The point $P=(1,2,3)$ is reflected in the $xy$-plane, then its image $Q$ is rotated by $180^\circ$ about the $x$-axis to produce $R$, and finally, $R$ is translated by 5 units in the positive-$y$ direction to produce $S$. What are the coordinates of $S$?
Reflecting the point $(1,2,3)$ in the $xy$-plane produces $(1,2,-3)$. A $180^\circ$ rotation about the $x$-axis yields $(1,-2,3)$. Finally, the translation gives $\boxed{(1,3,3)}$.

[asy]
import three;

size(250);
currentprojection = perspective(6,3,2);

triple I = (1,0,0), J = (0,1,0), K = (0,0,1), O = (0,0,0);
triple P = (1,2,3), Q = (1,2,-3), R = (1,-2,3), S = (1,3,3);

draw(O--4*I, Arrow3(6));
draw(O--4*J, Arrow3(6));
draw(O--4*K, Arrow3(6));
draw(O--P);
draw(O--Q);
draw(O--R);
draw(P--Q--R--S,dashed);

label("$x$", 4.5*I);
label("$y$", 4.5*J);
label("$z$", 4.5*K);

dot("$P = (1,2,3)$", P, N);
dot("$Q = (1,2,-3)$", Q, SE);
dot("$R = (1,-2,3)$", R, NW);
dot("$S = (1,3,3)$", S, SE);
[/asy]