CmdPal: Add option to return to home automatically after a delay (#43551)

## Summary of the Pull Request

This PR replaces the Go home when activated setting with a new
Automatically return home option. This allows users to specify how long
the Command Palette should wait after being dismissed before
automatically returning to the home page. It also introduces migration
logic to transition from the old setting to the new one.

## Pictures? Pictures!

<img width="1337" height="762" alt="image"
src="https://github.com/user-attachments/assets/c649ef03-b3ee-40ba-ac67-485bc40efa73"
/>


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## PR Checklist

- [x] Closes: #43355 
<!-- - [ ] Closes: #yyy (add separate lines for additional resolved
issues) -->
- [ ] **Communication:** I've discussed this with core contributors
already. If the work hasn't been agreed, this work might be rejected
- [ ] **Tests:** Added/updated and all pass
- [ ] **Localization:** All end-user-facing strings can be localized
- [ ] **Dev docs:** Added/updated
- [ ] **New binaries:** Added on the required places
- [ ] [JSON for
signing](https://github.com/microsoft/PowerToys/blob/main/.pipelines/ESRPSigning_core.json)
for new binaries
- [ ] [WXS for
installer](https://github.com/microsoft/PowerToys/blob/main/installer/PowerToysSetup/Product.wxs)
for new binaries and localization folder
- [ ] [YML for CI
pipeline](https://github.com/microsoft/PowerToys/blob/main/.pipelines/ci/templates/build-powertoys-steps.yml)
for new test projects
- [ ] [YML for signed
pipeline](https://github.com/microsoft/PowerToys/blob/main/.pipelines/release.yml)
- [ ] **Documentation updated:** If checked, please file a pull request
on [our docs
repo](https://github.com/MicrosoftDocs/windows-uwp/tree/docs/hub/powertoys)
and link it here: #xxx

<!-- Provide a more detailed description of the PR, other things fixed,
or any additional comments/features here -->
## Detailed Description of the Pull Request / Additional comments

<!-- Describe how you validated the behavior. Add automated tests
wherever possible, but list manual validation steps taken as well -->
## Validation Steps Performed

src/modules/cmdpal/Microsoft.CmdPal.UI/MainWindow.xaml.cs

@@ -58,6 +58,7 @@ public sealed partial class MainWindow : WindowEx,
     [System.Diagnostics.CodeAnalysis.SuppressMessage("StyleCop.CSharp.NamingRules", "SA1306:Field names should begin with lower-case letter", Justification = "Stylistically, window messages are WM_")]
     private readonly uint WM_TASKBAR_RESTART;
     private readonly HWND _hwnd;
+    private readonly DispatcherTimer _autoGoHomeTimer;
     private readonly WNDPROC? _hotkeyWndProc;
     private readonly WNDPROC? _originalWndProc;
     private readonly List<TopLevelHotkey> _hotkeys = [];
@@ -68,6 +69,7 @@ public sealed partial class MainWindow : WindowEx,
 
     private DesktopAcrylicController? _acrylicController;
     private SystemBackdropConfiguration? _configurationSource;
+    private TimeSpan _autoGoHomeInterval = Timeout.InfiniteTimeSpan;
 
     private WindowPosition _currentWindowPosition = new();
 
@@ -75,6 +77,9 @@ public sealed partial class MainWindow : WindowEx,
     {
         InitializeComponent();
 
+        _autoGoHomeTimer = new DispatcherTimer();
+        _autoGoHomeTimer.Tick += OnAutoGoHomeTimerOnTick;
+
         _hwnd = new HWND(WinRT.Interop.WindowNative.GetWindowHandle(this).ToInt32());
 
         unsafe
@@ -141,6 +146,15 @@ public sealed partial class MainWindow : WindowEx,
         HideWindow();
     }
 
+    private void OnAutoGoHomeTimerOnTick(object? s, object e)
+    {
+        _autoGoHomeTimer.Stop();
+
+        // BEAR LOADING: Focus Search must be suppressed here; otherwise it may steal focus (for example, from the system tray icon)
+        // and prevent the user from opening its context menu.
+        WeakReferenceMessenger.Default.Send(new GoHomeMessage(WithAnimation: false, FocusSearch: false));
+    }
+
     private static void LocalKeyboardListener_OnKeyPressed(object? sender, LocalKeyboardListenerKeyPressedEventArgs e)
     {
         if (e.Key == VirtualKey.GoBack)
@@ -220,6 +234,9 @@ public sealed partial class MainWindow : WindowEx,
         App.Current.Services.GetService<TrayIconService>()!.SetupTrayIcon(settings.ShowSystemTrayIcon);
 
         _ignoreHotKeyWhenFullScreen = settings.IgnoreShortcutWhenFullscreen;
+
+        _autoGoHomeInterval = settings.AutoGoHomeInterval;
+        _autoGoHomeTimer.Interval = _autoGoHomeInterval;
     }
 
     // We want to use DesktopAcrylicKind.Thin and custom colors as this is the default material
@@ -279,6 +296,8 @@ public sealed partial class MainWindow : WindowEx,
 
     private void ShowHwnd(IntPtr hwndValue, MonitorBehavior target)
     {
+        StopAutoGoHome();
+
         var hwnd = new HWND(hwndValue != 0 ? hwndValue : _hwnd);
 
         // Remember, IsIconic == "minimized", which is entirely different state
@@ -533,6 +552,25 @@ public sealed partial class MainWindow : WindowEx,
             // If the window was not cloaked, then leave it hidden.
             // Sure, it's not ideal, but at least it's not visible.
         }
+
+        // Start auto-go-home timer
+        RestartAutoGoHome();
+    }
+
+    private void StopAutoGoHome()
+    {
+        _autoGoHomeTimer.Stop();
+    }
+
+    private void RestartAutoGoHome()
+    {
+        if (_autoGoHomeInterval == Timeout.InfiniteTimeSpan)
+        {
+            return;
+        }
+
+        _autoGoHomeTimer.Stop();
+        _autoGoHomeTimer.Start();
     }
 
     private bool Cloak()