Trying to prove semidecidability of an undecidable language
We know that the halting problem is semidecidable, but the non-halting problem is not semidecidable (since otherwise the halting problem would be decidable). Hence one way to show that a language is not semidecidable is by reduction from the non-halting problem.

Let $\langle M,x \rangle$ be an input the the non-halting problem. We construct a new Turing machine $M'$ which on inputs $1,2,3,4,5$ halts, on input $6$ simulates $M$ on $x$, and on all other inputs never halts. Then $M' \in S$ iff $M$ doesn't halt on $x$.

We can also construct another Turing machine $M''$ which on inputs $1,2,3,4$ halts, on input $5$ simulates $M$ on $x$, and on all other inputs never halts. Then $M'' \notin S$ iff $M$ doesn't halt on $x$.

These reductions show that neither $S$ nor its complement are semidecidable.