Dispatcher.PushFrame on a background thread can cause exceptions via the InputManager

### Description

Sometimes when we start a new thread and do a Dispatcher.PushFrame we get the following exception:

` at System.Windows.Threading.Dispatcher.VerifyAccess()
at System.Windows.DependencyObject.SetValue(DependencyProperty dp, Object value)
at System.Windows.RoutedEventArgs.InvokeHandler(Delegate handler, Object target)
at System.Windows.EventRoute.InvokeHandlersImpl(Object source, RoutedEventArgs args, Boolean reRaised)
at System.Windows.UIElement.RaiseEventImpl(DependencyObject sender, RoutedEventArgs args)
at System.Windows.UIElement.RaiseTrustedEvent(RoutedEventArgs args)
at System.Windows.Input.InputManager.ProcessStagingArea()
at System.Windows.Input.InputProviderSite.ReportInput(InputReport inputReport)
at System.Windows.Interop.HwndKeyboardInputProvider.ProcessKeyAction(MSG& msg, Boolean& handled)
at System.Windows.Interop.HwndSource.CriticalTranslateAccelerator(MSG& msg, ModifierKeys modifiers)
at System.Windows.Interop.HwndSource.OnPreprocessMessage(Object param)
at System.Windows.Threading.ExceptionWrapper.InternalRealCall(Delegate callback, Object args, Int32 numArgs)
at System.Windows.Threading.ExceptionWrapper.TryCatchWhen(Object source, Delegate callback, Object args, Int32 numArgs, Delegate catchHandler)
at System.Windows.Threading.Dispatcher.LegacyInvokeImpl(DispatcherPriority priority, TimeSpan timeout, Delegate method, Object args, Int32 numArgs)
at System.Windows.Interop.HwndSource.OnPreprocessMessageThunk(MSG& msg, Boolean& handled)
at System.Windows.Interop.ComponentDispatcherThread.RaiseThreadMessage(MSG& msg)
at System.Windows.Threading.Dispatcher.PushFrameImpl(DispatcherFrame frame)
at System.Windows.Threading.Dispatcher.PushFrame(DispatcherFrame frame)`

### Reproduction Steps

I don't have repro steps but I believe someone is typing when the thread is created and the Dispatcher.PushFrame is called.
Looking at InputManager.ProcessInput, it also does not appear to be thread safe due to access to non-thread safe data structures like Stack.

### Expected behavior

No exception

### Actual behavior

Calling Dispatcher.PushFrame on a background thread should not cause exceptions.

### Regression?

_No response_

### Known Workarounds

_No response_

### Impact

_No response_

### Configuration

_No response_

### Other information

_No response_