feat(antigravity): add FetchAntigravityProjectID function and integrate project ID retrieval

internal/api/handlers/management/auth_files.go

@@ -1472,6 +1472,17 @@ func (h *Handler) RequestAntigravityToken(c *gin.Context) {
 			}
 		}
 
+		projectID := ""
+		if strings.TrimSpace(tokenResp.AccessToken) != "" {
+			fetchedProjectID, errProject := sdkAuth.FetchAntigravityProjectID(ctx, tokenResp.AccessToken, httpClient)
+			if errProject != nil {
+				log.Warnf("antigravity: failed to fetch project ID: %v", errProject)
+			} else {
+				projectID = fetchedProjectID
+				log.Infof("antigravity: obtained project ID %s", projectID)
+			}
+		}
+
 		now := time.Now()
 		metadata := map[string]any{
 			"type":          "antigravity",
@@ -1484,6 +1495,9 @@ func (h *Handler) RequestAntigravityToken(c *gin.Context) {
 		if email != "" {
 			metadata["email"] = email
 		}
+		if projectID != "" {
+			metadata["project_id"] = projectID
+		}
 
 		fileName := sanitizeAntigravityFileName(email)
 		label := strings.TrimSpace(email)
@@ -1507,6 +1521,9 @@ func (h *Handler) RequestAntigravityToken(c *gin.Context) {
 
 		delete(oauthStatus, state)
 		fmt.Printf("Authentication successful! Token saved to %s\n", savedPath)
+		if projectID != "" {
+			fmt.Printf("Using GCP project: %s\n", projectID)
+		}
 		fmt.Println("You can now use Antigravity services through this CLI")
 	}()