fix(launcher): extract startup subsystem and harden IPC detection

Co-authored-by: Cursor <cursoragent@cursor.com>

LanMountainDesktop.Launcher/Startup/PublicIpcConnection.cs

@@ -0,0 +1,57 @@
+using LanMountainDesktop.Launcher.Services;
+using LanMountainDesktop.Shared.IPC;
+
+namespace LanMountainDesktop.Launcher.Startup;
+
+internal static class PublicIpcConnection
+{
+    public static async Task<bool> TryConnectAsync(
+        LanMountainDesktopIpcClient ipcClient,
+        TimeSpan timeout,
+        CancellationToken cancellationToken = default)
+    {
+        if (ipcClient.IsConnected)
+        {
+            return true;
+        }
+
+        try
+        {
+            var connectTask = ipcClient.ConnectAsync();
+            var completedTask = await Task.WhenAny(connectTask, Task.Delay(timeout, cancellationToken)).ConfigureAwait(false);
+            if (completedTask != connectTask)
+            {
+                return false;
+            }
+
+            await connectTask.ConfigureAwait(false);
+            return ipcClient.IsConnected;
+        }
+        catch (Exception ex) when (ex is not OperationCanceledException)
+        {
+            Logger.Info($"Public IPC is not ready yet: {ex.Message}");
+            return false;
+        }
+    }
+
+    public static async Task<bool> TryConnectWithBackoffAsync(
+        LanMountainDesktopIpcClient ipcClient,
+        IReadOnlyList<TimeSpan> attemptTimeouts,
+        CancellationToken cancellationToken = default)
+    {
+        if (ipcClient.IsConnected)
+        {
+            return true;
+        }
+
+        foreach (var timeout in attemptTimeouts)
+        {
+            if (await TryConnectAsync(ipcClient, timeout, cancellationToken).ConfigureAwait(false))
+            {
+                return true;
+            }
+        }
+
+        return ipcClient.IsConnected;
+    }
+}