default and added config-level enforcement.
- Adjusted failure logging to include detailed metadata in `UsageReporter`.
- Updated multiple executors to pass error details to `PublishFailure` for better debugging.
- Enhanced unit tests to validate `count` behavior and usage statistics enforcement across components.

internal/home/client_test.go

@@ -0,0 +1,32 @@
+package home
+
+import (
+	"encoding/json"
+	"net/http"
+	"testing"
+)
+
+func TestAuthDispatchRequestIncludesCount(t *testing.T) {
+	req := newAuthDispatchRequest("gpt-5.4", "session-1", http.Header{"Authorization": {"Bearer test"}}, 2)
+
+	raw, err := json.Marshal(&req)
+	if err != nil {
+		t.Fatalf("marshal auth dispatch request: %v", err)
+	}
+
+	var payload map[string]any
+	if err := json.Unmarshal(raw, &payload); err != nil {
+		t.Fatalf("unmarshal auth dispatch request: %v", err)
+	}
+	if got := int(payload["count"].(float64)); got != 2 {
+		t.Fatalf("count = %d, want 2", got)
+	}
+}
+
+func TestAuthDispatchRequestDefaultsCountToOne(t *testing.T) {
+	req := newAuthDispatchRequest("gpt-5.4", "", nil, 0)
+
+	if req.Count != 1 {
+		t.Fatalf("count = %d, want 1", req.Count)
+	}
+}