Prussia Joins the 3rd Coalition

Feb 2019
What if, in December 1805 or January 1806 Prussia joined the 3rd coalition against Napoleon? In real life they did plan to do it if Napoleon rejected the coalition peace terms, however after Austerlitz the coalition collapsed before Prussia could join, rendering the threat moot.
What if Napoleon doesn't achieve a decisive win at Austerlitz or the coalition survives and Prussia joins it? Would Napoleon be able to defeat Prussia, Austria, Russia and Britain and if yes how and would it be possible to win the war quickly.