feat: implement Codex CLI authentication check and integrate with provider

- Added a new utility for checking Codex CLI authentication status using the 'codex login status' command.
- Integrated the authentication check into the CodexProvider's installation detection and authentication methods.
- Updated Codex CLI status display in the UI to reflect authentication status and method.
- Enhanced error handling and logging for better debugging during authentication checks.
- Refactored related components to ensure consistent handling of authentication across the application.

apps/ui/src/components/views/settings-view/providers/codex-settings-tab.tsx

@@ -54,7 +54,7 @@ export function CodexSettingsTab() {
         }
       : null);
 
-  // Load Codex CLI status on mount
+  // Load Codex CLI status and auth status on mount
   useEffect(() => {
     const checkCodexStatus = async () => {
       const api = getElectronAPI();
@@ -158,11 +158,13 @@ export function CodexSettingsTab() {
   );
 
   const showUsageTracking = codexAuthStatus?.authenticated ?? false;
+  const authStatusToDisplay = codexAuthStatus;
 
   return (
     <div className="space-y-6">
       <CodexCliStatus
         status={codexCliStatus}
+        authStatus={authStatusToDisplay}
         isChecking={isCheckingCodexCli}
         onRefresh={handleRefreshCodexCli}
       />