This isn't a PHP issue but the way browsers work.
When using method="get", ALL URL data must be in the form. The browser looks at the action attribute and then collects all the form elements and slaps them up on the URL and then sends it to the server.
With the post method, you CAN have URL data in the action attribute. So you could do what you're asking, but you'd have to reverse your PHP code to look for the form data in the post array and the URL data in the get array.