If a +2b +3c = 4 then the least value of $a^2+b^2+c^2$

$$\begin{gathered} \stackrel{\mathrm{‾}}{\mathrm{a}}=(\mathrm{a},\mathrm{b},\mathrm{c})\stackrel{\mathrm{‾}}{,\mathrm{b}}=(1,2,3), \\ \sin ce (\stackrel{\mathrm{‾}}{\mathrm{a}}\cdot \stackrel{\mathrm{‾}}{\mathrm{b}}{)}^2\le {\left|\overline{a}\right|}^2{\left|\overline{b}\right|}^2 \\ \Rightarrow (\mathrm{a}+2\mathrm{b}+3\mathrm{c}{)}^2\le \left({\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2\right)\left(1^2+2^2+3^2\right) \\ \Rightarrow (4{)}^2\le \left({\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2\right)(14) \\ \Rightarrow \frac{16}{14}\le {\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2 \\ \Rightarrow {\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2\ge \frac{16}{14}=\frac{8}{7} \end{gathered}$$