Let $a , b , c$ and $d$ be real numbers satisfying $a < b < c < d$. Suppose that the two subsets of real numbers
$$A = \{ x \mid a \leqq x \leqq c \} , \quad B = \{ x \mid b \leqq x \leqq d \}$$
satisfy
$$A \cap B = \left\{ x \mid x ^ { 2 } - 4 x + 3 \leqq 0 \right\} .$$
Then, answer the questions for cases (1) and (2).
(1) Let the union of $A$ and $B$ be
$$A \cup B = \left\{ x \mid x ^ { 2 } - 5 x - 24 \leqq 0 \right\} .$$
Then the values of $a , b , c$ and $d$ are
$$a = \mathbf { \text { NO } } , \quad b = \mathbf { P } , \quad c = \mathbf { Q } , \quad d = \mathbf { Q } .$$
(2) Let the intersection of $A$ and the complement $\bar { B }$ of $B$ be
$$A \cap \bar { B } = \left\{ x \mid x ^ { 2 } + 5 x - 6 \leqq 0 \text { and } x \neq 1 \right\} ,$$
and let the intersection of the complement $\bar { A }$ of $A$ and $B$ be
$$\bar { A } \cap B = \left\{ x \mid x ^ { 2 } - 9 x + 18 \leqq 0 \text { and } x \neq 3 \right\} .$$
Then the values of $a , b , c$ and $d$ are
$$a = \mathbf { S T } , \quad b = \mathbf { U } , \quad c = \mathbf { V } , \quad d = \mathbf { W } .$$

For any $x$ and $y$ satisfying $x > 0$ and $y > 0$, let $m$ be the smallest value among $\frac { y } { x } , x$ and $\frac { 8 } { y }$.
Also, let $A$ be the set of points $( x , y )$ where $m = \frac { y } { x }$, and let $B$ be the set of points $( x , y )$ where $m = \frac { 8 } { y }$.
(1) For $\mathbf { M } \sim$ S in the following sentence, choose the correct answer from among choices (0) $\sim$ (7) below. $A$ and $B$ can be expressed as follows:
$$\begin{aligned} & A = \{ ( x , y ) \mid \mathbf { M } \leqq \mathbf { N } , \quad \mathbf { O } \leqq 8\mathbf { P } \} \\ & B = \{ ( x , y ) \mid 8 \mathbf { Q } \leqq \mathbf { R } , \quad 8 \leqq \mathbf { S } \} . \end{aligned}$$
(0) $x$
(1) $y$
(2) $x + y$
(3) $x - y$
(4) $x ^ { 2 }$
(5) $x y$ (6) $y ^ { 2 }$ (7) $x ^ { 2 } + y ^ { 2 }$
(2) For $\mathbf { T }$ and $\mathbf { U }$ in the following sentence, choose the correct answer from among choices (0) $\sim$ (8).
When sets $A$ and $B$ are indicated on the $xy$-plane, $A$ is the shaded portion of graph $\mathbf{T}$ and $B$ is the shaded portion of graph $\mathbf{U}$. Note that the $x$ and $y$ axes are not included in the shaded portions.
(3) We are to find the maximum value of $m$ when a point $\mathrm { P } ( x , y )$ moves within $A \cup B$.
When $\mathrm { P } ( x , y ) \in A$, since $y = m x$, we need to find the point P which maximizes the slope of the straight line passing through the origin O and P.
Also, when $\mathrm { P } ( x , y ) \in B$, since $m = \frac { 8 } { y }$, we need to find the point P at which the $y$ coordinate of P is minimized.
From the above, at $( x , y ) = ( \mathbf { V } , \mathbf { W } ) , m$ takes the maximum value $\mathbf { X }$.

For positive integers $n$, the subsets of the set $R$ of real numbers are defined as
$$A _ { n } = \left\{ x \in R : \frac { ( - 1 ) ^ { n } } { n } < x < \frac { 2 } { n } \right\}$$
Accordingly, $$A _ { 1 } \cap A _ { 2 } \cap A _ { 3 }$$
the intersection set is equal to which of the following?
A) $\left( \frac { 1 } { 2 } , \frac { 2 } { 3 } \right)$
B) $\left( \frac { 1 } { 2 } , 2 \right)$
C) $\left( \frac { - 1 } { 3 } , \frac { 2 } { 3 } \right)$
D) $\left( \frac { - 1 } { 3 } , 1 \right)$
E) $\left( - 1 , \frac { 2 } { 3 } \right)$