# Turing recognizability and Reduction Mapping on pairs of related Turing machines

I am interested in computation and I am lost on undecidability and reductions. I have the following two problems I am stuck on.

Let us call 2 Turing machines related if there is an input $$w$$ on which both halt (after finitely many steps). Let $$h(M )$$ be the set of inputs on which the TM $$M$$ halts.

Show that the problem $$\text{RELATED} = \{\langle M1, M2\rangle \mid h(M1) \cap h(M2) \neq \emptyset\}$$ is Turing recognizable.

Assume, you know that the halting problem $$H$$ is undecidable. Show that the problem $$\text{RELATED}$$ is undecidable. Show this result by defining an appropriate mapping reduction $$g$$.

To show $$\mbox{RELATED} \in \mathsf{RE}$$, you can enumerate all pairs $$\langle w,t \rangle \in \Sigma^* \times \mathbb{Z}^+$$ in dovetail fashion. For each pair check whether both $$M_1$$ and $$M_2$$ halt on $$w$$ within $$t$$ steps.
Given $$x \in \Sigma^*$$, you can compute a description of a Turing machine $$M_x$$ that halts if and only if its input is $$x$$. Since $$\langle T, x \rangle \in H \iff \langle T, M_x\rangle \in \mbox{RELATED}$$, we must have $$\mbox{RELATED} \not\in \mathsf{R}$$.