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