The lines $\frac{\mathrm{x}-2}{2}=\frac{\mathrm{y}}{-2}=\frac{\mathrm{z}-7}{16}$ and $\frac{\mathrm{x}+3}{4}=\frac{\mathrm{y}+2}{3}=\frac{\mathrm{z}+2}{1}$ intersect at the point $\mathrm{P}$. If the distance of $\mathrm{P}$ from the line $\frac{\mathrm{x}+1}{2}=\frac{\mathrm{y}-1}{3}=\frac{\mathrm{z}-1}{1}$ is $l$, then $14l^2$ is equal to _____.

Three lines are given by

$\overrightarrow{r}=\lambda \hat{i}, \lambda \in R$,

$\overrightarrow{r}=\mu \left(\widehat{i}+\widehat{j}\right),\mu \in R$ and

$\overrightarrow{r}=\nu \left(\widehat{i}+\widehat{j}+\widehat{k}\right),\nu \in R.$

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 $\mathrm{\Delta }$ then the value of ${\left(6\mathrm{\Delta }\right)}^2$ equals_________