Here’s the question:

Assume there exists a polynomial time machine $ M$ that receives two formulas $ \varphi_1,\varphi_2$ and satisfies the following:

- If $ \varphi_1 \in \mathrm{SAT}$ and $ \varphi_2 \notin \mathrm{SAT}$ , then $ M(\varphi_1,\varphi_2)=1$
- If $ \varphi_1 \notin \mathrm{SAT}$ and $ \varphi_2 \in \mathrm{SAT}$ , then $ M(\varphi_1,\varphi_2)=0$
Show that there also exists a polynomial time algorithm for SAT.

Note that we have no guarantee for the output of $ M(\varphi_1,\varphi_2)$ whenever $ \varphi_1,\varphi_2$ are both satisfiable (or when both are not).

Can I get any hint on how to start the solution?