Let the set $M = \{ ( i , j , s , t ) \mid i \in \{ 1,2 \} , j \in \{ 3,4 \} , s \in \{ 5,6 \} , t \in \{ 7,8 \} \}$. For a given finite sequence $A$ and sequence $\Omega : \omega _ { 1 } , \omega _ { 2 } , \cdots , \omega _ { k } , \omega _ { k } = \left( i _ { k } , j _ { k } , s _ { k } , t _ { k } \right) \in M$, define transformation $T$: add 1 to columns $i _ { 1 } , j _ { 1 } , s _ { 1 } , t _ { 1 }$ of sequence $A$ to obtain sequence $T _ { 1 } ( A )$; add 1 to columns $i _ { 2 } , j _ { 2 } , s _ { 2 } , t _ { 2 }$ of sequence $T _ { 1 } ( A )$ to obtain sequence $T _ { 2 } T _ { 1 } ( A )$; repeat the above operations to obtain sequence $T _ { k } \cdots T _ { 2 } T _ { 1 } ( A )$, denoted as $\Omega ( A )$.
(3) If $a _ { 1 } + a _ { 3 } + a _ { 5 } + a _ { 7 }$ is even, prove that ``$\Omega ( A )$ is a constant sequence'' is a necessary and sufficient condition for ``$a _ { 1 } + a _ { 2 } = a _ { 3 } + a _ { 4 } = a _ { 5 } + a _ { 6 } = a _ { 7 } + a _ { 8 }$''.