Any general point on the line is $\left(\lambda +2,\mathrm{}2\lambda +3,\mathrm{}\lambda +4\right)$

$\overrightarrow{PR}=<\lambda +1,\mathrm{}2\lambda ,\mathrm{}3\lambda +2>$
$\overrightarrow{RQ}=<4-\lambda ,\mathrm{}4-2\lambda ,\mathrm{}1-3\lambda >$
From the diagram, $\overrightarrow{PR}\cdot \overrightarrow{RQ}=0$

$\Rightarrow \left(\lambda +1\right)\left(4-\lambda \right)+2\lambda \left(4-2\lambda \right)+\left(3\lambda +2\right)\left(1-3\lambda \right)=0$
$\Rightarrow {\lambda }^2-3\lambda -4+4{\lambda }^2-8\lambda +9{\lambda }^2+3\lambda -2=0$
$\Rightarrow 14{\lambda }^2-8\lambda -6=0$
$\Rightarrow \lambda =1,\mathrm{}-\frac{3}{7}$
Coordinates of $R$ are $\left(3,5,7\right)$ or $\left(\frac{11}{7},\frac{15}{7},\frac{19}{7}\right)$
Least value of $7\left(a+b+c\right)=11+15+19=45$