### Instruction: The point $(1,1,1)$ is rotated $180^\circ$ about the $y$-axis, then reflected through the $yz$-plane, reflected through the $xz$-plane, rotated $180^\circ$ about the $y$-axis, and reflected through the $xz$-plane.  Find the coordinates of the point now.### Response: This problem involves a sequence of transformations on a point in three-dimensional space.
To keep track of the changes, I will label the point with its coordinates after each transformation, starting with $P(1,1,1)$.
The first transformation is a rotation of $180^\circ$ about the $y$-axis.
This means that the $y$-coordinate stays the same, but the $x$ and $z$ coordinates switch signs.
So, the point becomes $Q(-1,1,-1)$.
The second transformation is a reflection through the $yz$-plane.
This means that the $x$-coordinate changes sign, but the $y$ and $z$ coordinates stay the same.
So, the point becomes $R(1,1,-1)$.
The third transformation is a reflection through the $xz$-plane.
This means that the $y$-coordinate changes sign, but the $x$ and $z$ coordinates stay the same.
So, the point becomes $S(1,-1,-1)$.
The fourth transformation is another rotation of $180^\circ$ about the $y$-axis.
This is the same as the first transformation, so the point becomes $T(-1,-1,1)$.
The fifth and final transformation is another reflection through the $xz$-plane.
This is the same as the third transformation, so the point becomes $U(-1,1,1)$.