Consider the following multiple conditions on $x , y , z \in \mathbb { R }$.
$$\left\{ \begin{array} { c c c c c } 0 & < & z - x y & < & 1 \\ 0 & < & z - ( x + y ) ^ { 2 } & < & - x y \end{array} \right.$$
Let $\Omega$ be the set of points $( x , y )$ for which at least one $z$ exists satisfying the above conditions. Note that the set $\Omega$ can be seen in the three-dimensional Cartesian coordinate system as the orthogonal projection of points $( x , y , z )$ satisfying the above conditions onto the $x y$-plane. Answer the following questions.
(1) Find the inequalities on $x$ and $y$ representing $\Omega$.
(2) Draw a figure of $\Omega$ in the $x y$-plane. If the boundary of $\Omega$ intersects with the $x$-axis or the $y$-axis, write down the coordinates at each intersection.
(3) The curved segments of the boundary of $\Omega$ correspond to the linear transformation of arcs of the unit circle with a matrix $\mathbf { X }$. Find one such $\mathbf { X }$. Note that the point $( 1,0 )$ on the unit circle must be transformed to a point where the curvature is maximized in the curved segments.
(4) Calculate the determinant of $\mathbf { X }$ found in (3).
(5) Calculate the area of the set $\Omega$. Note that the absolute value of the determinant of a matrix is the area scale factor of the transformation with that matrix.