Is that really enough info?

Begin by noticing:

$\dfrac{7 \sqrt{3}}{12}= \dfrac{1}{2} \times \dfrac {7 \sqrt{3}}{6}= \dfrac{1}{2} \times \dfrac{7}{2 \sqrt{3}}$

The Euler inequality gives $R \geq 2r \Rightarrow r \leq \dfrac{R}{2}$ so:

$r \leq \dfrac{1}{2} \times \dfrac{7}{2 \sqrt{3}}=\dfrac{7 \sqrt{3}}{12}$

But we also have: $r \geq \dfrac{7 \sqrt{3}}{12}$ so therefore $r=\dfrac{7 \sqrt{3}}{12}$

Note that $R=2r$ if and only if the triangle is equilateral so $\triangle ABC$ is equilateral. Let a side length be $x$ :

$2R=\dfrac{a}{\sin(A)} \Rightarrow 2 \times \dfrac{7}{2 \sqrt{3}}= \dfrac{x}{\sin(60^{\circ})} \Rightarrow \dfrac{7}{\sqrt{3}}=\dfrac{2x}{\sqrt{3}} \Rightarrow x=\dfrac{7}{2}$

As all the sides are the same length this is also a maximal side so:

$\sqrt{4x+2}=\sqrt{4 \times \dfrac{7}{2} +2}=\sqrt{14+2}=\sqrt{16}=\boxed{4}$