$\overrightarrow{\mathrm{a}},\overrightarrow{\mathrm{b},}\overrightarrow{\mathrm{c}}$ are three coplanar unit vectors such that  $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=0$ if three vectors $\overrightarrow{p}$,$\overrightarrow{\mathrm{q}}$ and $\overrightarrow{\mathrm{r}}$ are parallel to $\overrightarrow{\mathrm{a}},\overrightarrow{\mathrm{b}}$and $\overrightarrow{\mathrm{c}}$ respectively and have integral but different magnitudes, then among the following operations $|\overrightarrow{\mathrm{p}}+\overrightarrow{\mathrm{q}}+\overrightarrow{\mathrm{r}}|$ can take a value equal  to

Let  $\overrightarrow{a},\overrightarrow{b},$ and  $\overrightarrow{c}$ lies in x-y plane 

    Let  $\overrightarrow{a}=\widehat{i},\overrightarrow{b}=\frac{-1}{2}\widehat{i}+\frac{\sqrt{3}}{2}\widehat{j},\overrightarrow{c}=\frac{-1}{2}\widehat{i}-\frac{\sqrt{3}}{2}\widehat{j}$

$\therefore \left|\overrightarrow{p}+\overrightarrow{q}+\overrightarrow{r}\right|=\left|\lambda \overrightarrow{a}+\mu \overrightarrow{b}+\gamma \overrightarrow{c}\right|$

$\left|\lambda \widehat{i}+\mu \left(\frac{-1}{2}\widehat{i}+\frac{\sqrt{3}}{2}\widehat{j}\right)+\gamma \left(\frac{-1}{2}\widehat{i}-\frac{\sqrt{3}}{2}\widehat{j}\right)\right|$

$\Rightarrow \left(\lambda -\frac{\mu }{2}-\frac{\gamma }{2}\right)\widehat{i}+\frac{\sqrt{3}}{2}(\mu -\gamma )\widehat{j}$

$=\sqrt{{\left(\lambda -\frac{\mu }{2}-\frac{\gamma }{2}\right)}^2+\frac{3}{4}{(u-\gamma )}^2}$

$\Rightarrow \sqrt{{\lambda }^2+{\mu }^2+{\gamma }^2-\lambda \mu -\lambda \gamma -\mu \gamma }$

$=\frac{1}{\sqrt{2}}\sqrt{{(\lambda -\mu )}^2+{(\mu -\gamma )}^2+{(\gamma -\lambda )}^2}$$\ge \frac{1}{\sqrt{2}}\sqrt{1+1+4}=\sqrt{3}$

Hence $\left|\overrightarrow{p}+\overrightarrow{q}+\overrightarrow{r}\right|$ can take a value equal to  $\sqrt{3}$ and 2