When God hardened Pharaoh's heart, did He remove Pharaoh's free will?

Read in isolation, Romans 9 makes God look like the only actor and even seems unjust, and when the objection is raised, Paul answers only that God hardens whom He hardens and who are you to reply against God. But you can never draw a conclusion from one passage of Scripture in isolation, because all Scripture is relevant at once. To build a theodicy here you must go back to the story of the plagues, where God keeps sending them expecting a response, which shows Pharaoh has free will. And the decisive text is Exodus 9:34-35, which says Pharaoh hardened his own heart, before the later statements that God hardened it. Both are true in the same context, and the order matters: Pharaoh hardened his own heart first, and then God came in and sealed it, saying in effect, you have been tested many times and refused, so now it is over. This is what we call being hardened in sin: a person spurns the call to repent so many times that his conscience is seared and he can no longer respond, and God confirms it. So the blame rests on us, not on God; God is just and no liar, and since Scripture says God cannot lie, there are indeed things God cannot do. His hardening implies a heart we already hardened ourselves.