Simplify the following expression.

$4 ÷[2 + 2 ÷ \begin{Bmatrix}2 + 2 ÷ (2 + \frac{2}{3})\end{Bmatrix}]$

$\frac{22}{15}$

$4 ÷[2 + 2 ÷ \begin{Bmatrix}2 + 2 ÷ (2 + \frac{2}{3})\end{Bmatrix}]$

= $4 ÷[2 + 2 ÷ \begin{Bmatrix}2 + 2 ÷ (\frac{8}{3})\end{Bmatrix}]$

= $4 ÷[2 + 2 ÷  \frac{11}{4}$]

= $4 ÷[2 + \frac{8}{11}]$

= $4 ÷[\frac{30}{11}]$

= $\frac{22}{15}$