Three lines are given by
$$\begin{aligned}
& \vec { r } = \lambda \hat { i } , \lambda \in \mathbb { R } \\
& \vec { r } = \mu ( \hat { i } + \hat { j } ) , \quad \mu \in \mathbb { R } \text { and } \\
& \vec { r } = v ( \hat { i } + \hat { j } + \hat { k } ) , \quad v \in \mathbb { R }
\end{aligned}$$
Let the lines cut the plane $x + y + z = 1$ at the points $A , B$ and $C$ respectively. If the area of the triangle $A B C$ is $\triangle$ then the value of $( 6 \Delta ) ^ { 2 }$ equals