Questions on proving and manipulating inequalities.

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How prove this inequality $\sqrt{(2a+1)^2+(2b-\frac{\sqrt{3}}{3})^2}+\sqrt{(2a-1)^2+(2b-\frac{\sqrt{3}}{3})^2}+\cdots$

Question: let $a,b\in R$, show that $$\sqrt{(2a+1)^2+(2b-\dfrac{\sqrt{3}}{3})^2}+\sqrt{(2a-1)^2+(2b-\dfrac{\sqrt{3}}{3})^2}+\sqrt{4a^2+(2b+\dfrac{2\sqrt{3}}{3})^2}\ge ...