Given the function

$$f ( x ) = x ^ { 3 } - 3 a x ^ { 2 } - 3 ( 2 a + 1 ) x + a + 2 ,$$

answer the following questions.

(1) For $\mathbf { G }$ $\sim$ $\mathbf { K }$, choose the correct answers from among (0) $\sim$ (5) below, and for the other $\square$, enter the correct numbers.

Since

$$f ^ { \prime } ( x ) = \mathbf { A } ( x - \mathbf { B } a - \mathbf { C } ) ( x + \mathbf { D } ) ,$$

we see that\\
(i) when $a > \mathbf { EF }$, $f ( x )$ is $\mathbf { G }$ at $x = - \square \mathbf { D }$ and is $\square$ H at $x =$ $\square$ B $a +$ $\square$ C;\\
(ii) when $a =$ $\square$ EF, $f ( x )$ is always $\square$ I;\\
(iii) when $a < \mathbf{EF}$, $f ( x )$ is $\square$ J at $x = -$ $\square$ D and is $\square$ K at $x =$ $\square$ B $a +$ $\square$ C.\\
(0) locally maximized\\
(1) locally minimized\\
(2) increasing\\
(3) decreasing\\
(4) maximized\\
(5) minimized

(2) When we express the minimum value $m$ of $f ( x )$ over the range $- 1 \leqq x \leqq 1$ in terms of $a$, we have that\\
(i) when $a \geqq \mathbf { L }$, $m = \mathbf { MN } a$;\\
(ii) when $\mathbf { OP } \leqq a < \mathbf { L }$, $m = \mathbf { QR } \left( a ^ { 3 } + \mathbf { S } a ^ { 2 } + \mathbf { T } a \right)$;\\
(iii) when $a < \mathbf{OP}$, $m = \mathbf { U } a + \mathbf { V }$.

(3) The value of $m$ in (2) is maximized at $a = \frac { - \mathbf { W } + \sqrt { \mathbf { X } } } { \square \mathbf { Y } }$.