Build X from H and S

Build a one-qubit circuit whose unitary equals $X=\begin{pmatrix}0&1\\1&0\end{pmatrix}$ up to a global phase, using only $H=\tfrac1{\sqrt2}\begin{pmatrix}1&1\\1&-1\end{pmatrix}$ and $S=\begin{pmatrix}1&0\\0&i\end{pmatrix}$, at most four gates in total. The grader compares the circuit unitary $U$ with $X$ through $F_U=\lvert\mathrm{Tr}(X^\dagger U)\rvert/2$, which equals $1$ exactly when $U=e^{i\gamma}X$ for some real $\gamma$. Gates placed left to right on the wire multiply right to left in the formula: the wire $A$ then $B$ implements $BA$.