Let the plane a x+ b y+ c z+ d =0 bisect the line joining the points (4, -3, 1) and (2, 3, -5) at the right angles. If a, b, c, d are integers, then…

Let the plane $\displaystyle \mathrm{a} x+\mathrm{b} y+\mathrm{c} z+\mathrm{d}=0$ bisect the line joining the points ($\displaystyle 4$, -$\displaystyle 3$, $\displaystyle 1$) and ($\displaystyle 2$, $\displaystyle 3$, -$\displaystyle 5$) at the right angles. If $\displaystyle a, b, c, d$ are integers, then the minimum value of $\displaystyle \left(\mathrm{a}^2+\mathrm{b}^2+\mathrm{c}^2+\mathrm{d}^2\right)$ is \_\_\_\_.